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class="mbox-text-span">This article <b>may be <a href="/wiki/Wikipedia:Vagueness" title="Wikipedia:Vagueness">confusing or unclear</a> to readers</b>. In particular, it is unclear when the article is restricted to smooth algebraic surfaces over the complexes, and when more general cases are considered.<span class="hide-when-compact"> Please help <a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarify the article</a>. There might be a discussion about this on <a href="/wiki/Talk:Algebraic_surface" title="Talk:Algebraic surface">the talk page</a>.</span> <span class="date-container"><i>(<span class="date">May 2023</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> </div> </div><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>, an <b>algebraic surface</b> is an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> of <a href="/wiki/Dimension_of_an_algebraic_variety" title="Dimension of an algebraic variety">dimension</a> two. In the case of geometry over the field of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, an algebraic surface has complex dimension two (as a <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifold</a>, when it is <a href="/wiki/Algebraic_curve#Singularities" title="Algebraic curve">non-singular</a>) and so of dimension four as a <a href="/wiki/Smooth_manifold" class="mw-redirect" title="Smooth manifold">smooth manifold</a>. </p><p>The theory of algebraic surfaces is much more complicated than that of <a href="/wiki/Algebraic_curve" title="Algebraic curve">algebraic curves</a> (including the <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Riemann_surface" title="Riemann surface">Riemann surfaces</a>, which are genuine <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a> of (real) dimension two). Many results were obtained, but, in the <a href="/wiki/Italian_school_of_algebraic_geometry" title="Italian school of algebraic geometry">Italian school of algebraic geometry </a>, and are up to 100 years old. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Classification_by_the_Kodaira_dimension">Classification by the Kodaira dimension</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_surface&action=edit&section=1" title="Edit section: Classification by the Kodaira dimension"><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/Enriques%E2%80%93Kodaira_classification" title="Enriques–Kodaira classification">Enriques–Kodaira classification</a></div> <p>In the case of dimension one, varieties are classified by only the <a href="/wiki/Topological_genus" class="mw-redirect" title="Topological genus">topological genus</a>, but, in dimension two, one needs to distinguish the <a href="/wiki/Arithmetic_genus" title="Arithmetic genus">arithmetic genus</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 p_{a}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{a}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c0b14a6cab7974f68d39ef402066d46ac314f87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.361ex; height:2.009ex;" alt="{\displaystyle p_{a}}" /></span> and the <a href="/wiki/Geometric_genus" title="Geometric genus">geometric genus</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 p_{g}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>g</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{g}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6d9377860f66afd38c35fd80600e5f5d6d6329" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:2.28ex; height:2.343ex;" alt="{\displaystyle p_{g}}" /></span> because one cannot distinguish birationally only the topological genus. Then, <a href="/wiki/Irregularity_of_a_surface" title="Irregularity of a surface">irregularity</a> is introduced for the classification of varieties. A summary of the results (in detail, for each kind of surface refers to each redirection), follows: </p><p>Examples of algebraic surfaces include (κ is the <a href="/wiki/Kodaira_dimension" title="Kodaira dimension">Kodaira dimension</a>): </p> <ul><li>κ = −∞: the <a href="/wiki/Complex_projective_plane" title="Complex projective plane">projective plane</a>, <a href="/wiki/Quadric" title="Quadric">quadrics</a> in <b>P</b><sup>3</sup>, <a href="/wiki/Cubic_surface" title="Cubic surface">cubic surfaces</a>, <a href="/wiki/Veronese_surface" title="Veronese surface">Veronese surface</a>, <a href="/wiki/Del_Pezzo_surface" title="Del Pezzo surface">del Pezzo surfaces</a>, <a href="/wiki/Ruled_surface" title="Ruled surface">ruled surfaces</a></li> <li>κ = 0 : <a href="/wiki/K3_surface" title="K3 surface">K3 surfaces</a>, <a href="/wiki/Abelian_surface" title="Abelian surface">abelian surfaces</a>, <a href="/wiki/Enriques_surface" title="Enriques surface">Enriques surfaces</a>, <a href="/wiki/Hyperelliptic_surface" title="Hyperelliptic surface">hyperelliptic surfaces</a></li> <li>κ = 1: <a href="/wiki/Elliptic_surface" title="Elliptic surface">elliptic surfaces</a></li> <li>κ = 2: <a href="/wiki/Surface_of_general_type" title="Surface of general type">surfaces of general type</a>.</li></ul> <p>For more examples see the <a href="/wiki/List_of_algebraic_surfaces" class="mw-redirect" title="List of algebraic surfaces">list of algebraic surfaces</a>. </p><p>The first five examples are in fact <a href="/wiki/Birationally_equivalent" class="mw-redirect" title="Birationally equivalent">birationally equivalent</a>. That is, for example, a cubic surface has a <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a> isomorphic to that of the <a href="/wiki/Projective_plane" title="Projective plane">projective plane</a>, being the <a href="/wiki/Rational_function" title="Rational function">rational functions</a> in two indeterminates. The Cartesian product of two curves also provides examples. </p> <div class="mw-heading mw-heading2"><h2 id="Birational_geometry_of_surfaces">Birational geometry of surfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_surface&action=edit&section=2" title="Edit section: Birational geometry of surfaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Birational_geometry" title="Birational geometry">birational geometry</a> of algebraic surfaces is rich, because of <a href="/wiki/Blowing_up" title="Blowing up">blowing up</a> (also known as a <a href="/wiki/Monoidal_transformation" class="mw-redirect" title="Monoidal transformation">monoidal transformation</a>), under which a point is replaced by the <i>curve</i> of all limiting tangent directions coming into it (a <a href="/wiki/Projective_line" title="Projective line">projective line</a>). Certain curves may also be blown <i>down</i>, but there is a restriction (self-intersection number must be −1). </p> <div class="mw-heading mw-heading3"><h3 id="Castelnuovo's_Theorem"><span id="Castelnuovo.27s_Theorem"></span>Castelnuovo's Theorem</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_surface&action=edit&section=3" title="Edit section: Castelnuovo's Theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One of the fundamental theorems for the birational geometry of surfaces is <a href="/wiki/Castelnuovo%27s_contraction_theorem" title="Castelnuovo's contraction theorem"><b>Castelnuovo's theorem</b></a>. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_surface&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Ample_line_bundle#Intersection_theorem" title="Ample line bundle"><b>Nakai criterion</b></a> says that: </p> <dl><dd>A Divisor <i>D</i> on a surface <i>S</i> is ample if and only if <i>D<sup>2</sup> > 0</i> and for all irreducible curve <i>C</i> on <i>S</i> <i>D•C > 0.</i></dd></dl> <p>Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let <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 {\mathcal {D}}(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67e3574b7f657cf615281304defb16f0370c1d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.1ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(S)}" /></span> be the abelian group consisting of all the divisors on <i>S</i>. Then due to the <a href="/wiki/Intersection_number" title="Intersection number">intersection theorem</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 {\mathcal {D}}(S)\times {\mathcal {D}}(S)\rightarrow \mathbb {Z} :(X,Y)\mapsto X\cdot Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>X</mi> <mo>⋅</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}(S)\times {\mathcal {D}}(S)\rightarrow \mathbb {Z} :(X,Y)\mapsto X\cdot Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9c2c31042426ebbdf503a6f47da358428f235ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.786ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}(S)\times {\mathcal {D}}(S)\rightarrow \mathbb {Z} :(X,Y)\mapsto X\cdot Y}" /></span></dd></dl> <p>is viewed as a <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a>. Let </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 {\mathcal {D}}_{0}(S):=\{D\in {\mathcal {D}}(S)|D\cdot X=0,{\text{for all }}X\in {\mathcal {D}}(S)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>D</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mo>⋅</mo> <mi>X</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mi>X</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}_{0}(S):=\{D\in {\mathcal {D}}(S)|D\cdot X=0,{\text{for all }}X\in {\mathcal {D}}(S)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99e75b150c94562fba83c29635831124851ec32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.94ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}_{0}(S):=\{D\in {\mathcal {D}}(S)|D\cdot X=0,{\text{for all }}X\in {\mathcal {D}}(S)\}}" /></span></dd></dl> <p>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 {\mathcal {D}}/{\mathcal {D}}_{0}(S):=Num(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mi>N</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}/{\mathcal {D}}_{0}(S):=Num(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d4baec0f71210963109a8e94c336dafc157231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.597ex; height:2.843ex;" alt="{\displaystyle {\mathcal {D}}/{\mathcal {D}}_{0}(S):=Num(S)}" /></span> becomes to be a <b>numerical equivalent class group</b> of <i>S</i> and </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 Num(S)\times Num(S)\mapsto \mathbb {Z} =({\bar {D}},{\bar {E}})\mapsto D\cdot E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>×</mo> <mi>N</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>E</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>D</mi> <mo>⋅</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Num(S)\times Num(S)\mapsto \mathbb {Z} =({\bar {D}},{\bar {E}})\mapsto D\cdot E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2fc56a9c35ade91b874e3f7197ea3bd1ddedbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.124ex; height:3.009ex;" alt="{\displaystyle Num(S)\times Num(S)\mapsto \mathbb {Z} =({\bar {D}},{\bar {E}})\mapsto D\cdot E}" /></span></dd></dl> <p>also becomes to be a quadratic form 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 Num(S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Num(S)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f97d07db182bc2ce7ed88e9eaa035b0a8f5ed75a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.742ex; height:2.843ex;" alt="{\displaystyle Num(S)}" /></span>, 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 {\bar {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a899b65dfb3ab3073b4f7f257be4491141bee64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.509ex;" alt="{\displaystyle {\bar {D}}}" /></span> is the image of a divisor <i>D</i> on <i>S</i>. (In the below the image <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 {\bar {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>D</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a899b65dfb3ab3073b4f7f257be4491141bee64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.509ex;" alt="{\displaystyle {\bar {D}}}" /></span> is abbreviated with <i>D</i>.) </p><p>For an ample line bundle <i>H</i> on <i>S</i>, the definition </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 \{H\}^{\perp }:=\{D\in Num(S)|D\cdot H=0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>H</mi> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>D</mi> <mo>∈</mo> <mi>N</mi> <mi>u</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mo>⋅</mo> <mi>H</mi> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{H\}^{\perp }:=\{D\in Num(S)|D\cdot H=0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99d81102819691c7cf1258b00bfe3f5a81ee137a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.698ex; height:3.176ex;" alt="{\displaystyle \{H\}^{\perp }:=\{D\in Num(S)|D\cdot H=0\}.}" /></span></dd></dl> <p>is used in the surface version of the <b>Hodge index theorem</b>: </p> <dl><dd>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 D\in \{\{H\}^{\perp }|D\neq 0\},D\cdot D<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mo fence="false" stretchy="false">{</mo> <mi>H</mi> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>D</mi> <mo>≠</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>D</mi> <mo>⋅</mo> <mi>D</mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D\in \{\{H\}^{\perp }|D\neq 0\},D\cdot D<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef9bc97230468e7ff6a1f09388d2e9e4f959e34d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.644ex; height:3.176ex;" alt="{\displaystyle D\in \{\{H\}^{\perp }|D\neq 0\},D\cdot D<0}" /></span>, i.e. the restriction of the intersection form 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 \{H\}^{\perp }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>H</mi> <msup> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{H\}^{\perp }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e742b303ca3c2c9d3cd45b8e8c4202bd84084abf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.899ex; height:3.176ex;" alt="{\displaystyle \{H\}^{\perp }}" /></span> is a negative definite quadratic form.</dd></dl> <p>This theorem is proven using the Nakai criterion and the Riemann-Roch theorem for surfaces. The Hodge index theorem is used in Deligne's proof of the <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjecture</a>. </p><p>Basic results on algebraic surfaces include the <a href="/wiki/Hodge_index_theorem" title="Hodge index theorem">Hodge index theorem</a>, and the division into five groups of birational equivalence classes called the <a href="/wiki/Classification_of_algebraic_surfaces" class="mw-redirect" title="Classification of algebraic surfaces">classification of algebraic surfaces</a>. The <i>general type</i> class, of <a href="/wiki/Kodaira_dimension" title="Kodaira dimension">Kodaira dimension</a> 2, is very large (degree 5 or larger for a non-singular surface in <b>P</b><sup>3</sup> lies in it, for example). </p><p>There are essential three <a href="/wiki/Hodge_number" class="mw-redirect" title="Hodge number">Hodge number</a> invariants of a surface. Of those, <i>h</i><sup>1,0</sup> was classically called the <b>irregularity</b> and denoted by <i>q</i>; and <i>h</i><sup>2,0</sup> was called the <b>geometric genus</b> <i>p</i><sub><i>g</i></sub>. The third, <i>h</i><sup>1,1</sup>, is not a <a href="/wiki/Birational_invariant" title="Birational invariant">birational invariant</a>, because <a href="/wiki/Blowing_up" title="Blowing up">blowing up</a> can add whole curves, with classes in <i>H</i><sup>1,1</sup>. It is known that <a href="/wiki/Hodge_cycle" title="Hodge cycle">Hodge cycles</a> are algebraic and that <a href="/wiki/Algebraic_equivalence" class="mw-redirect" title="Algebraic equivalence">algebraic equivalence</a> coincides with <a href="/w/index.php?title=Homological_equivalence&action=edit&redlink=1" class="new" title="Homological equivalence (page does not exist)">homological equivalence</a>, so that <i>h</i><sup>1,1</sup> is an upper bound for ρ, the rank of the <a href="/wiki/N%C3%A9ron-Severi_group" class="mw-redirect" title="Néron-Severi group">Néron-Severi group</a>. The <a href="/wiki/Arithmetic_genus" title="Arithmetic genus">arithmetic genus</a> <i>p</i><sub><i>a</i></sub> is the difference </p> <dl><dd>geometric genus − irregularity.</dd></dl> <p>This explains why the irregularity got its name, as a kind of 'error term'. </p> <div class="mw-heading mw-heading2"><h2 id="Riemann-Roch_theorem_for_surfaces">Riemann-Roch theorem for surfaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_surface&action=edit&section=5" title="Edit section: Riemann-Roch theorem for surfaces"><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/Riemann-Roch_theorem_for_surfaces" class="mw-redirect" title="Riemann-Roch theorem for surfaces">Riemann-Roch theorem for surfaces</a></div> <p>The <a href="/wiki/Riemann-Roch_theorem_for_surfaces" class="mw-redirect" title="Riemann-Roch theorem for surfaces">Riemann-Roch theorem for surfaces</a> was first formulated by <a href="/wiki/Max_Noether" title="Max Noether">Max Noether</a>. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry. </p> <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_surface&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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="CITEREFDolgachev2001" class="citation cs2">Dolgachev, I.V. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Algebraic_surface">"Algebraic surface"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Algebraic+surface&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.aulast=Dolgachev&rft.aufirst=I.V.&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DAlgebraic_surface&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+surface" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFZariski1995" class="citation cs2"><a href="/wiki/Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (1995), <i>Algebraic surfaces</i>, Classics in Mathematics, 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/978-3-540-58658-6" title="Special:BookSources/978-3-540-58658-6"><bdi>978-3-540-58658-6</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=1336146">1336146</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+surfaces&rft.place=Berlin%2C+New+York&rft.series=Classics+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-3-540-58658-6&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1336146%23id-name%3DMR&rft.aulast=Zariski&rft.aufirst=Oscar&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebraic+surface" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebraic_surface&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://imaginary.org/program/surfer">Free program SURFER</a> to visualize algebraic surfaces in real-time, including a user gallery.</li> <li><a rel="nofollow" class="external text" href="http://www.singsurf.org/singsurf/SingSurf.html">SingSurf</a> an interactive 3D viewer for algebraic surfaces.</li> <li><a rel="nofollow" class="external text" href="http://www.bru.hlphys.jku.at/surf/index.html">Page on Algebraic Surfaces started in 2008</a></li> <li><a rel="nofollow" class="external text" href="http://maxwelldemon.com/2009/03/29/surfaces-2-algebraic-surfaces/">Overview and thoughts on designing Algebraic surfaces</a></li></ul> <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 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