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href="#Line_extended_by_a_point_at_infinity"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Line extended by a point at infinity</span> </div> </a> <ul id="toc-Line_extended_by_a_point_at_infinity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</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-Real_projective_line" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_projective_line"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Real projective line</span> </div> </a> <ul id="toc-Real_projective_line-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_projective_line:_the_Riemann_sphere" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_projective_line:_the_Riemann_sphere"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Complex projective line: the Riemann sphere</span> </div> </a> <ul id="toc-Complex_projective_line:_the_Riemann_sphere-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-For_a_finite_field" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#For_a_finite_field"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>For a finite field</span> </div> </a> <ul id="toc-For_a_finite_field-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Symmetry_group" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Symmetry_group"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Symmetry group</span> </div> </a> <ul id="toc-Symmetry_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-As_algebraic_curve" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#As_algebraic_curve"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>As algebraic curve</span> </div> </a> <ul id="toc-As_algebraic_curve-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" 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href="https://de.wikipedia.org/wiki/Projektive_Gerade" title="Projektive Gerade – German" lang="de" hreflang="de" data-title="Projektive Gerade" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Recta_proyectiva" title="Recta proyectiva – Spanish" lang="es" hreflang="es" data-title="Recta proyectiva" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Droite_projective" title="Droite projective – French" lang="fr" hreflang="fr" data-title="Droite projective" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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 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src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Projective_line" title="Special:EditPage/Projective line">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Projective+line%22">"Projective line"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Projective+line%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Projective+line%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Projective+line%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Projective+line%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Projective+line%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2009</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>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>projective line</b> is, roughly speaking, the extension of a usual <a href="/wiki/Line_(geometry)" title="Line (geometry)">line</a> by a point called a <i><a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a></i>. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a> meet in exactly one point (there is no "parallel" case). </p><p>There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i>, commonly denoted <b>P</b><sup>1</sup>(<i>K</i>), as the set of one-dimensional <a href="/wiki/Linear_subspace" title="Linear subspace">subspaces</a> of a two-dimensional <i>K</i>-<a href="/wiki/Vector_space" title="Vector space">vector space</a>. This definition is a special instance of the general definition of a <a href="/wiki/Projective_space" title="Projective space">projective space</a>. </p><p>The projective line over the <a href="/wiki/Real_number" title="Real number">reals</a> is a <a href="/wiki/Manifold" title="Manifold">manifold</a>; see <i><a href="/wiki/Real_projective_line" title="Real projective line">Real projective line</a></i> for details. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Homogeneous_coordinates">Homogeneous coordinates</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=1" title="Edit section: Homogeneous coordinates"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An arbitrary point in the projective line <b>P</b><sup>1</sup>(<i>K</i>) may be represented by an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of <i><a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a></i>, which take the form of a pair </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 [x_{1}:x_{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09404e16ef9ccccd05414dabeedac84b306fe9e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.999ex; height:2.843ex;" alt="{\displaystyle [x_{1}:x_{2}]}"></span></dd></dl> <p>of elements of <i>K</i> that are not both zero. Two such pairs are <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalent</a> if they differ by an overall nonzero factor <i>λ</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 [x_{1}:x_{2}]\sim [\lambda x_{1}:\lambda x_{2}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>∼</mo> <mo stretchy="false">[</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <mi>λ</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]\sim [\lambda x_{1}:\lambda x_{2}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e149dfacd2fd2bfcc157bc3da4e654227498373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.453ex; height:2.843ex;" alt="{\displaystyle [x_{1}:x_{2}]\sim [\lambda x_{1}:\lambda x_{2}].}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Line_extended_by_a_point_at_infinity">Line extended by a point at infinity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=2" title="Edit section: Line extended by a point at infinity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The projective line may be identified with the line <i>K</i> extended by a <a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a>. More precisely, the line <i>K</i> may be identified with the subset of <b>P</b><sup>1</sup>(<i>K</i>) given by </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 \left\{[x:1]\in \mathbf {P} ^{1}(K)\mid x\in K\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>:</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>∣</mo> <mi>x</mi> <mo>∈</mo> <mi>K</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{[x:1]\in \mathbf {P} ^{1}(K)\mid x\in K\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/086c5d6b3f2d24eccfa35f0847c01d70c75b8c57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.238ex; height:3.343ex;" alt="{\displaystyle \left\{[x:1]\in \mathbf {P} ^{1}(K)\mid x\in K\right\}.}"></span></dd></dl> <p>This subset covers all points in <b>P</b><sup>1</sup>(<i>K</i>) except one, which is called the <i>point at infinity</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 \infty =[1:0].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>:</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \infty =[1:0].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a6746d720a3849c5aeeeb5b0cb7fb0f12eac33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.625ex; height:2.843ex;" alt="{\displaystyle \infty =[1:0].}"></span></dd></dl> <p>This allows to extend the arithmetic on <i>K</i> to <b>P</b><sup>1</sup>(<i>K</i>) by the formulas </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 {\frac {1}{0}}=\infty ,\qquad {\frac {1}{\infty }}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>0</mn> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi mathvariant="normal">∞</mi> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{0}}=\infty ,\qquad {\frac {1}{\infty }}=0,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92fb1813885e723a28a5dd43cb095740370857fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.168ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{0}}=\infty ,\qquad {\frac {1}{\infty }}=0,}"></span></dd> <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 x\cdot \infty =\infty \quad {\text{if}}\quad x\not =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot \infty =\infty \quad {\text{if}}\quad x\not =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3e7d5ea1269dff1e9dc0e8b219d7eb8c18719af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.349ex; height:2.676ex;" alt="{\displaystyle x\cdot \infty =\infty \quad {\text{if}}\quad x\not =0}"></span></dd> <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 x+\infty =\infty \quad {\text{if}}\quad x\not =\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo>=</mo> <mi mathvariant="normal">∞</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>if</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>≠</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+\infty =\infty \quad {\text{if}}\quad x\not =\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4dd2d88379caf16a1f8229dfe60c1690283bee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.672ex; height:2.676ex;" alt="{\displaystyle x+\infty =\infty \quad {\text{if}}\quad x\not =\infty }"></span></dd></dl> <p>Translating this arithmetic in terms of homogeneous coordinates gives, when <span class="nowrap">[0 : 0]</span> does not occur: </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 [x_{1}:x_{2}]+[y_{1}:y_{2}]=[(x_{1}y_{2}+y_{1}x_{2}):x_{2}y_{2}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>+</mo> <mo stretchy="false">[</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]+[y_{1}:y_{2}]=[(x_{1}y_{2}+y_{1}x_{2}):x_{2}y_{2}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74a9ac91bf482d9d4a59fd0e81fed72cea4384ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.815ex; height:2.843ex;" alt="{\displaystyle [x_{1}:x_{2}]+[y_{1}:y_{2}]=[(x_{1}y_{2}+y_{1}x_{2}):x_{2}y_{2}],}"></span></dd> <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 [x_{1}:x_{2}]\cdot [y_{1}:y_{2}]=[x_{1}y_{1}:x_{2}y_{2}],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>⋅</mo> <mo stretchy="false">[</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]\cdot [y_{1}:y_{2}]=[x_{1}y_{1}:x_{2}y_{2}],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fa88c74d83459420ba532b4549908fd14cbab9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.427ex; height:2.843ex;" alt="{\displaystyle [x_{1}:x_{2}]\cdot [y_{1}:y_{2}]=[x_{1}y_{1}:x_{2}y_{2}],}"></span></dd> <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 [x_{1}:x_{2}]^{-1}=[x_{2}:x_{1}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]^{-1}=[x_{2}:x_{1}].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8afadaa9e638cbce6c51e27ae575b77755bc3cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.075ex; height:3.176ex;" alt="{\displaystyle [x_{1}:x_{2}]^{-1}=[x_{2}:x_{1}].}"></span></dd></dl> <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=Projective_line&action=edit&section=3" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Real_projective_line">Real projective line</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=4" title="Edit section: Real projective line"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Real_projective_line" title="Real projective line">Real projective line</a></div> <p>The projective line over the <a href="/wiki/Real_number" title="Real number">real numbers</a> is called the <b>real projective line</b>. It may also be thought of as the line <i>K</i> together with an idealised <i><a href="/wiki/Point_at_infinity" title="Point at infinity">point at infinity</a></i> ∞; the point connects to both ends of <i>K</i> creating a closed loop or topological circle. </p><p>An example is obtained by projecting points in <b>R</b><sup>2</sup> onto the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a> and then <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">identifying</a> <a href="/wiki/Diametrically_opposite" class="mw-redirect" title="Diametrically opposite">diametrically opposite</a> points. In terms of <a href="/wiki/Group_theory" title="Group theory">group theory</a> we can take the quotient by the <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> <span class="nowrap">{1, −1}</span>. </p><p>Compare the <a href="/wiki/Extended_real_number_line" title="Extended real number line">extended real number line</a>, which distinguishes ∞ and −∞. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_projective_line:_the_Riemann_sphere">Complex projective line: the Riemann sphere</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=5" title="Edit section: Complex projective line: the Riemann sphere"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Adding a point at infinity to the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a> results in a space that is topologically a <a href="/wiki/Sphere" title="Sphere">sphere</a>. Hence the complex projective line is also known as the <b><a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></b> (or sometimes the <i>Gauss sphere</i>). It is in constant use in <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a> theory, as the simplest example of a <a href="/wiki/Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">compact Riemann surface</a>. </p> <div class="mw-heading mw-heading3"><h3 id="For_a_finite_field">For a finite field</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=6" title="Edit section: For a finite field"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The projective line over a <a href="/wiki/Finite_field" title="Finite field">finite field</a> <i>F</i><sub><i>q</i></sub> of <i>q</i> elements has <span class="nowrap"><i>q</i> + 1</span> points. In all other respects it is no different from projective lines defined over other types of fields. In the terms of homogeneous coordinates <span class="nowrap">[<i>x</i> : <i>y</i>]</span>, <i>q</i> of these points have the form: </p> <dl><dd><span class="texhtml">[<i>a</i> : 1]</span> for each <span class="texhtml mvar" style="font-style:italic;"><i>a</i></span> in <span class="texhtml mvar" style="font-style:italic;"><i>F</i><sub><i>q</i></sub></span>,</dd></dl> <p>and the remaining <a href="/wiki/Point_at_infinity" title="Point at infinity">point <i>at infinity</i></a> may be represented as <span class="nowrap">[1 : 0]</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Symmetry_group">Symmetry group</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=7" title="Edit section: Symmetry group"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Quite generally, the group of <a href="/wiki/Homography" title="Homography">homographies</a> with <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> in <i>K</i> acts on the projective line <b>P</b><sup>1</sup>(<i>K</i>). This <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> is <a href="/wiki/Group_action_(mathematics)#Types_of_actions" class="mw-redirect" title="Group action (mathematics)">transitive</a>, so that <b>P</b><sup>1</sup>(<i>K</i>) is a <a href="/wiki/Homogeneous_space" title="Homogeneous space">homogeneous space</a> for the group, often written PGL<sub>2</sub>(<i>K</i>) to emphasise the projective nature of these transformations. <i>Transitivity</i> says that there exists a homography that will transform any point <i>Q</i> to any other point <i>R</i>. The <i>point at infinity</i> on <b>P</b><sup>1</sup>(<i>K</i>) is therefore an <i>artifact</i> of choice of coordinates: <a href="/wiki/Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</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 [X:Y]\sim [\lambda X:\lambda Y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>X</mi> <mo>:</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>∼</mo> <mo stretchy="false">[</mo> <mi>λ</mi> <mi>X</mi> <mo>:</mo> <mi>λ</mi> <mi>Y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [X:Y]\sim [\lambda X:\lambda Y]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6578d7b00b891341414ba7439d96deba71b24318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.777ex; height:2.843ex;" alt="{\displaystyle [X:Y]\sim [\lambda X:\lambda Y]}"></span></dd></dl> <p>express a one-dimensional subspace by a single non-zero point <span class="nowrap">(<i>X</i>, <i>Y</i>)</span> lying in it, but the symmetries of the projective line can move the point <span class="nowrap">∞ = [1 : 0]</span> to any other, and it is in no way distinguished. </p><p>Much more is true, in that some transformation can take any given <a href="/wiki/Distinct_(mathematics)" class="mw-redirect" title="Distinct (mathematics)">distinct</a> points <i>Q</i><sub><i>i</i></sub> for <span class="nowrap"><i>i</i> = 1, 2, 3</span> to any other 3-tuple <i>R</i><sub><i>i</i></sub> of distinct points (<i>triple transitivity</i>). This amount of specification 'uses up' the three dimensions of PGL<sub>2</sub>(<i>K</i>); in other words, the group action is <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">sharply 3-transitive</a>. The computational aspect of this is the <a href="/wiki/Cross-ratio" title="Cross-ratio">cross-ratio</a>. Indeed, a generalized converse is true: a sharply 3-transitive group action is always (isomorphic to) a generalized form of a PGL<sub>2</sub>(<i>K</i>) action on a projective line, replacing "field" by "KT-field" (generalizing the inverse to a weaker kind of involution), and "PGL" by a corresponding generalization of projective linear maps.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="As_algebraic_curve">As algebraic curve</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Projective_line&action=edit&section=8" title="Edit section: As algebraic curve"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The projective line is a fundamental example of an <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curve</a>. From the point of view of algebraic geometry, <b>P</b><sup>1</sup>(<i>K</i>) is a <a href="/wiki/Algebraic_curve#Singularities" title="Algebraic curve">non-singular</a> curve of <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> 0. If <i>K</i> is <a href="/wiki/Algebraically_closed" class="mw-redirect" title="Algebraically closed">algebraically closed</a>, it is the unique such curve over <i>K</i>, up to <a href="/wiki/Rational_equivalence" class="mw-redirect" title="Rational equivalence">rational equivalence</a>. In general a (non-singular) curve of genus 0 is rationally equivalent over <i>K</i> to a <a href="/wiki/Conic" class="mw-redirect" title="Conic">conic</a> <i>C</i>, which is itself birationally equivalent to projective line if and only if <i>C</i> has a point defined over <i>K</i>; geometrically such a point <i>P</i> can be used as origin to make explicit the birational equivalence. </p><p>The <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a> of the projective line is the field <i>K</i>(<i>T</i>) of <a href="/wiki/Rational_function" title="Rational function">rational functions</a> over <i>K</i>, in a single indeterminate <i>T</i>. The <a href="/wiki/Field_automorphism" class="mw-redirect" title="Field automorphism">field automorphisms</a> of <i>K</i>(<i>T</i>) over <i>K</i> are precisely the group PGL<sub>2</sub>(<i>K</i>) discussed above. </p><p>Any function field <i>K</i>(<i>V</i>) of an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> <i>V</i> over <i>K</i>, other than a single point, has a subfield isomorphic with <i>K</i>(<i>T</i>). From the point of view of <a href="/wiki/Birational_geometry" title="Birational geometry">birational geometry</a>, this means that there will be a <a href="/wiki/Rational_map" class="mw-redirect" title="Rational map">rational map</a> from <i>V</i> to <b>P</b><sup>1</sup>(<i>K</i>), that is not constant. The image will omit only finitely many points of <b>P</b><sup>1</sup>(<i>K</i>), and the inverse image of a typical point <i>P</i> will be of dimension <span class="nowrap">dim <i>V</i> − 1</span>. This is the beginning of methods in algebraic geometry that are inductive on dimension. The rational maps play a role analogous to the <a href="/wiki/Meromorphic_function" title="Meromorphic function">meromorphic functions</a> of <a href="/wiki/Complex_analysis" title="Complex analysis">complex analysis</a>, and indeed in the case of <a href="/wiki/Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">compact Riemann surfaces</a> the two concepts coincide. </p><p>If <i>V</i> is now taken to be of dimension 1, we get a picture of a typical algebraic curve <i>C</i> presented 'over' <b>P</b><sup>1</sup>(<i>K</i>). Assuming <i>C</i> is non-singular (which is no loss of generality starting with <i>K</i>(<i>C</i>)), it can be shown that such a rational map from <i>C</i> to <b>P</b><sup>1</sup>(<i>K</i>) will in fact be everywhere defined. (That is not the case if there are singularities, since for example a <i><a href="/wiki/Double_point" class="mw-redirect" title="Double point">double point</a></i> where a curve <i>crosses itself</i> may give an indeterminate result after a rational map.) This gives a picture in which the main geometric feature is <a href="/wiki/Ramification_(mathematics)" title="Ramification (mathematics)">ramification</a>. </p><p>Many curves, for example <a href="/wiki/Hyperelliptic_curve" title="Hyperelliptic curve">hyperelliptic curves</a>, may be presented abstractly, as <a href="/wiki/Ramified_cover" class="mw-redirect" title="Ramified cover">ramified covers</a> of the projective line. According to the <a href="/wiki/Riemann%E2%80%93Hurwitz_formula" title="Riemann–Hurwitz formula">Riemann–Hurwitz formula</a>, the genus then depends only on the type of ramification. </p><p>A <b>rational curve</b> is a curve that is <a href="/wiki/Birational_equivalence" class="mw-redirect" title="Birational equivalence">birationally equivalent</a> to a projective line (see <a href="/wiki/Rational_variety" title="Rational variety">rational variety</a>); its <a href="/wiki/Genus_(mathematics)" title="Genus (mathematics)">genus</a> is 0. A <a href="/wiki/Rational_normal_curve" title="Rational normal curve">rational normal curve</a> in projective space <b>P</b><sup><i>n</i></sup> is a rational curve that lies in no proper linear subspace; it is known that there is only one example (up to projective equivalence),<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> given parametrically in homogeneous coordinates as </p> <dl><dd>[1 : <i>t</i> : <i>t</i><sup>2</sup> : ... : <i>t</i><sup><i>n</i></sup>].</dd></dl> <p>See <i><a href="/wiki/Twisted_cubic" title="Twisted cubic">Twisted cubic</a></i> for the first interesting case. </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=Projective_line&action=edit&section=9" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebraic_curve" title="Algebraic curve">Algebraic curve</a></li> <li><a href="/wiki/Cross-ratio" title="Cross-ratio">Cross-ratio</a></li> <li><a href="/wiki/M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a></li> <li><a href="/wiki/Projective_line_over_a_ring" title="Projective line over a ring">Projective line over a ring</a></li> <li><a href="/wiki/Projectively_extended_real_line" title="Projectively extended real line">Projectively extended real line</a></li> <li><a href="/wiki/Projective_range" title="Projective range">Projective range</a></li> <li><a href="/wiki/Wheel_theory" title="Wheel theory">Wheel theory</a></li></ul> <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=Projective_line&action=edit&section=10" title="Edit section: References"><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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/66865">Action of PGL(2) on Projective Space</a> – see comment and cited paper.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></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="CITEREFHarris1992" class="citation cs2">Harris, Joe (1992), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=_XxZdhbtf1sC&pg=PA10"><i>Algebraic Geometry: A First Course</i></a>, Graduate Texts in Mathematics, vol. 133, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780387977164" title="Special:BookSources/9780387977164"><bdi>9780387977164</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%3A+A+First+Course&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1992&rft.isbn=9780387977164&rft.aulast=Harris&rft.aufirst=Joe&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_XxZdhbtf1sC%26pg%3DPA10&rfr_id=info%3Asid%2Fen.wikipedia.org%3AProjective+line" class="Z3988"></span>.</span> </li> </ol></div></div> <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 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href="/wiki/Template:Algebraic_curves_navbox" title="Template:Algebraic curves navbox"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_curves_navbox" title="Template talk:Algebraic curves navbox"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_curves_navbox" title="Special:EditPage/Template:Algebraic curves navbox"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topics_in_algebraic_curves" style="font-size:114%;margin:0 4em">Topics in <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Rational_curve" class="mw-redirect" title="Rational curve">Rational curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Five_points_determine_a_conic" title="Five points determine a conic">Five points determine a conic</a></li> <li><a class="mw-selflink selflink">Projective line</a></li> <li><a href="/wiki/Rational_normal_curve" title="Rational normal curve">Rational normal curve</a></li> <li><a href="/wiki/Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li> <li><a href="/wiki/Twisted_cubic" title="Twisted cubic">Twisted cubic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Elliptic_curve" title="Elliptic curve">Elliptic curves</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Analytic theory</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_function" title="Elliptic function">Elliptic function</a></li> <li><a href="/wiki/Elliptic_integral" title="Elliptic integral">Elliptic integral</a></li> <li><a href="/wiki/Fundamental_pair_of_periods" title="Fundamental pair of periods">Fundamental pair of periods</a></li> <li><a href="/wiki/Modular_form" title="Modular form">Modular form</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Arithmetic theory</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Counting_points_on_elliptic_curves" title="Counting points on elliptic curves">Counting points on elliptic curves</a></li> <li><a href="/wiki/Division_polynomials" title="Division polynomials">Division polynomials</a></li> <li><a href="/wiki/Hasse%27s_theorem_on_elliptic_curves" title="Hasse's theorem on elliptic curves">Hasse's theorem on elliptic curves</a></li> <li><a href="/wiki/Mazur%27s_torsion_theorem" class="mw-redirect" title="Mazur's torsion theorem">Mazur's torsion theorem</a></li> <li><a href="/wiki/Modular_elliptic_curve" title="Modular elliptic curve">Modular elliptic curve</a></li> <li><a href="/wiki/Modularity_theorem" title="Modularity theorem">Modularity theorem</a></li> <li><a href="/wiki/Mordell%E2%80%93Weil_theorem" title="Mordell–Weil theorem">Mordell–Weil theorem</a></li> <li><a href="/wiki/Nagell%E2%80%93Lutz_theorem" title="Nagell–Lutz theorem">Nagell–Lutz theorem</a></li> <li><a href="/wiki/Supersingular_elliptic_curve" title="Supersingular elliptic curve">Supersingular elliptic curve</a></li> <li><a href="/wiki/Schoof%27s_algorithm" title="Schoof's algorithm">Schoof's algorithm</a></li> <li><a href="/wiki/Schoof%E2%80%93Elkies%E2%80%93Atkin_algorithm" title="Schoof–Elkies–Atkin algorithm">Schoof–Elkies–Atkin algorithm</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">Elliptic curve cryptography</a></li> <li><a href="/wiki/Elliptic_curve_primality" title="Elliptic curve primality">Elliptic curve primality</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Higher genus</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/De_Franchis_theorem" title="De Franchis theorem">De Franchis theorem</a></li> <li><a href="/wiki/Faltings%27s_theorem" title="Faltings's theorem">Faltings's theorem</a></li> <li><a href="/wiki/Hurwitz%27s_automorphisms_theorem" title="Hurwitz's automorphisms theorem">Hurwitz's automorphisms theorem</a></li> <li><a href="/wiki/Hurwitz_surface" title="Hurwitz surface">Hurwitz surface</a></li> <li><a href="/wiki/Hyperelliptic_curve" title="Hyperelliptic curve">Hyperelliptic curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plane_curve" title="Plane curve">Plane curves</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/AF%2BBG_theorem" title="AF+BG theorem">AF+BG theorem</a></li> <li><a href="/wiki/B%C3%A9zout%27s_theorem" title="Bézout's theorem">Bézout's theorem</a></li> <li><a href="/wiki/Bitangent" title="Bitangent">Bitangent</a></li> <li><a href="/wiki/Cayley%E2%80%93Bacharach_theorem" title="Cayley–Bacharach theorem">Cayley–Bacharach theorem</a></li> <li><a href="/wiki/Conic_section" title="Conic section">Conic section</a></li> <li><a href="/wiki/Cramer%27s_paradox" title="Cramer's paradox">Cramer's paradox</a></li> <li><a href="/wiki/Cubic_plane_curve" title="Cubic plane curve">Cubic plane curve</a></li> <li><a href="/wiki/Fermat_curve" title="Fermat curve">Fermat curve</a></li> <li><a href="/wiki/Genus%E2%80%93degree_formula" title="Genus–degree formula">Genus–degree formula</a></li> <li><a href="/wiki/Hilbert%27s_sixteenth_problem" title="Hilbert's sixteenth problem">Hilbert's sixteenth problem</a></li> <li><a href="/wiki/Nagata%27s_conjecture_on_curves" title="Nagata's conjecture on curves">Nagata's conjecture on curves</a></li> <li><a href="/wiki/Pl%C3%BCcker_formula" title="Plücker formula">Plücker formula</a></li> <li><a href="/wiki/Quartic_plane_curve" title="Quartic plane curve">Quartic plane curve</a></li> <li><a href="/wiki/Real_plane_curve" title="Real plane curve">Real plane curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Belyi%27s_theorem" title="Belyi's theorem">Belyi's theorem</a></li> <li><a href="/wiki/Bring%27s_curve" title="Bring's curve">Bring's curve</a></li> <li><a href="/wiki/Bolza_surface" title="Bolza surface">Bolza surface</a></li> <li><a href="/wiki/Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">Compact Riemann surface</a></li> <li><a href="/wiki/Dessin_d%27enfant" title="Dessin d'enfant">Dessin d'enfant</a></li> <li><a href="/wiki/Differential_of_the_first_kind" title="Differential of the first kind">Differential of the first kind</a></li> <li><a href="/wiki/Klein_quartic" title="Klein quartic">Klein quartic</a></li> <li><a href="/wiki/Riemann%27s_existence_theorem" class="mw-redirect" title="Riemann's existence theorem">Riemann's existence theorem</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Teichm%C3%BCller_space" title="Teichmüller space">Teichmüller space</a></li> <li><a href="/wiki/Torelli_theorem" title="Torelli theorem">Torelli theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_curve" title="Dual curve">Dual curve</a></li> <li><a href="/wiki/Polar_curve" title="Polar curve">Polar curve</a></li> <li><a href="/wiki/Smooth_completion" title="Smooth completion">Smooth completion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Structure of curves</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Divisors on curves</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abel%E2%80%93Jacobi_map" title="Abel–Jacobi map">Abel–Jacobi map</a></li> <li><a href="/wiki/Brill%E2%80%93Noether_theory" title="Brill–Noether theory">Brill–Noether theory</a></li> <li><a href="/wiki/Clifford%27s_theorem_on_special_divisors" title="Clifford's theorem on special divisors">Clifford's theorem on special divisors</a></li> <li><a href="/wiki/Gonality_of_an_algebraic_curve" title="Gonality of an algebraic curve">Gonality of an algebraic curve</a></li> <li><a href="/wiki/Jacobian_variety" title="Jacobian variety">Jacobian variety</a></li> <li><a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li> <li><a href="/wiki/Weierstrass_point" title="Weierstrass point">Weierstrass point</a></li> <li><a href="/wiki/Weil_reciprocity_law" title="Weil reciprocity law">Weil reciprocity law</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Moduli</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/ELSV_formula" title="ELSV formula">ELSV formula</a></li> <li><a href="/wiki/Gromov%E2%80%93Witten_invariant" title="Gromov–Witten invariant">Gromov–Witten invariant</a></li> <li><a href="/wiki/Hodge_bundle" title="Hodge bundle">Hodge bundle</a></li> <li><a href="/wiki/Moduli_of_algebraic_curves" title="Moduli of algebraic curves">Moduli of algebraic curves</a></li> <li><a href="/wiki/Stable_curve" title="Stable curve">Stable curve</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Morphisms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hasse%E2%80%93Witt_matrix" title="Hasse–Witt matrix">Hasse–Witt matrix</a></li> <li><a href="/wiki/Riemann%E2%80%93Hurwitz_formula" title="Riemann–Hurwitz formula">Riemann–Hurwitz formula</a></li> <li><a href="/wiki/Prym_variety" title="Prym variety">Prym variety</a></li> <li><a href="/wiki/Weber%27s_theorem_(Algebraic_curves)" title="Weber's theorem (Algebraic curves)">Weber's theorem (Algebraic curves)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Singular_point_of_a_curve" title="Singular point of a curve">Singularities</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ak_singularity" title="Ak singularity"><i>A<sub>k</sub></i> singularity</a></li> <li><a href="/wiki/Acnode" title="Acnode">Acnode</a></li> <li><a href="/wiki/Crunode" title="Crunode">Crunode</a></li> <li><a href="/wiki/Cusp_(singularity)" title="Cusp (singularity)">Cusp</a></li> <li><a href="/wiki/Delta_invariant" class="mw-redirect" title="Delta invariant">Delta invariant</a></li> <li><a href="/wiki/Tacnode" title="Tacnode">Tacnode</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_bundle" title="Vector bundle">Vector bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Birkhoff%E2%80%93Grothendieck_theorem" title="Birkhoff–Grothendieck theorem">Birkhoff–Grothendieck theorem</a></li> <li><a href="/wiki/Stable_vector_bundle" title="Stable vector bundle">Stable vector bundle</a></li> <li><a href="/wiki/Vector_bundles_on_algebraic_curves" title="Vector bundles on algebraic curves">Vector bundles on algebraic curves</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" 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