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<ul id="toc-Overview_and_definitions-sublist" class="vector-toc-list"> <li id="toc-Affine_varieties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Affine varieties</span> </div> </a> <ul id="toc-Affine_varieties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_varieties_and_quasi-projective_varieties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_varieties_and_quasi-projective_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Projective varieties and quasi-projective varieties</span> </div> </a> <ul id="toc-Projective_varieties_and_quasi-projective_varieties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Abstract_varieties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Abstract_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Abstract varieties</span> </div> </a> <ul id="toc-Abstract_varieties-sublist" class="vector-toc-list"> <li id="toc-Existence_of_non-quasiprojective_abstract_algebraic_varieties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Existence_of_non-quasiprojective_abstract_algebraic_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3.1</span> <span>Existence of non-quasiprojective abstract algebraic varieties</span> </div> </a> <ul id="toc-Existence_of_non-quasiprojective_abstract_algebraic_varieties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Subvariety" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subvariety"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Subvariety</span> </div> </a> <ul id="toc-Subvariety-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Affine_variety" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Affine_variety"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Affine variety</span> </div> </a> <ul id="toc-Affine_variety-sublist" class="vector-toc-list"> <li id="toc-Example_1" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_1"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Example 1</span> </div> </a> <ul id="toc-Example_1-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>Example 2</span> </div> </a> <ul id="toc-Example_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_3" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_3"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.3</span> <span>Example 3</span> </div> </a> <ul id="toc-Example_3-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_linear_group" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#General_linear_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.4</span> <span>General linear group</span> </div> </a> <ul id="toc-General_linear_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characteristic_variety" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Characteristic_variety"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.5</span> <span>Characteristic variety</span> </div> </a> <ul id="toc-Characteristic_variety-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Projective_variety" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_variety"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Projective variety</span> </div> </a> <ul id="toc-Projective_variety-sublist" class="vector-toc-list"> <li id="toc-Example_1_2" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_1_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Example 1</span> </div> </a> <ul id="toc-Example_1_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Example_2:_Grassmannian" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Example_2:_Grassmannian"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Example 2: Grassmannian</span> </div> </a> <ul id="toc-Example_2:_Grassmannian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jacobian_variety_and_abelian_variety" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Jacobian_variety_and_abelian_variety"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Jacobian variety and abelian variety</span> </div> </a> <ul id="toc-Jacobian_variety_and_abelian_variety-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Moduli_varieties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Moduli_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.4</span> <span>Moduli varieties</span> </div> </a> <ul id="toc-Moduli_varieties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Non-affine_and_non-projective_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-affine_and_non-projective_example"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>Non-affine and non-projective example</span> </div> </a> <ul id="toc-Non-affine_and_non-projective_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>Non-examples</span> </div> </a> <ul id="toc-Non-examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_results" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_results"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Basic results</span> </div> </a> <ul id="toc-Basic_results-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Isomorphism_of_algebraic_varieties" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Isomorphism_of_algebraic_varieties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Isomorphism of algebraic varieties</span> </div> </a> <ul id="toc-Isomorphism_of_algebraic_varieties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Discussion_and_generalizations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Discussion_and_generalizations"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Discussion and generalizations</span> </div> </a> <ul id="toc-Discussion_and_generalizations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_manifolds" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Algebraic_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Algebraic manifolds</span> </div> </a> <ul id="toc-Algebraic_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algebraic variety</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 37 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-37" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">37 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D9%86%D9%88%D8%B9_%D8%AC%D8%A8%D8%B1%D9%8A" title="تنوع جبري – Arabic" lang="ar" hreflang="ar" data-title="تنوع جبري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Varied%C3%A1_alxebraica" title="Variedá alxebraica – Asturian" lang="ast" hreflang="ast" data-title="Variedá alxebraica" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D0%BA_%D0%BA%D2%AF%D0%BF_%D1%82%D3%A9%D1%80%D0%BB%D3%A9%D0%BB%D3%A9%D0%BA" title="Алгебраик күп төрлөлөк – Bashkir" lang="ba" hreflang="ba" data-title="Алгебраик күп төрлөлөк" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target"><span>Башҡортса</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B8%D1%87%D0%BD%D0%BE_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Алгебрично многообразие – Bulgarian" lang="bg" hreflang="bg" data-title="Алгебрично многообразие" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Varietat_algebraica" title="Varietat algebraica – Catalan" lang="ca" hreflang="ca" data-title="Varietat algebraica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algebraick%C3%A1_varieta" title="Algebraická varieta – Czech" lang="cs" hreflang="cs" data-title="Algebraická varieta" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebraische_Variet%C3%A4t" title="Algebraische Varietät – German" lang="de" hreflang="de" data-title="Algebraische Varietät" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Algebraline_muutkond" title="Algebraline muutkond – Estonian" lang="et" hreflang="et" data-title="Algebraline muutkond" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%91%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B9%CE%BA%CE%AE_%CF%80%CE%BF%CE%B9%CE%BA%CE%B9%CE%BB%CE%AF%CE%B1" title="Αλγεβρική ποικιλία – Greek" lang="el" hreflang="el" data-title="Αλγεβρική ποικιλία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Variedad_algebraica" title="Variedad algebraica – Spanish" lang="es" hreflang="es" data-title="Variedad algebraica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Algebra_varia%C4%B5o" title="Algebra variaĵo – Esperanto" lang="eo" hreflang="eo" data-title="Algebra variaĵo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Barietate_aljebraiko" title="Barietate aljebraiko – Basque" lang="eu" hreflang="eu" data-title="Barietate aljebraiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%88%D8%A7%D8%B1%DB%8C%D8%AA%D9%87_%D8%AC%D8%A8%D8%B1%DB%8C" title="واریته جبری – Persian" lang="fa" hreflang="fa" data-title="واریته جبری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Vari%C3%A9t%C3%A9_alg%C3%A9brique" title="Variété algébrique – French" lang="fr" hreflang="fr" data-title="Variété algébrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Variedade_alx%C3%A9brica" title="Variedade alxébrica – Galician" lang="gl" hreflang="gl" data-title="Variedade alxébrica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EC%88%98%EB%8B%A4%EC%96%91%EC%B2%B4" title="대수다양체 – Korean" lang="ko" hreflang="ko" data-title="대수다양체" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Varietas_aljabar" title="Varietas aljabar – Indonesian" lang="id" hreflang="id" data-title="Varietas aljabar" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Variet%C3%A0_algebrica" title="Varietà algebrica – Italian" lang="it" hreflang="it" data-title="Varietà algebrica" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99%D7%AA" title="יריעה אלגברית – Hebrew" lang="he" hreflang="he" data-title="יריעה אלגברית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%BB%D1%8B%D0%BA_%D1%82%D0%B0%D0%BB%D0%B0%D0%B0" title="Алгебралык талаа – Kyrgyz" lang="ky" hreflang="ky" data-title="Алгебралык талаа" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Algebrin%C4%97_atmaina" title="Algebrinė atmaina – Lithuanian" lang="lt" hreflang="lt" data-title="Algebrinė atmaina" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra%C3%AFsche_vari%C3%ABteit" title="Algebraïsche variëteit – Dutch" lang="nl" hreflang="nl" data-title="Algebraïsche variëteit" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BB%A3%E6%95%B0%E5%A4%9A%E6%A7%98%E4%BD%93" title="代数多様体 – Japanese" lang="ja" hreflang="ja" data-title="代数多様体" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Varietet_(matematikk)" title="Varietet (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Varietet (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozmaito%C5%9B%C4%87_algebraiczna" title="Rozmaitość algebraiczna – Polish" lang="pl" hreflang="pl" data-title="Rozmaitość algebraiczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Variedade_alg%C3%A9brica" title="Variedade algébrica – Portuguese" lang="pt" hreflang="pt" data-title="Variedade algébrica" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Алгебраическое многообразие – Russian" lang="ru" hreflang="ru" data-title="Алгебраическое многообразие" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Algebraic_variety" title="Algebraic variety – Simple English" lang="en-simple" hreflang="en-simple" data-title="Algebraic variety" data-language-autonym="Simple English" data-language-local-name="Simple English" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical object studied in the field of algebraic geometry</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about algebraic varieties. For the term "variety of algebras", see <a href="/wiki/Variety_(universal_algebra)" title="Variety (universal algebra)">Variety (universal algebra)</a>.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Twisted_cubic_curve.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Twisted_cubic_curve.png/220px-Twisted_cubic_curve.png" decoding="async" width="220" height="218" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/89/Twisted_cubic_curve.png/330px-Twisted_cubic_curve.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/89/Twisted_cubic_curve.png/440px-Twisted_cubic_curve.png 2x" data-file-width="541" data-file-height="536" /></a><figcaption>The <a href="/wiki/Twisted_cubic" title="Twisted cubic">twisted cubic</a> is a projective algebraic variety.</figcaption></figure> <p><b>Algebraic varieties</b> are the central objects of study in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a sub-field of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>. Classically, an algebraic variety is defined as the <a href="/wiki/Solution_set" title="Solution set">set of solutions</a> of a <a href="/wiki/System_of_polynomial_equations" title="System of polynomial equations">system of polynomial equations</a> over the <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.<sup id="cite_ref-Hartshorne_1-0" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 58">: 58 </span></sup> </p><p>Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be <a href="/wiki/Irreducible_component" title="Irreducible component">irreducible</a>, which means that it is not the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of two smaller <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">sets</a> that are <a href="/wiki/Closed_set" title="Closed set">closed</a> in the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a>. Under this definition, non-irreducible algebraic varieties are called <b>algebraic sets</b>. Other conventions do not require irreducibility. </p><p>The <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> establishes a link between <a href="/wiki/Algebra" title="Algebra">algebra</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a> by showing that a <a href="/wiki/Monic_polynomial" title="Monic polynomial">monic polynomial</a> (an algebraic object) in one variable with complex number coefficients is determined by the set of its <a href="/wiki/Zero_of_a_function" title="Zero of a function">roots</a> (a geometric object) in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. Generalizing this result, <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a> provides a fundamental correspondence between <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> of <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> and algebraic sets. Using the <i>Nullstellensatz</i> and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>. This correspondence is a defining feature of algebraic geometry. </p><p>Many algebraic varieties are <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a>, but an algebraic variety may have <a href="/wiki/Singular_point_of_an_algebraic_variety" title="Singular point of an algebraic variety">singular points</a> while a differentiable manifold cannot. Algebraic varieties can be characterized by their <a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">dimension</a>. Algebraic varieties of dimension one are called <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a> and algebraic varieties of dimension two are called <a href="/wiki/Algebraic_surface" title="Algebraic surface">algebraic surfaces</a>. </p><p>In the context of modern <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> theory, an algebraic variety over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is an integral (irreducible and reduced) scheme over that field whose <a href="/wiki/Structure_morphism" class="mw-redirect" title="Structure morphism">structure morphism</a> is separated and of finite type. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Overview_and_definitions">Overview and definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=1" title="Edit section: Overview and definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <i>affine variety</i> over an <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious that one can construct genuinely new examples of varieties in this way, but <a href="/wiki/Masayoshi_Nagata" title="Masayoshi Nagata">Nagata</a> gave an example of such a new variety in the 1950s. </p> <div class="mw-heading mw-heading3"><h3 id="Affine_varieties">Affine varieties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=2" title="Edit section: Affine varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Affine_variety" title="Affine variety">Affine variety</a></div> <p>For an algebraically closed field <span class="texhtml mvar" style="font-style:italic;">K</span> and a <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml mvar" style="font-style:italic;">n</span>, let <span class="texhtml"><b>A</b><sup><i>n</i></sup></span> be an <a href="/wiki/Affine_space" title="Affine space">affine <span class="texhtml"><i>n</i></span>-space</a> over <span class="texhtml"><i>K</i></span>, identified to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d63366b3d00300e06eee81786182062b98775c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.312ex; height:2.343ex;" alt="{\displaystyle K^{n}}"></span> through the choice of an <a href="/wiki/Affine_coordinate_system" class="mw-redirect" title="Affine coordinate system">affine coordinate system</a>. The polynomials <span class="texhtml"> <i>f</i> </span> in the ring <span class="texhtml"><i>K</i>[<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>]</span> can be viewed as <i>K</i>-valued functions on <span class="texhtml"><b>A</b><sup><i>n</i></sup></span> by evaluating <span class="texhtml"> <i>f</i> </span> at the points in <span class="texhtml"><b>A</b><sup><i>n</i></sup></span>, i.e. by choosing values in <i>K</i> for each <i>x<sub>i</sub></i>. For each set <i>S</i> of polynomials in <span class="texhtml"><i>K</i>[<i>x</i><sub>1</sub>, ..., <i>x<sub>n</sub></i>]</span>, define the zero-locus <i>Z</i>(<i>S</i>) to be the set of points in <span class="texhtml"><b>A</b><sup><i>n</i></sup></span> on which the functions in <i>S</i> simultaneously vanish, that is to say </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(S)=\left\{x\in \mathbf {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>f</mi> <mo>∈</mo> <mi>S</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(S)=\left\{x\in \mathbf {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c75da95ce86ba1b26492033e94de7d872289c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.073ex; height:2.843ex;" alt="{\displaystyle Z(S)=\left\{x\in \mathbf {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\right\}.}"></span></dd></dl> <p>A subset <i>V</i> of <span class="texhtml"><b>A</b><sup><i>n</i></sup></span> is called an <b>affine algebraic set</b> if <i>V</i> = <i>Z</i>(<i>S</i>) for some <i>S</i>.<sup id="cite_ref-Hartshorne_1-1" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 2">: 2 </span></sup> A nonempty affine algebraic set <i>V</i> is called <b>irreducible</b> if it cannot be written as the union of two <a href="/wiki/Subset" title="Subset">proper</a> algebraic subsets.<sup id="cite_ref-Hartshorne_1-2" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 3">: 3 </span></sup> An irreducible affine algebraic set is also called an <b>affine variety</b>.<sup id="cite_ref-Hartshorne_1-3" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 3">: 3 </span></sup> (Some authors use the phrase <i>affine variety</i> to refer to any affine algebraic set, irreducible or not.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>) </p><p>Affine varieties can be given a <a href="/wiki/Natural_topology" title="Natural topology">natural topology</a> by declaring the <a href="/wiki/Closed_set" title="Closed set">closed sets</a> to be precisely the affine algebraic sets. This topology is called the Zariski topology.<sup id="cite_ref-Hartshorne_1-4" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 2">: 2 </span></sup> </p><p>Given a subset <i>V</i> of <span class="texhtml"><b>A</b><sup><i>n</i></sup></span>, we define <i>I</i>(<i>V</i>) to be the ideal of all polynomial functions vanishing on <i>V</i>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(V)=\left\{f\in K[x_{1},\ldots ,x_{n}]\mid f(x)=0{\text{ for all }}x\in V\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mi>f</mi> <mo>∈</mo> <mi>K</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>x</mi> <mo>∈</mo> <mi>V</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(V)=\left\{f\in K[x_{1},\ldots ,x_{n}]\mid f(x)=0{\text{ for all }}x\in V\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e401fa9a6b3bc2616ed05a01fbd98adc8a388cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.372ex; height:2.843ex;" alt="{\displaystyle I(V)=\left\{f\in K[x_{1},\ldots ,x_{n}]\mid f(x)=0{\text{ for all }}x\in V\right\}.}"></span></dd></dl> <p>For any affine algebraic set <i>V</i>, the <b>coordinate ring</b> or <b>structure ring</b> of <i>V</i> is the <a href="/wiki/Quotient_ring" title="Quotient ring">quotient</a> of the polynomial ring by this ideal.<sup id="cite_ref-Hartshorne_1-5" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 4">: 4 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Projective_varieties_and_quasi-projective_varieties">Projective varieties and quasi-projective varieties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=3" title="Edit section: Projective varieties and quasi-projective varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Projective_variety" title="Projective variety">Projective variety</a> and <a href="/wiki/Quasi-projective_variety" title="Quasi-projective variety">Quasi-projective variety</a></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">k</span> be an algebraically closed field and let <span class="texhtml"><b>P</b><sup><i>n</i></sup></span> be the <a href="/wiki/Algebraic_geometry_of_projective_spaces" title="Algebraic geometry of projective spaces">projective <i>n</i>-space</a> over <span class="texhtml mvar" style="font-style:italic;">k</span>. Let <span class="texhtml"> <i>f</i> </span> in <span class="texhtml"><i>k</i>[<i>x</i><sub>0</sub>, ..., <i>x<sub>n</sub></i>]</span> be a <a href="/wiki/Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomial</a> of degree <i>d</i>. It is not well-defined to evaluate <span class="texhtml"> <i>f</i> </span> on points in <span class="texhtml"><b>P</b><sup><i>n</i></sup></span> in <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>. However, because <span class="texhtml"> <i>f</i> </span> is homogeneous, meaning that <span class="texhtml"> <i>f</i>  (<i>λx</i><sub>0</sub>, ..., <i>λx<sub>n</sub></i>) = <i>λ<sup>d</sup></i> <i>f</i>  (<i>x</i><sub>0</sub>, ..., <i>x<sub>n</sub></i>)</span>, it <i>does</i> make sense to ask whether <span class="texhtml"> <i>f</i> </span> vanishes at a point <span class="texhtml">[<i>x</i><sub>0</sub> : ... : <i>x<sub>n</sub></i>]</span>. For each set <i>S</i> of homogeneous polynomials, define the zero-locus of <i>S</i> to be the set of points in <span class="texhtml"><b>P</b><sup><i>n</i></sup></span> on which the functions in <i>S</i> vanish: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(S)=\{x\in \mathbf {P} ^{n}\mid f(x)=0{\text{ for all }}f\in S\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∣</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>f</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(S)=\{x\in \mathbf {P} ^{n}\mid f(x)=0{\text{ for all }}f\in S\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0a785e9f06715d40f0c010733ecc48edbaa1db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.493ex; height:2.843ex;" alt="{\displaystyle Z(S)=\{x\in \mathbf {P} ^{n}\mid f(x)=0{\text{ for all }}f\in S\}.}"></span></dd></dl> <p>A subset <i>V</i> of <span class="texhtml"><b>P</b><sup><i>n</i></sup></span> is called a <b>projective algebraic set</b> if <i>V</i> = <i>Z</i>(<i>S</i>) for some <i>S</i>.<sup id="cite_ref-Hartshorne_1-6" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 9">: 9 </span></sup> An irreducible projective algebraic set is called a <b>projective variety</b>.<sup id="cite_ref-Hartshorne_1-7" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 10">: 10 </span></sup> </p><p>Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed. </p><p>Given a subset <i>V</i> of <span class="texhtml"><b>P</b><sup><i>n</i></sup></span>, let <i>I</i>(<i>V</i>) be the ideal generated by all homogeneous polynomials vanishing on <i>V</i>. For any projective algebraic set <i>V</i>, the <b><a href="/wiki/Homogeneous_coordinate_ring" title="Homogeneous coordinate ring">coordinate ring</a></b> of <i>V</i> is the quotient of the polynomial ring by this ideal.<sup id="cite_ref-Hartshorne_1-8" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 10">: 10 </span></sup> </p><p>A <b><a href="/wiki/Quasi-projective_variety" title="Quasi-projective variety">quasi-projective variety</a></b> is a <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski open</a> subset of a projective variety. Notice that every affine variety is quasi-projective.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variety is usually not called a variety but a <a href="/wiki/Constructible_set_(topology)" title="Constructible set (topology)">constructible set</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Abstract_varieties">Abstract varieties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=4" title="Edit section: Abstract varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In classical algebraic geometry, all varieties were by definition <a href="/wiki/Quasiprojective_variety" class="mw-redirect" title="Quasiprojective variety">quasi-projective varieties</a>, meaning that they were open subvarieties of closed subvarieties of a <a href="/wiki/Projective_space" title="Projective space">projective space</a>. For example, in Chapter 1 of <a href="/wiki/Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Hartshorne</a> a <i>variety</i> over an algebraically closed field is defined to be a <a href="/wiki/Quasi-projective_variety" title="Quasi-projective variety">quasi-projective variety</a>,<sup id="cite_ref-Hartshorne_1-9" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 15">: 15 </span></sup> but from Chapter 2 onwards, the term <b>variety</b> (also called an <b>abstract variety</b>) refers to a more general object, which locally is a quasi-projective variety, but when viewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into <a href="/wiki/Projective_space" title="Projective space">projective space</a>.<sup id="cite_ref-Hartshorne_1-10" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 105">: 105 </span></sup> So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the <a href="/wiki/Regular_function" class="mw-redirect" title="Regular function">regular functions</a> on the variety. The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. For example, under this definition, the product <span class="texhtml"><b>P</b><sup>1</sup> × <b>P</b><sup>1</sup></span> is not a variety until it is embedded into a larger projective space; this is usually done by the <a href="/wiki/Segre_embedding" title="Segre embedding">Segre embedding</a>. Furthermore, any variety that admits one embedding into projective space admits many others, for example by composing the embedding with the <a href="/wiki/Veronese_embedding" class="mw-redirect" title="Veronese embedding">Veronese embedding</a>; thus many notions that should be intrinsic, such as that of a regular function, are not obviously so. </p><p>The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by <a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a>. In his <i><a href="/wiki/Foundations_of_Algebraic_Geometry" title="Foundations of Algebraic Geometry">Foundations of Algebraic Geometry</a>,</i> using <a href="/wiki/Valuation_(algebra)" title="Valuation (algebra)">valuations</a>. <a href="/wiki/Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a> made a definition of a <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>, which served a similar purpose, but was more general. However, <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a>'s definition of a scheme is more general still and has received the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be an <a href="/wiki/Glossary_of_scheme_theory#integral" class="mw-redirect" title="Glossary of scheme theory">integral</a>, <a href="/wiki/Glossary_of_scheme_theory#separated" class="mw-redirect" title="Glossary of scheme theory">separated</a> scheme of <a href="/wiki/Glossary_of_algebraic_geometry#finite_type_(locally)" title="Glossary of algebraic geometry">finite type</a> over an algebraically closed field,<sup id="cite_ref-Hartshorne_1-11" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Pages: 104–105">: 104–105 </span></sup> although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>note 2<span class="cite-bracket">]</span></a></sup> Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field. </p> <div class="mw-heading mw-heading4"><h4 id="Existence_of_non-quasiprojective_abstract_algebraic_varieties">Existence of non-quasiprojective abstract algebraic varieties</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=5" title="Edit section: Existence of non-quasiprojective abstract algebraic varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata.<sup id="cite_ref-Nagata56_5-0" class="reference"><a href="#cite_note-Nagata56-5"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Nagata's example was not <a href="/wiki/Complete_variety" title="Complete variety">complete</a> (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non-projective.<sup id="cite_ref-Nagata57_6-0" class="reference"><a href="#cite_note-Nagata57-6"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Hartshorne_1-12" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Remark 4.10.2 p.105">: Remark 4.10.2 p.105 </span></sup> Since then other examples have been found: for example, it is straightforward to construct <a href="/wiki/Toric_variety" title="Toric variety">toric varieties</a> that are not quasi-projective but complete.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=6" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Subvariety">Subvariety</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=7" title="Edit section: Subvariety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>subvariety</b> is a subset of a variety that is itself a variety (with respect to the topological structure induced by the ambient variety). For example, every open subset of a variety is a variety. See also <a href="/wiki/Closed_immersion" title="Closed immersion">closed immersion</a>. </p><p><a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a> says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or non-irrelevant homogeneous prime ideals of the coordinate ring of the variety. </p> <div class="mw-heading mw-heading3"><h3 id="Affine_variety">Affine variety</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=8" title="Edit section: Affine variety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Example_1">Example 1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=9" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>k</i> = <b>C</b></span>, and <b>A</b><sup>2</sup> be the two-dimensional <a href="/wiki/Affine_space" title="Affine space">affine space</a> over <b>C</b>. Polynomials in the ring <b>C</b>[<i>x</i>, <i>y</i>] can be viewed as complex valued functions on <b>A</b><sup>2</sup> by evaluating at the points in <b>A</b><sup>2</sup>. Let subset <i>S</i> of <b>C</b>[<i>x</i>, <i>y</i>] contain a single element <span class="texhtml"> <i>f</i>  (<i>x</i>, <i>y</i>)</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=x+y-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=x+y-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a7a5556df91f0674d74f110a53073f09e7a6c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.681ex; height:2.843ex;" alt="{\displaystyle f(x,y)=x+y-1.}"></span></dd></dl> <p>The zero-locus of <span class="texhtml"> <i>f</i>  (<i>x</i>, <i>y</i>)</span> is the set of points in <b>A</b><sup>2</sup> on which this function vanishes: it is the set of all pairs of complex numbers (<i>x</i>, <i>y</i>) such that <i>y</i> = 1 − <i>x</i>. This is called a <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> in the affine plane. (In the <b>classical topology</b> coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.) This is the set <span class="texhtml"><i>Z</i>( <i>f</i> )</span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z(f)=\{(x,1-x)\in \mathbf {C} ^{2}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z(f)=\{(x,1-x)\in \mathbf {C} ^{2}\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7ee8a81b95954c3d27280189ff9bcff5f7a5e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.17ex; height:3.176ex;" alt="{\displaystyle Z(f)=\{(x,1-x)\in \mathbf {C} ^{2}\}.}"></span></dd></dl> <p>Thus the subset <span class="texhtml"><i>V</i> = <i>Z</i>( <i>f</i> )</span> of <b>A</b><sup>2</sup> is an <a href="#Affine_varieties">algebraic set</a>. The set <i>V</i> is not empty. It is irreducible, as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety. </p> <div class="mw-heading mw-heading4"><h4 id="Example_2">Example 2</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=10" title="Edit section: Example 2"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml"><i>k</i> = <b>C</b></span>, and <b>A</b><sup>2</sup> be the two-dimensional affine space over <b>C</b>. Polynomials in the ring <b>C</b>[<i>x</i>, <i>y</i>] can be viewed as complex valued functions on <b>A</b><sup>2</sup> by evaluating at the points in <b>A</b><sup>2</sup>. Let subset <i>S</i> of <b>C</b>[<i>x</i>, <i>y</i>] contain a single element <i>g</i>(<i>x</i>, <i>y</i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x,y)=x^{2}+y^{2}-1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x,y)=x^{2}+y^{2}-1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3f5fda37ab33ddf664630a45e2d5635e87da22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.632ex; height:3.176ex;" alt="{\displaystyle g(x,y)=x^{2}+y^{2}-1.}"></span></dd></dl> <p>The zero-locus of <i>g</i>(<i>x</i>, <i>y</i>) is the set of points in <b>A</b><sup>2</sup> on which this function vanishes, that is the set of points (<i>x</i>,<i>y</i>) such that <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 1. As <i>g</i>(<i>x</i>, <i>y</i>) is an <a href="/wiki/Absolutely_irreducible" class="mw-redirect" title="Absolutely irreducible">absolutely irreducible</a> polynomial, this is an algebraic variety. The set of its real points (that is the points for which <i>x</i> and <i>y</i> are real numbers), is known as the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>; this name is also often given to the whole variety. </p> <div class="mw-heading mw-heading4"><h4 id="Example_3">Example 3</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=11" title="Edit section: Example 3"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The following example is neither a <a href="/wiki/Hypersurface" title="Hypersurface">hypersurface</a>, nor a <a href="/wiki/Vector_space" title="Vector space">linear space</a>, nor a single point. Let <b>A</b><sup>3</sup> be the three-dimensional affine space over <b>C</b>. The set of points (<i>x</i>, <i>x</i><sup>2</sup>, <i>x</i><sup>3</sup>) for <i>x</i> in <b>C</b> is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>note 3<span class="cite-bracket">]</span></a></sup> It is the <a href="/wiki/Twisted_cubic" title="Twisted cubic">twisted cubic</a> shown in the above figure. It may be defined by the equations </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}y-x^{2}&=0\\z-x^{3}&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>y</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> <mo>−</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}y-x^{2}&=0\\z-x^{3}&=0\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ab06079479a1485abb2cc0ce507e4a37f592aca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.392ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}y-x^{2}&=0\\z-x^{3}&=0\end{aligned}}}"></span></dd></dl> <p>The irreducibility of this algebraic set needs a proof. One approach in this case is to check that the projection (<i>x</i>, <i>y</i>, <i>z</i>) → (<i>x</i>, <i>y</i>) is <a href="/wiki/Injective_function" title="Injective function">injective</a> on the set of the solutions and that its image is an irreducible plane curve. </p><p>For more difficult examples, a similar proof may always be given, but may imply a difficult computation: first a <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a> computation to compute the dimension, followed by a random linear change of variables (not always needed); then a <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a> computation for another <a href="/wiki/Monomial_order" title="Monomial order">monomial ordering</a> to compute the projection and to prove that it is <a href="/wiki/Generic_property" title="Generic property">generically</a> injective and that its image is a <a href="/wiki/Hypersurface" title="Hypersurface">hypersurface</a>, and finally a <a href="/wiki/Polynomial_factorization" class="mw-redirect" title="Polynomial factorization">polynomial factorization</a> to prove the irreducibility of the image. </p> <div class="mw-heading mw-heading4"><h4 id="General_linear_group">General linear group</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=12" title="Edit section: General linear group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The set of <i>n</i>-by-<i>n</i> matrices over the base field <i>k</i> can be identified with the affine <i>n</i><sup>2</sup>-space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71bf24d6e7a7b322a2b51c886aaac84b012ae73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} ^{n^{2}}}"></span> with coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27e97949cb2cc8f2d4c2a9421477a65f839db11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.807ex; height:2.343ex;" alt="{\displaystyle x_{ij}}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{ij}(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{ij}(A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e69d70d08afe55569bab0d1f0e6fa4670b949737" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.359ex; height:3.009ex;" alt="{\displaystyle x_{ij}(A)}"></span> is the (<i>i</i>, <i>j</i>)-th entry of the matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>. The <a href="/wiki/Determinant_of_a_matrix" class="mw-redirect" title="Determinant of a matrix">determinant</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/742df4b24488fb61ad6fc7b4cf229a9168f08690" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.23ex; height:2.176ex;" alt="{\displaystyle \det }"></span> is then a polynomial in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27e97949cb2cc8f2d4c2a9421477a65f839db11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.807ex; height:2.343ex;" alt="{\displaystyle x_{ij}}"></span> and thus defines the hypersurface <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=V(\det )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=V(\det )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4292be1ad9e18fbea8fe626a08840b2e80f0e9c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.988ex; height:2.843ex;" alt="{\displaystyle H=V(\det )}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71bf24d6e7a7b322a2b51c886aaac84b012ae73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} ^{n^{2}}}"></span>. The complement of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"></span> is then an open subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71bf24d6e7a7b322a2b51c886aaac84b012ae73b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.728ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} ^{n^{2}}}"></span> that consists of all the invertible <i>n</i>-by-<i>n</i> matrices, the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc9e8d8cf24c036c3b17aba368840366cb313fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.516ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(k)}"></span>. It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4867bbc96499a4eac4e5d1f5ab33a4430b481b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.301ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}"></span> where the affine line is given coordinate <i>t</i>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc9e8d8cf24c036c3b17aba368840366cb313fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.516ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(k)}"></span> amounts to the zero-locus in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </msup> <mo>×</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4867bbc96499a4eac4e5d1f5ab33a4430b481b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.301ex; height:3.009ex;" alt="{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}"></span> of the polynomial in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{ij},t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{ij},t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce20d241f4399f2d40cfac6d30cb9c9084c89aaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.68ex; height:2.676ex;" alt="{\displaystyle x_{ij},t}"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\cdot \det[x_{ij}]-1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>⋅</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\cdot \det[x_{ij}]-1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4eebf11e099dae629b56fdef0ddbe6a0190ada2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.498ex; height:3.009ex;" alt="{\displaystyle t\cdot \det[x_{ij}]-1,}"></span></dd></dl> <p>i.e., the set of matrices <i>A</i> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\det(A)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\det(A)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961287a186444b53213c85ae425518696f639be0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.27ex; height:2.843ex;" alt="{\displaystyle t\det(A)=1}"></span> has a solution. This is best seen algebraically: the coordinate ring of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc9e8d8cf24c036c3b17aba368840366cb313fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.516ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(k)}"></span> is the <a href="/wiki/Localization_(commutative_algebra)" title="Localization (commutative algebra)">localization</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{ij}\mid 0\leq i,j\leq n][{\det }^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∣</mo> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">det</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[x_{ij}\mid 0\leq i,j\leq n][{\det }^{-1}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906251a2585beae9160026b614821edf0e9d3d6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.653ex; height:3.343ex;" alt="{\displaystyle k[x_{ij}\mid 0\leq i,j\leq n][{\det }^{-1}]}"></span>, which can be identified with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x_{ij},t\mid 0\leq i,j\leq n]/(t\det -1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mi>t</mi> <mo>∣</mo> <mn>0</mn> <mo>≤</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo movablelimits="true" form="prefix">det</mo> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[x_{ij},t\mid 0\leq i,j\leq n]/(t\det -1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3d4682930efbb6a3665071a1a0df6a5b6e4cdf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:31.457ex; height:3.009ex;" alt="{\displaystyle k[x_{ij},t\mid 0\leq i,j\leq n]/(t\det -1)}"></span>. </p><p>The multiplicative group k<sup>*</sup> of the base field <i>k</i> is the same as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{1}(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{1}(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efc2f13ab6ad99615859ba9c807c0cfbe575bc22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.352ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{1}(k)}"></span> and thus is an affine variety. A finite product of it <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k^{*})^{r}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k^{*})^{r}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/540225d48edac7e12e4dccbaae0f51ffa6e89eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.049ex; height:2.843ex;" alt="{\displaystyle (k^{*})^{r}}"></span> is an <a href="/wiki/Algebraic_torus" title="Algebraic torus">algebraic torus</a>, which is again an affine variety. </p><p>A general linear group is an example of a <a href="/wiki/Linear_algebraic_group" title="Linear algebraic group">linear algebraic group</a>, an affine variety that has a structure of a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> in such a way the group operations are morphism of varieties. </p> <div class="mw-heading mw-heading4"><h4 id="Characteristic_variety">Characteristic variety</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=13" title="Edit section: Characteristic variety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Characteristic_variety" title="Characteristic variety">Characteristic variety</a></div> <p>Let <i>A</i> be a not-necessarily-commutative algebra over a field <i>k</i>. Even if <i>A</i> is not commutative, it can still happen that <i>A</i> has a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>-filtration so that the <a href="/wiki/Associated_graded_ring" title="Associated graded ring">associated ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} A=\bigoplus _{i=-\infty }^{\infty }A_{i}/{A_{i-1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>A</mi> <mo>=</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} A=\bigoplus _{i=-\infty }^{\infty }A_{i}/{A_{i-1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dccbea842f16b9d459de3c988f56f4a296c6ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.806ex; height:7.009ex;" alt="{\displaystyle \operatorname {gr} A=\bigoplus _{i=-\infty }^{\infty }A_{i}/{A_{i-1}}}"></span> is commutative, reduced and finitely generated as a <i>k</i>-algebra; i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a7c1f11bc58808f27ef9a4f13f1defdd56da3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.204ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} A}"></span> is the coordinate ring of an affine (reducible) variety <i>X</i>. For example, if <i>A</i> is the <a href="/wiki/Universal_enveloping_algebra" title="Universal enveloping algebra">universal enveloping algebra</a> of a finite-dimensional <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.172ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {g}}}"></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a7c1f11bc58808f27ef9a4f13f1defdd56da3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.204ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} A}"></span> is a polynomial ring (the <a href="/wiki/PBW_theorem" class="mw-redirect" title="PBW theorem">PBW theorem</a>); more precisely, the coordinate ring of the dual vector space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {g}}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">g</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {g}}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0acfc869f2ed2da044f5c582042afd3065e12bdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.226ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {g}}^{*}}"></span>. </p><p>Let <i>M</i> be a filtered module over <i>A</i> (i.e., <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i}M_{j}\subset M_{i+j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⊂</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}M_{j}\subset M_{i+j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a534fbe54d7d4b34975a4ecfa3e66f49f08e761" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.815ex; height:2.843ex;" alt="{\displaystyle A_{i}M_{j}\subset M_{i+j}}"></span>). If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74600b40dbc52661865dfe9e25e9b7908f05f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.903ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} M}"></span> is fintiely generated as a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a7c1f11bc58808f27ef9a4f13f1defdd56da3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.204ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} A}"></span>-algebra, then the <a href="/wiki/Support_of_a_module" title="Support of a module">support</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74600b40dbc52661865dfe9e25e9b7908f05f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.903ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} M}"></span> in <i>X</i>; i.e., the locus where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>gr</mi> <mo>⁡</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d74600b40dbc52661865dfe9e25e9b7908f05f86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.903ex; height:2.509ex;" alt="{\displaystyle \operatorname {gr} M}"></span> does not vanish is called the <a href="/wiki/Characteristic_variety" title="Characteristic variety">characteristic variety</a> of <i>M</i>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The notion plays an important role in the theory of <a href="/wiki/D-module" title="D-module"><i>D</i>-modules</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Projective_variety">Projective variety</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=14" title="Edit section: Projective variety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> is a closed subvariety of a projective space. That is, it is the zero locus of a set of <a href="/wiki/Homogeneous_polynomials" class="mw-redirect" title="Homogeneous polynomials">homogeneous polynomials</a> that generate a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Example_1_2">Example 1</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=15" title="Edit section: Example 1"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Elliptic_curve2.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/0c/Elliptic_curve2.png" decoding="async" width="206" height="204" class="mw-file-element" data-file-width="206" data-file-height="204" /></a><figcaption>The affine plane curve <span class="nowrap"><i>y</i><sup>2</sup> = <i>x</i><sup>3</sup> − <i>x</i></span>. The corresponding projective curve is called an elliptic curve.</figcaption></figure> <p>A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The <a href="/wiki/Projective_line" title="Projective line">projective line</a> <b>P</b><sup>1</sup> is an example of a projective curve; it can be viewed as the curve in the projective plane <span class="nowrap"><b>P</b><sup>2</sup> = {[<i>x</i>, <i>y</i>, <i>z</i>]</span>} defined by <span class="nowrap"><i>x</i> = 0</span>. For another example, first consider the affine cubic curve </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}=x^{3}-x.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}=x^{3}-x.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0627eabf66ffd684829247628b403bec9310c64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.514ex; height:3.009ex;" alt="{\displaystyle y^{2}=x^{3}-x.}"></span></dd></dl> <p>in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{2}z=x^{3}-xz^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>z</mi> <mo>=</mo> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>−</mo> <mi>x</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y^{2}z=x^{3}-xz^{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70f4b749e0e903b2b5e672c37971cd48d74dd6cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.747ex; height:3.009ex;" alt="{\displaystyle y^{2}z=x^{3}-xz^{2},}"></span></dd></dl> <p>which defines a curve in <b>P</b><sup>2</sup> called an <a href="/wiki/Elliptic_curve" title="Elliptic curve">elliptic curve</a>. The curve has genus one (<a href="/wiki/Genus_formula" class="mw-redirect" title="Genus formula">genus formula</a>); in particular, it is not isomorphic to the projective line <b>P</b><sup>1</sup>, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of <a href="/wiki/Moduli_of_algebraic_curves" title="Moduli of algebraic curves">moduli of algebraic curves</a>). </p> <div class="mw-heading mw-heading4"><h4 id="Example_2:_Grassmannian">Example 2: Grassmannian</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=16" title="Edit section: Example 2: Grassmannian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>V</i> be a finite-dimensional vector space. The <a href="/wiki/Grassmannian_variety" class="mw-redirect" title="Grassmannian variety">Grassmannian variety</a> <i>G<sub>n</sub></i>(<i>V</i>) is the set of all <i>n</i>-dimensional subspaces of <i>V</i>. It is a projective variety: it is embedded into a projective space via the <a href="/wiki/Pl%C3%BCcker_embedding" title="Plücker embedding">Plücker embedding</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \mathbf {P} \left(\wedge ^{n}V\right)\\\langle b_{1},\ldots ,b_{n}\rangle \mapsto [b_{1}\wedge \cdots \wedge b_{n}]\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↪</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>V</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false">⟨</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∧</mo> <mo>⋯</mo> <mo>∧</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \mathbf {P} \left(\wedge ^{n}V\right)\\\langle b_{1},\ldots ,b_{n}\rangle \mapsto [b_{1}\wedge \cdots \wedge b_{n}]\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49e905092e65ff3c107d6d5a880404396231d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.814ex; height:6.176ex;" alt="{\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \mathbf {P} \left(\wedge ^{n}V\right)\\\langle b_{1},\ldots ,b_{n}\rangle \mapsto [b_{1}\wedge \cdots \wedge b_{n}]\end{cases}}}"></span></dd></dl> <p>where <i>b<sub>i</sub></i> are any set of linearly independent vectors in <i>V</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge ^{n}V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mo>∧</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \wedge ^{n}V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7138ef9134549b40ecd503094a0e0a02d6018362" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.556ex; height:2.343ex;" alt="{\displaystyle \wedge ^{n}V}"></span> is the <i>n</i>-th <a href="/wiki/Exterior_power" class="mw-redirect" title="Exterior power">exterior power</a> of <i>V</i>, and the bracket [<i>w</i>] means the line spanned by the nonzero vector <i>w</i>. </p><p>The Grassmannian variety comes with a natural <a href="/wiki/Vector_bundle" title="Vector bundle">vector bundle</a> (or <a href="/wiki/Locally_free_sheaf" class="mw-redirect" title="Locally free sheaf">locally free sheaf</a> in other terminology) called the <a href="/wiki/Tautological_bundle" title="Tautological bundle">tautological bundle</a>, which is important in the study of <a href="/wiki/Characteristic_class" title="Characteristic class">characteristic classes</a> such as <a href="/wiki/Chern_class" title="Chern class">Chern classes</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Jacobian_variety_and_abelian_variety">Jacobian variety and abelian variety</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=17" title="Edit section: Jacobian variety and abelian variety"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>C</i> be a smooth complete curve and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Pic} (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Pic</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Pic} (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389b3854d40a72f523a426763c9af6e1b0406fbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.838ex; height:2.843ex;" alt="{\displaystyle \operatorname {Pic} (C)}"></span> the <a href="/wiki/Picard_group" title="Picard group">Picard group</a> of it; i.e., the group of isomorphism classes of line bundles on <i>C</i>. Since <i>C</i> is smooth, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Pic} (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Pic</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Pic} (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389b3854d40a72f523a426763c9af6e1b0406fbd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.838ex; height:2.843ex;" alt="{\displaystyle \operatorname {Pic} (C)}"></span> can be identified as the <a href="/wiki/Divisor_class_group" class="mw-redirect" title="Divisor class group">divisor class group</a> of <i>C</i> and thus there is the degree homomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {deg} :\operatorname {Pic} (C)\to \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>deg</mi> <mo>:</mo> <mi>Pic</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {deg} :\operatorname {Pic} (C)\to \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11a2d60027752a504fb594f39d22349e5f3ef925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.427ex; height:2.843ex;" alt="{\displaystyle \operatorname {deg} :\operatorname {Pic} (C)\to \mathbb {Z} }"></span>. The <a href="/wiki/Jacobian_variety" title="Jacobian variety">Jacobian variety</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Jac} (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Jac} (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b004a0342cf4acb68460b0ef0290bf615fcdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.965ex; height:2.843ex;" alt="{\displaystyle \operatorname {Jac} (C)}"></span> of <i>C</i> is the kernel of this degree map; i.e., the group of the divisor classes on <i>C</i> of degree zero. A Jacobian variety is an example of an <a href="/wiki/Abelian_variety" title="Abelian variety">abelian variety</a>, a complete variety with a compatible <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (in short, algebraic <a href="/wiki/Theta_function" title="Theta function">theta functions</a> give an embedding into a projective space. See <a href="/wiki/Equations_defining_abelian_varieties" title="Equations defining abelian varieties">equations defining abelian varieties</a>); thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Jac} (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Jac} (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b004a0342cf4acb68460b0ef0290bf615fcdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.965ex; height:2.843ex;" alt="{\displaystyle \operatorname {Jac} (C)}"></span> is a projective variety. The tangent space to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Jac} (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Jac} (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b004a0342cf4acb68460b0ef0290bf615fcdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.965ex; height:2.843ex;" alt="{\displaystyle \operatorname {Jac} (C)}"></span> at the identity element is naturally isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fbb3bbeb8368c79fe7b50c89ee5943f36cdc316" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.385ex; height:3.176ex;" alt="{\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});}"></span><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> hence, the dimension of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Jac} (C)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Jac} (C)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93b004a0342cf4acb68460b0ef0290bf615fcdb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.965ex; height:2.843ex;" alt="{\displaystyle \operatorname {Jac} (C)}"></span> is the genus of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. </p><p>Fix a point <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/671bd891701e0d6cfa6da0114a5dd64233b58709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.547ex; height:2.509ex;" alt="{\displaystyle P_{0}}"></span> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. For each integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n>0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27a6a5d982d54202a14f111cb8a49210501b2c96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}"></span>, there is a natural morphism<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{n}\to \operatorname {Jac} (C),\,(P_{1},\dots ,P_{r})\mapsto [P_{1}+\cdots +P_{n}-nP_{0}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mo stretchy="false">[</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>n</mi> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{n}\to \operatorname {Jac} (C),\,(P_{1},\dots ,P_{r})\mapsto [P_{1}+\cdots +P_{n}-nP_{0}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d95415e42a4d1ac4cb41a7199523cccd098ffa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.368ex; height:2.843ex;" alt="{\displaystyle C^{n}\to \operatorname {Jac} (C),\,(P_{1},\dots ,P_{r})\mapsto [P_{1}+\cdots +P_{n}-nP_{0}]}"></span></dd></dl> <p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50cc4309121472bbd901a7e54c365829cd150d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.016ex; height:2.343ex;" alt="{\displaystyle C^{n}}"></span> is the product of <i>n</i> copies of <i>C</i>. For <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba0027652d8e5f694d4aa1c71ce16c9380ce1186" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g=1}"></span> (i.e., <i>C</i> is an elliptic curve), the above morphism for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span> turns out to be an isomorphism;<sup id="cite_ref-Hartshorne_1-13" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Location: Ch. IV, Example 1.3.7.">: Ch. IV, Example 1.3.7. </span></sup> in particular, an elliptic curve is an abelian variety. </p> <div class="mw-heading mw-heading4"><h4 id="Moduli_varieties">Moduli varieties</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=18" title="Edit section: Moduli varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given an integer <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9beeeef79ee73a89cef1e1986fc88d7d248f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 0}"></span>, the set of isomorphism classes of smooth complete curves of genus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> is called the <a href="/wiki/Moduli_of_curves" class="mw-redirect" title="Moduli of curves">moduli of curves</a> of genus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> and is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {M}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {M}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b598f80fbad1b8e1cd191ff9df1700ba8c5c8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.461ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {M}}_{g}}"></span>. There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to use <a href="/wiki/Geometric_invariant_theory" title="Geometric invariant theory">geometric invariant theory</a> which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Moduli such as the moduli of curves of fixed genus is typically not a projective variety; roughly the reason is that a degeneration (limit) of a smooth curve tends to be non-smooth or reducible. This leads to the notion of a <a href="/wiki/Stable_curve" title="Stable curve">stable curve</a> of genus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84b2394f64ae9927e09ee51b848be213edecafe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}"></span>, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathfrak {M}}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathfrak {M}}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4147e255ee32f5037a2c4c57f8a185abd0504155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.576ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathfrak {M}}}_{g}}"></span>, the set of isomorphism classes of stable curves of genus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84b2394f64ae9927e09ee51b848be213edecafe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}"></span>, is then a projective variety which contains <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {M}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {M}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b598f80fbad1b8e1cd191ff9df1700ba8c5c8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.461ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {M}}_{g}}"></span> as an open dense subset. Since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathfrak {M}}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathfrak {M}}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4147e255ee32f5037a2c4c57f8a185abd0504155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.576ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathfrak {M}}}_{g}}"></span> is obtained by adding boundary points to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {M}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {M}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b598f80fbad1b8e1cd191ff9df1700ba8c5c8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.461ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {M}}_{g}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathfrak {M}}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathfrak {M}}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4147e255ee32f5037a2c4c57f8a185abd0504155" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.576ex; height:3.676ex;" alt="{\displaystyle {\overline {\mathfrak {M}}}_{g}}"></span> is colloquially said to be a <a href="/w/index.php?title=Compactification_(algebraic_geometry)&action=edit&redlink=1" class="new" title="Compactification (algebraic geometry) (page does not exist)">compactification</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {M}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {M}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b598f80fbad1b8e1cd191ff9df1700ba8c5c8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.461ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {M}}_{g}}"></span>. Historically a paper of Mumford and Deligne<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> introduced the notion of a stable curve to show <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {M}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">M</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {M}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b598f80fbad1b8e1cd191ff9df1700ba8c5c8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.461ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {M}}_{g}}"></span> is irreducible when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>≥</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84b2394f64ae9927e09ee51b848be213edecafe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}"></span>. </p><p>The moduli of curves exemplifies a typical situation: a moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here, there are the notions of <a href="/wiki/Stable_vector_bundle" title="Stable vector bundle">stable</a> and semistable vector bundles on a smooth complete curve <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. The moduli of semistable vector bundles of a given rank <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and a given degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> (degree of the determinant of the bundle) is then a projective variety denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle SU_{C}(n,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle SU_{C}(n,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402d1496b171e2f860583696256d00fca96a52ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.022ex; height:2.843ex;" alt="{\displaystyle SU_{C}(n,d)}"></span>, which contains the set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{C}(n,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>C</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{C}(n,d)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ad4ace4170463b331fbf507aa7503312099daa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.523ex; height:2.843ex;" alt="{\displaystyle U_{C}(n,d)}"></span> of isomorphism classes of stable vector bundles of rank <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> and degree <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> as an open subset.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Since a line bundle is stable, such a moduli is a generalization of the Jacobian variety of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. </p><p>In general, in contrast to the case of moduli of curves, a compactification of a moduli need not be unique and, in some cases, different non-equivalent compactifications are constructed using different methods and by different authors. An example over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> is the problem of compactifying <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e36d6fca890fb2cf80bec0dcb1191a5535df42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.54ex; height:2.843ex;" alt="{\displaystyle D/\Gamma }"></span>, the quotient of a bounded symmetric domain <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> by an action of an arithmetic discrete group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> A basic example of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e36d6fca890fb2cf80bec0dcb1191a5535df42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.54ex; height:2.843ex;" alt="{\displaystyle D/\Gamma }"></span> is when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\mathfrak {H}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">H</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\mathfrak {H}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512447d1587bd55930bd37886f48ccefcc1a3237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.718ex; height:2.843ex;" alt="{\displaystyle D={\mathfrak {H}}_{g}}"></span>, <a href="/wiki/Siegel%27s_upper_half-space" class="mw-redirect" title="Siegel's upper half-space">Siegel's upper half-space</a> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cfde86a3f7ec967af9955d0988592f0693d2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }"></span> <a href="/wiki/Commensurability_(group_theory)" title="Commensurability (group theory)">commensurable</a> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sp} (2g,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>g</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sp} (2g,\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d07869bdae034830dcf276ad7989c2080e86bbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.257ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sp} (2g,\mathbb {Z} )}"></span>; in that case, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e36d6fca890fb2cf80bec0dcb1191a5535df42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.54ex; height:2.843ex;" alt="{\displaystyle D/\Gamma }"></span> has an interpretation as the moduli <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {A}}_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">A</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {A}}_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/077d4daff49c6e91c17dd741ddc2323df981fe1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.69ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {A}}_{g}}"></span> of principally polarized complex abelian varieties of dimension <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span> (a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e36d6fca890fb2cf80bec0dcb1191a5535df42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.54ex; height:2.843ex;" alt="{\displaystyle D/\Gamma }"></span>, a <a href="/w/index.php?title=Toroidal_compactification&action=edit&redlink=1" class="new" title="Toroidal compactification (page does not exist)">toroidal compactification</a> of it.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> But there are other ways to compactify <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e36d6fca890fb2cf80bec0dcb1191a5535df42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.54ex; height:2.843ex;" alt="{\displaystyle D/\Gamma }"></span>; for example, there is the <a href="/wiki/Minimal_compactification" class="mw-redirect" title="Minimal compactification">minimal compactification</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D/\Gamma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Γ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D/\Gamma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31e36d6fca890fb2cf80bec0dcb1191a5535df42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.54ex; height:2.843ex;" alt="{\displaystyle D/\Gamma }"></span> due to Baily and Borel: it is the <a href="/wiki/Proj_construction" title="Proj construction">projective variety associated to the graded ring</a> formed by <a href="/wiki/Modular_form" title="Modular form">modular forms</a> (in the Siegel case, <a href="/wiki/Siegel_modular_form" title="Siegel modular form">Siegel modular forms</a>;<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> see also <a href="/wiki/Siegel_modular_variety" title="Siegel modular variety">Siegel modular variety</a>). The non-uniqueness of compactifications is due to the lack of moduli interpretations of those compactifications; i.e., they do not represent (in the category-theory sense) any natural moduli problem or, in the precise language, there is no natural <a href="/wiki/Moduli_stack" class="mw-redirect" title="Moduli stack">moduli stack</a> that would be an analog of moduli stack of stable curves. </p> <div class="mw-heading mw-heading3"><h3 id="Non-affine_and_non-projective_example">Non-affine and non-projective example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=19" title="Edit section: Non-affine and non-projective example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An algebraic variety can be neither affine nor projective. To give an example, let <span class="nowrap"><i>X</i> = <b>P</b><sup>1</sup> × <b>A</b><sup>1</sup></span> and <span class="nowrap"><i>p</i>: <i>X</i> → <b>A</b><sup>1</sup></span> the projection. Here <i>X</i> is an algebraic variety since it is a product of varieties. It is not affine since <b>P</b><sup>1</sup> is a closed subvariety of <i>X</i> (as the zero locus of <i>p</i>), but an affine variety cannot contain a projective variety of positive dimension as a closed subvariety. It is not projective either, since there is a nonconstant <a href="/wiki/Regular_function" class="mw-redirect" title="Regular function">regular function</a> on <i>X</i>; namely, <i>p</i>. </p><p>Another example of a non-affine non-projective variety is <span class="nowrap"><i>X</i> = <b>A</b><sup>2</sup> − (0, 0)</span> (cf. <i><a href="/wiki/Morphism_of_varieties#Examples" class="mw-redirect" title="Morphism of varieties">Morphism of varieties § Examples</a></i>.) </p> <div class="mw-heading mw-heading3"><h3 id="Non-examples">Non-examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=20" title="Edit section: Non-examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider the affine line <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67530a5bd8c23c0be226ac63ddf5f6b2619e682d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{1}}"></span> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. The complement of the circle <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{z\in \mathbb {C} {\text{ with }}|z|^{2}=1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> with </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{z\in \mathbb {C} {\text{ with }}|z|^{2}=1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8d411bf776a44a0f8def619538f0a8a50af03f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.312ex; height:3.343ex;" alt="{\displaystyle \{z\in \mathbb {C} {\text{ with }}|z|^{2}=1\}}"></span> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f62bdd01640ec554d3d7b70c9874f291475ff0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }"></span> is not an algebraic variety (nor even an algebraic set). Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |z|^{2}-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>z</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |z|^{2}-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f27bc4c8d06c68af94acd1c0de1e5b5d2533a62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.439ex; height:3.343ex;" alt="{\displaystyle |z|^{2}-1}"></span> is not a polynomial in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span> (although it is a polynomial in the real coordinates <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea0abffd33a692ded22accc104515a032851dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.519ex; height:2.009ex;" alt="{\displaystyle x,y}"></span>). On the other hand, the complement of the origin in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">A</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f62bdd01640ec554d3d7b70c9874f291475ff0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.509ex; height:2.676ex;" alt="{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }"></span> is an algebraic (affine) variety, since the origin is the zero-locus of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>. This may be explained as follows: the affine line has dimension one and so any subvariety of it other than itself must have strictly less dimension; namely, zero. </p><p>For similar reasons, a <a href="/wiki/Unitary_group" title="Unitary group">unitary group</a> (over the complex numbers) is not an algebraic variety, while the special linear group <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {SL} _{n}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>SL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {SL} _{n}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57faaf2eac1922e859d1a2047edb26f8fb2fbee1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.451ex; height:2.843ex;" alt="{\displaystyle \operatorname {SL} _{n}(\mathbb {C} )}"></span> is a closed subvariety of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c81fb83c9a3c1639fe166eaa1c72180c64410b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.983ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}"></span>, the zero-locus of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det -1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/604944a692a3584043dd1d5e06ec9941e4815307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.587ex; height:2.343ex;" alt="{\displaystyle \det -1}"></span>. (Over a different base field, a unitary group can however be given a structure of a variety.) </p> <div class="mw-heading mw-heading2"><h2 id="Basic_results">Basic results</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=21" title="Edit section: Basic results"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>An affine algebraic set <i>V</i> is a variety <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <i>I</i>(<i>V</i>) is a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>; equivalently, <i>V</i> is a variety if and only if its coordinate ring is an <span class="nowrap"><a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>.<sup id="cite_ref-Harris_19-0" class="reference"><a href="#cite_note-Harris-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 52">: 52 </span></sup></span><sup id="cite_ref-Hartshorne_1-14" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 4">: 4 </span></sup></li> <li>Every nonempty affine algebraic set may be written uniquely as a finite union of algebraic varieties (where none of the varieties in the decomposition is a subvariety of any other).<sup id="cite_ref-Hartshorne_1-15" class="reference"><a href="#cite_note-Hartshorne-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 5">: 5 </span></sup></li> <li>The <b>dimension</b> of a variety may be defined in various equivalent ways. See <a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">Dimension of an algebraic variety</a> for details.</li> <li>A product of finitely many algebraic varieties (over an algebraically closed field) is an algebraic variety. A finite product of affine varieties is affine<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> and a finite product of projective varieties is projective.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Isomorphism_of_algebraic_varieties">Isomorphism of algebraic varieties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=22" title="Edit section: Isomorphism of algebraic varieties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Morphism_of_varieties" class="mw-redirect" title="Morphism of varieties">Morphism of varieties</a></div> <p>Let <span class="texhtml"><i>V</i><sub>1</sub>, <i>V</i><sub>2</sub></span> be algebraic varieties. We say <span class="texhtml"><i>V</i><sub>1</sub></span> and <span class="texhtml"><i>V</i><sub>2</sub></span> are <a href="/wiki/Graph_isomorphism" title="Graph isomorphism">isomorphic</a>, and write <span class="texhtml"><i>V</i><sub>1</sub> ≅ <i>V</i><sub>2</sub></span>, if there are <a href="/wiki/Regular_map_(algebraic_geometry)" class="mw-redirect" title="Regular map (algebraic geometry)">regular maps</a> <span class="texhtml"><i>φ</i> : <i>V</i><sub>1</sub> → <i>V</i><sub>2</sub></span> and <span class="texhtml"><i>ψ</i> : <i>V</i><sub>2</sub> → <i>V</i><sub>1</sub></span> such that the <a href="/wiki/Function_composition" title="Function composition">compositions</a> <span class="texhtml"><i>ψ</i> ∘ <i>φ</i></span> and <span class="texhtml"><i>φ</i> ∘ <i>ψ</i></span> are the <a href="/wiki/Identity_function" title="Identity function">identity maps</a> on <span class="texhtml"><i>V</i><sub>1</sub></span> and <span class="texhtml"><i>V</i><sub>2</sub></span> respectively. </p> <div class="mw-heading mw-heading2"><h2 id="Discussion_and_generalizations">Discussion and generalizations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=23" title="Edit section: Discussion and generalizations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-No_footnotes plainlinks metadata ambox ambox-style ambox-No_footnotes" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/40px-Text_document_with_red_question_mark.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/60px-Text_document_with_red_question_mark.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a4/Text_document_with_red_question_mark.svg/80px-Text_document_with_red_question_mark.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section includes a <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">list of references</a>, <a href="/wiki/Wikipedia:Further_reading" title="Wikipedia:Further reading">related reading</a>, or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a>, <b>but its sources remain unclear because it lacks <a href="/wiki/Wikipedia:Citing_sources#Inline_citations" title="Wikipedia:Citing sources">inline citations</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:WikiProject_Fact_and_Reference_Check" class="mw-redirect" title="Wikipedia:WikiProject Fact and Reference Check">improve</a> this section by <a href="/wiki/Wikipedia:When_to_cite" title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">March 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed</a> — some foundational changes are required. The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An <i>abstract algebraic variety</i> is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a <a href="/wiki/Locally_ringed_space" class="mw-redirect" title="Locally ringed space">locally ringed space</a> such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a>. Basically, a variety over <span class="texhtml mvar" style="font-style:italic;">k</span> is a scheme whose <a href="/wiki/Structure_sheaf" class="mw-redirect" title="Structure sheaf">structure sheaf</a> is a <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> of <span class="texhtml mvar" style="font-style:italic;">k</span>-algebras with the property that the rings <i>R</i> that occur above are all <a href="/wiki/Integral_domain" title="Integral domain">integral domains</a> and are all finitely generated <span class="texhtml mvar" style="font-style:italic;">k</span>-algebras, that is to say, they are quotients of <a href="/wiki/Polynomial_algebra" class="mw-redirect" title="Polynomial algebra">polynomial algebras</a> by <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a>. </p><p>This definition works over any field <span class="texhtml mvar" style="font-style:italic;">k</span>. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be <i>separated</i>. (Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.) </p><p>Some modern researchers also remove the restriction on a variety having <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> affine charts, and when speaking of a variety only require that the affine charts have trivial <a href="/wiki/Nilradical_of_a_ring" title="Nilradical of a ring">nilradical</a>. </p><p>A <a href="/wiki/Complete_variety" title="Complete variety">complete variety</a> is a variety such that any map from an open subset of a nonsingular <a href="/wiki/Algebraic_curve" title="Algebraic curve">curve</a> into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa. </p><p>These varieties have been called "varieties in the sense of Serre", since <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre</a>'s foundational paper <a href="/wiki/Faisceaux_alg%C3%A9briques_coh%C3%A9rents" class="mw-redirect" title="Faisceaux algébriques cohérents">FAC</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> on <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way. </p><p>One way that leads to generalizations is to allow reducible algebraic sets (and fields <span class="texhtml mvar" style="font-style:italic;">k</span> that aren't algebraically closed), so the rings <i>R</i> may not be integral domains. A more significant modification is to allow <a href="/wiki/Nilpotent" title="Nilpotent">nilpotents</a> in the sheaf of rings, that is, rings which are not <b>reduced</b>. This is one of several generalizations of classical algebraic geometry that are built into <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck</a>'s theory of schemes. </p><p>Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry. For example, the closed subscheme of the affine line defined by <i>x</i><sup>2</sup> = 0 is different from the subscheme defined by <i>x</i> = 0 (the origin). More generally, the <a href="/wiki/Fiber_product_of_schemes" title="Fiber product of schemes">fiber</a> of a morphism of schemes <i>X</i> → <i>Y</i> at a point of <i>Y</i> may be non-reduced, even if <i>X</i> and <i>Y</i> are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal" structure. </p><p>There are further generalizations called <a href="/wiki/Algebraic_space" title="Algebraic space">algebraic spaces</a> and <a href="/wiki/Algebraic_stack" title="Algebraic stack">stacks</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Algebraic_manifolds">Algebraic manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=24" title="Edit section: Algebraic manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Algebraic_manifold" title="Algebraic manifold">Algebraic manifold</a></div> <p>An algebraic manifold is an algebraic variety that is also an <i>m</i>-dimensional manifold, and hence every sufficiently small local patch is isomorphic to <i>k<sup>m</sup></i>. Equivalently, the variety is <a href="/wiki/Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> (free from singular points). When <span class="texhtml mvar" style="font-style:italic;">k</span> is the real numbers, <b>R</b>, algebraic manifolds are called <a href="/wiki/Nash_manifold" class="mw-redirect" title="Nash manifold">Nash manifolds</a>. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. <a href="/wiki/Projective_algebraic_manifold" class="mw-redirect" title="Projective algebraic manifold">Projective algebraic manifolds</a> are an equivalent definition for projective varieties. The <a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a> is one example. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=25" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Variety_(disambiguation)" class="mw-redirect mw-disambig" title="Variety (disambiguation)">Variety (disambiguation)</a> — listing also several mathematical meanings</li> <li><a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">Function field of an algebraic variety</a></li> <li><a href="/wiki/Birational_geometry" title="Birational geometry">Birational geometry</a></li> <li><a href="/wiki/Motive_(algebraic_geometry)" title="Motive (algebraic geometry)">Motive (algebraic geometry)</a></li> <li><a href="/wiki/Analytic_variety" class="mw-redirect" title="Analytic variety">Analytic variety</a></li> <li><a href="/wiki/Zariski%E2%80%93Riemann_space" title="Zariski–Riemann space">Zariski–Riemann space</a></li> <li><a href="/wiki/Semi-algebraic_set" class="mw-redirect" title="Semi-algebraic set">Semi-algebraic set</a></li> <li><a href="/wiki/Fano_variety" title="Fano variety">Fano variety</a></li> <li><a href="/wiki/Mn%C3%ABv%27s_universality_theorem" title="Mnëv's universality theorem">Mnëv's universality theorem</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=26" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Hartshorne, p.xv, Harris, p.3</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Liu, Qing. <i>Algebraic Geometry and Arithmetic Curves</i>, p. 55 Definition 2.3.47, and p. 88 Example 3.2.3</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Harris, p.9; that it is irreducible is stated as an exercise in Hartshorne p.7</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=27" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Hartshorne-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Hartshorne_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-10"><sup><i><b>k</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-11"><sup><i><b>l</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-12"><sup><i><b>m</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-13"><sup><i><b>n</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-14"><sup><i><b>o</b></i></sup></a> <a href="#cite_ref-Hartshorne_1-15"><sup><i><b>p</b></i></sup></a></span> <span class="reference-text"> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHartshorne1977" class="citation book cs1"><a href="/wiki/Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977). <i>Algebraic Geometry</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90244-9" title="Special:BookSources/0-387-90244-9"><bdi>0-387-90244-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry&rft.pub=Springer-Verlag&rft.date=1977&rft.isbn=0-387-90244-9&rft.aulast=Hartshorne&rft.aufirst=Robin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Hartshorne, Exercise I.2.9, p.12</span> </li> <li id="cite_note-Nagata56-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nagata56_5-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagata1956" class="citation journal cs1"><a href="/wiki/Masayoshi_Nagata" title="Masayoshi Nagata">Nagata, Masayoshi</a> (1956). <a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fkjm%2F1250777138">"On the imbedding problem of abstract varieties in projective varieties"</a>. <i>Memoirs of the College of Science, University of Kyoto. Series A: Mathematics</i>. <b>30</b>: <span class="nowrap">71–</span>82. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fkjm%2F1250777138">10.1215/kjm/1250777138</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0088035">0088035</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Memoirs+of+the+College+of+Science%2C+University+of+Kyoto.+Series+A%3A+Mathematics&rft.atitle=On+the+imbedding+problem+of+abstract+varieties+in+projective+varieties&rft.volume=30&rft.pages=%3Cspan+class%3D%22nowrap%22%3E71-%3C%2Fspan%3E82&rft.date=1956&rft_id=info%3Adoi%2F10.1215%2Fkjm%2F1250777138&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0088035%23id-name%3DMR&rft.aulast=Nagata&rft.aufirst=Masayoshi&rft_id=https%3A%2F%2Fdoi.org%2F10.1215%252Fkjm%252F1250777138&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-Nagata57-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Nagata57_6-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagata1957" class="citation journal cs1"><a href="/wiki/Masayoshi_Nagata" title="Masayoshi Nagata">Nagata, Masayoshi</a> (1957). <a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fkjm%2F1250777007">"On the imbeddings of abstract surfaces in projective varieties"</a>. <i>Memoirs of the College of Science, University of Kyoto. Series A: Mathematics</i>. <b>30</b> (3): <span class="nowrap">231–</span>235. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1215%2Fkjm%2F1250777007">10.1215/kjm/1250777007</a></span>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0094358">0094358</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118328992">118328992</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Memoirs+of+the+College+of+Science%2C+University+of+Kyoto.+Series+A%3A+Mathematics&rft.atitle=On+the+imbeddings+of+abstract+surfaces+in+projective+varieties&rft.volume=30&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E231-%3C%2Fspan%3E235&rft.date=1957&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0094358%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118328992%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1215%2Fkjm%2F1250777007&rft.aulast=Nagata&rft.aufirst=Masayoshi&rft_id=https%3A%2F%2Fdoi.org%2F10.1215%252Fkjm%252F1250777007&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">In page 65 of <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFulton1993" class="citation cs2"><a href="/wiki/William_Fulton_(mathematician)" title="William Fulton (mathematician)">Fulton, William</a> (1993), <i>Introduction to toric varieties</i>, <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-00049-7" title="Special:BookSources/978-0-691-00049-7"><bdi>978-0-691-00049-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+toric+varieties&rft.pub=Princeton+University+Press&rft.date=1993&rft.isbn=978-0-691-00049-7&rft.aulast=Fulton&rft.aufirst=William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span>, a remark describes a complete toric variety that has no non-trivial line bundle; thus, in particular, it has no ample line bundle.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Definition 1.1.12 in Ginzburg, V., 1998. Lectures on D-modules. University of Chicago.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilne2008">Milne 2008</a>, Proposition 2.1.</span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFMilne2008">Milne 2008</a>, The beginning of § 5.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFMFK1994">MFK 1994</a>, Theorem 5.11.</span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDeligneMumford1969" class="citation journal cs1"><a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Deligne, Pierre</a>; <a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a> (1969). <a rel="nofollow" class="external text" href="http://archive.numdam.org/article/PMIHES_1969__36__75_0.pdf">"The irreducibility of the space of curves of given genus"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a></i>. <b>36</b>: <span class="nowrap">75–</span>109. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.589.288">10.1.1.589.288</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02684599">10.1007/bf02684599</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16482150">16482150</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Publications+Math%C3%A9matiques+de+l%27IH%C3%89S&rft.atitle=The+irreducibility+of+the+space+of+curves+of+given+genus&rft.volume=36&rft.pages=%3Cspan+class%3D%22nowrap%22%3E75-%3C%2Fspan%3E109&rft.date=1969&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.589.288%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A16482150%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fbf02684599&rft.aulast=Deligne&rft.aufirst=Pierre&rft.au=Mumford%2C+David&rft_id=http%3A%2F%2Farchive.numdam.org%2Farticle%2FPMIHES_1969__36__75_0.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFMFK1994">MFK 1994</a>, Appendix C to Ch. 5.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">Mark Goresky. Compactifications and cohomology of modular varieties. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 551–582. Amer. Math. Soc., Providence, RI, 2005.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAshMumfordRapoportTai1975" class="citation cs2">Ash, A.; <a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a>; Rapoport, M.; Tai, Y. (1975), <a rel="nofollow" class="external text" href="http://www.uni-due.de/~mat903/sem/ss08/ash_mumford_rapoport_tai_Compactifications.pdf"><i>Smooth compactification of locally symmetric varieties</i></a> <span class="cs1-format">(PDF)</span>, Brookline, Mass.: Math. Sci. Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-73955-9" title="Special:BookSources/978-0-521-73955-9"><bdi>978-0-521-73955-9</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0457437">0457437</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Smooth+compactification+of+locally+symmetric+varieties&rft.place=Brookline%2C+Mass.&rft.pub=Math.+Sci.+Press&rft.date=1975&rft.isbn=978-0-521-73955-9&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0457437%23id-name%3DMR&rft.aulast=Ash&rft.aufirst=A.&rft.au=Mumford%2C+David&rft.au=Rapoport%2C+M.&rft.au=Tai%2C+Y.&rft_id=http%3A%2F%2Fwww.uni-due.de%2F~mat903%2Fsem%2Fss08%2Fash_mumford_rapoport_tai_Compactifications.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNamikawa1980" class="citation book cs1">Namikawa, Yukihiko (1980). <i>Toroidal Compactification of Siegel Spaces</i>. Lecture Notes in Mathematics. Vol. 812. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0091051">10.1007/BFb0091051</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-10021-8" title="Special:BookSources/978-3-540-10021-8"><bdi>978-3-540-10021-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Toroidal+Compactification+of+Siegel+Spaces&rft.series=Lecture+Notes+in+Mathematics&rft.date=1980&rft_id=info%3Adoi%2F10.1007%2FBFb0091051&rft.isbn=978-3-540-10021-8&rft.aulast=Namikawa&rft.aufirst=Yukihiko&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChai1986" class="citation book cs1">Chai, Ching-Li (1986). "Siegel Moduli Schemes and Their Compactifications over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>". <a rel="nofollow" class="external text" href="https://books.google.com/books?id=X8TkBwAAQBAJ&pg=PA237"><i>Arithmetic Geometry</i></a>. pp. <span class="nowrap">231–</span>251. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4613-8655-1_9">10.1007/978-1-4613-8655-1_9</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-8657-5" title="Special:BookSources/978-1-4613-8657-5"><bdi>978-1-4613-8657-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Siegel+Moduli+Schemes+and+Their+Compactifications+over+MATH+RENDER+ERROR&rft.btitle=Arithmetic+Geometry&rft.pages=%3Cspan+class%3D%22nowrap%22%3E231-%3C%2Fspan%3E251&rft.date=1986&rft_id=info%3Adoi%2F10.1007%2F978-1-4613-8655-1_9&rft.isbn=978-1-4613-8657-5&rft.aulast=Chai&rft.aufirst=Ching-Li&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DX8TkBwAAQBAJ%26pg%3DPA237&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-Harris-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-Harris_19-0">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarris1992" class="citation book cs1"><a href="/wiki/Joe_Harris_(mathematician)" title="Joe Harris (mathematician)">Harris, Joe</a> (1992). <i>Algebraic Geometry - A first course</i>. Graduate Texts in Mathematics. Vol. 133. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4757-2189-8">10.1007/978-1-4757-2189-8</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-97716-3" title="Special:BookSources/0-387-97716-3"><bdi>0-387-97716-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry+-+A+first+course&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1992&rft_id=info%3Adoi%2F10.1007%2F978-1-4757-2189-8&rft.isbn=0-387-97716-3&rft.aulast=Harris&rft.aufirst=Joe&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=-QMWR-x66XUC&pg=PA90"><i>Algebraic Geometry I</i></a>. Encyclopaedia of Mathematical Sciences. Vol. 23. 1994. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-57878-6">10.1007/978-3-642-57878-6</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-63705-9" title="Special:BookSources/978-3-540-63705-9"><bdi>978-3-540-63705-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Geometry+I&rft.series=Encyclopaedia+of+Mathematical+Sciences&rft.date=1994&rft_id=info%3Adoi%2F10.1007%2F978-3-642-57878-6&rft.isbn=978-3-540-63705-9&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-QMWR-x66XUC%26pg%3DPA90&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre1955" class="citation journal cs1">Serre, Jean-Pierre (1955). <a rel="nofollow" class="external text" href="https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf">"Faisceaux Algebriques Coherents"</a> <span class="cs1-format">(PDF)</span>. <i>Annals of Mathematics</i>. <b>61</b> (2): <span class="nowrap">197–</span>278. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969915">10.2307/1969915</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969915">1969915</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Faisceaux+Algebriques+Coherents&rft.volume=61&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E197-%3C%2Fspan%3E278&rft.date=1955&rft_id=info%3Adoi%2F10.2307%2F1969915&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969915%23id-name%3DJSTOR&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rft_id=https%3A%2F%2Fwww.college-de-france.fr%2Fmedia%2Fjean-pierre-serre%2FUPL5435398796951750634_Serre_FAC.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_variety&action=edit&section=28" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCoxJohn_LittleDon_O'Shea1997" class="citation book cs1"><a href="/wiki/David_Cox_(mathematician)" class="mw-redirect" title="David Cox (mathematician)">Cox, David</a>; John Little; Don O'Shea (1997). <i>Ideals, Varieties, and Algorithms</i> (second ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94680-2" title="Special:BookSources/0-387-94680-2"><bdi>0-387-94680-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ideals%2C+Varieties%2C+and+Algorithms&rft.edition=second&rft.pub=Springer-Verlag&rft.date=1997&rft.isbn=0-387-94680-2&rft.aulast=Cox&rft.aufirst=David&rft.au=John+Little&rft.au=Don+O%27Shea&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisenbud1999" class="citation book cs1"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1999). <i>Commutative Algebra with a View Toward Algebraic Geometry</i>. <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-94269-6" title="Special:BookSources/0-387-94269-6"><bdi>0-387-94269-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+Algebra+with+a+View+Toward+Algebraic+Geometry&rft.pub=Springer-Verlag&rft.date=1999&rft.isbn=0-387-94269-6&rft.aulast=Eisenbud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilne2008" class="citation web cs1"><a href="/wiki/James_Milne_(mathematician)" title="James Milne (mathematician)">Milne, James S.</a> (2008). <a rel="nofollow" class="external text" href="http://www.jmilne.org/math/CourseNotes/ag.html">"Algebraic Geometry"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2009-09-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Algebraic+Geometry&rft.date=2008&rft.aulast=Milne&rft.aufirst=James+S.&rft_id=http%3A%2F%2Fwww.jmilne.org%2Fmath%2FCourseNotes%2Fag.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></li> <li>Milne J., <a rel="nofollow" class="external text" href="https://www.jmilne.org/math/xnotes/JVs.pdf">Jacobian Varieties</a>, published as Chapter VII of Arithmetic geometry (Storrs, Conn., 1984), 167–212, Springer, New York, 1986.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMFK1994" class="citation book cs1"><a href="/wiki/David_Mumford" title="David Mumford">Mumford, David</a>; Fogarty, John; <a href="/wiki/Frances_Kirwan" title="Frances Kirwan">Kirwan, Frances</a> (1994). <i>Geometric invariant theory</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Vol. 34 (3rd ed.). Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-56963-3" title="Special:BookSources/978-3-540-56963-3"><bdi>978-3-540-56963-3</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1304906">1304906</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+invariant+theory&rft.place=Berlin%2C+New+York&rft.series=Ergebnisse+der+Mathematik+und+ihrer+Grenzgebiete+%282%29+%5BResults+in+Mathematics+and+Related+Areas+%282%29%5D&rft.edition=3rd&rft.pub=Springer-Verlag&rft.date=1994&rft.isbn=978-3-540-56963-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1304906%23id-name%3DMR&rft.aulast=Mumford&rft.aufirst=David&rft.au=Fogarty%2C+John&rft.au=Kirwan%2C+Frances&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://pi.lib.uchicago.edu/1001/cat/bib/11217270"><i>Algebraic geometry and arithmetic curves /</i></a>. Oxford science publications. Oxford University Press. 2006. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-154780-5" title="Special:BookSources/978-0-19-154780-5"><bdi>978-0-19-154780-5</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/646747871">646747871</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+geometry+and+arithmetic+curves+%2F&rft.series=Oxford+science+publications&rft.pub=Oxford+University+Press&rft.date=2006&rft_id=info%3Aoclcnum%2F646747871&rft.isbn=978-0-19-154780-5&rft_id=http%3A%2F%2Fpi.lib.uchicago.edu%2F1001%2Fcat%2Fbib%2F11217270&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+variety" class="Z3988"></span></li></ul> </div> <p><i>This article incorporates material from <a rel="nofollow" class="external text" href="https://planetmath.org/isomorphismofvarieties">Isomorphism of varieties</a> on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist 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