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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Projective line</span></span>

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<p>In <a href="Projective_geometry" title="Projective geometry">projective geometry</a> and <a href="Mathematics" title="Mathematics">mathematics</a> more generally, a <b>projective line</b> is, roughly speaking, the extension of a usual <a href="Line_(geometry)" title="Line (geometry)">line</a> by a point called a <i><a href="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 resulting elimination of special cases; for example, two distinct projective lines in a <a href="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="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="Linear_subspace" title="Linear subspace">subspaces</a> of a two-dimensional <i>K</i>-<a href="Vector_space" title="Vector space">vector space</a>. This definition is a special instance of the general definition of a <a href="Projective_space" title="Projective space">projective space</a>.

</p><p>The projective line over the <a href="Real_number" title="Real number">reals</a> is a <a href="Manifold" title="Manifold">manifold</a>; see <i><a href="Real_projective_line" title="Real projective line">Real projective line</a></i> for details.

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<div class="mw-heading mw-heading2"><h2 id="Homogeneous_coordinates">Homogeneous coordinates</h2></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="Equivalence_class" title="Equivalence class">equivalence class</a> of <i><a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a></i>, which take the form of a pair

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<annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]}</annotation>

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</math></span><img src="./09404e16ef9ccccd05414dabeedac84b306fe9e4.svg" 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}]}" loading="lazy"></span></dd></dl>

<p>of elements of <i>K</i> that are not both zero. Two such pairs are <a href="Equivalence_relation" title="Equivalence relation">equivalent</a> if they differ by an overall nonzero factor <i>λ</i>:

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<annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]\sim [\lambda x_{1}:\lambda x_{2}].}</annotation>

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</math></span><img src="./0e149dfacd2fd2bfcc157bc3da4e654227498373.svg" 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}].}" loading="lazy"></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></div>

<p>The projective line may be identified with the line <i>K</i> extended by a <a href="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

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<annotation encoding="application/x-tex">{\displaystyle \left\{[x:1]\in \mathbf {P} ^{1}(K)\mid x\in K\right\}.}</annotation>

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</math></span><img src="./086c5d6b3f2d24eccfa35f0847c01d70c75b8c57.svg" 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\}.}" loading="lazy"></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>:

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<annotation encoding="application/x-tex">{\displaystyle \infty =[1:0].}</annotation>

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</math></span><img src="./54a6746d720a3849c5aeeeb5b0cb7fb0f12eac33.svg" 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].}" loading="lazy"></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

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{0}}=\infty ,\qquad {\frac {1}{\infty }}=0,}</annotation>

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</math></span><img src="./92fb1813885e723a28a5dd43cb095740370857fd.svg" 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,}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle x\cdot \infty =\infty \quad {\text{if}}\quad x\not =0}</annotation>

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</math></span><img src="./a3e7d5ea1269dff1e9dc0e8b219d7eb8c18719af.svg" 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}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle x+\infty =\infty \quad {\text{if}}\quad x\not =\infty }</annotation>

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</math></span><img src="./1a4dd2d88379caf16a1f8229dfe60c1690283bee.svg" 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 }" loading="lazy"></span></dd></dl>

<p>Translating this arithmetic in terms of homogeneous coordinates gives, when <span class="nowrap">[0 : 0]</span> does not occur:

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<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>

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<annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]\cdot [y_{1}:y_{2}]=[x_{1}y_{1}:x_{2}y_{2}],}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [x_{1}:x_{2}]^{-1}=[x_{2}:x_{1}].}</annotation>

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</math></span><img src="./c8afadaa9e638cbce6c51e27ae575b77755bc3cc.svg" 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}].}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Real_projective_line">Real projective line</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Real_projective_line" title="Real projective line">Real projective line</a></div>

<p>The projective line over the <a href="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="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="Unit_circle" title="Unit circle">unit circle</a> and then <a href="Quotient_space_(topology)" title="Quotient space (topology)">identifying</a> <a href="Diametrically_opposite" class="mw-redirect" title="Diametrically opposite">diametrically opposite</a> points. In terms of <a href="Group_theory" title="Group theory">group theory</a> we can take the quotient by the <a href="Subgroup" title="Subgroup">subgroup</a> <span class="nowrap">{1, −1}</span> under multiplication.

</p><p>Compare the <a href="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></div>

<p>Adding a point at infinity to the <a href="Complex_plane" title="Complex plane">complex plane</a> results in a space that is topologically a <a href="Sphere" title="Sphere">sphere</a>. Hence the complex projective line is also known as the <b><a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a></b> (or sometimes the <i>Gauss sphere</i>). It is in constant use in <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and <a href="Complex_manifold" title="Complex manifold">complex manifold</a> theory, as the simplest example of a <a href="Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">compact Riemann surface</a>.

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<div class="mw-heading mw-heading3"><h3 id="For_a_finite_field">For a finite field</h3></div>

<p>The projective line over a <a href="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>

<p>and the remaining <a href="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></div>

<p>Quite generally, the group of <a href="Homography" title="Homography">homographies</a> with <a href="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="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">group action</a> is <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">transitive</a>, so that <b>P</b><sup>1</sup>(<i>K</i>) is a <a href="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="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a>

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<annotation encoding="application/x-tex">{\displaystyle [X:Y]\sim [\lambda X:\lambda Y]}</annotation>

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</math></span><img src="./6578d7b00b891341414ba7439d96deba71b24318.svg" 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]}" loading="lazy"></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="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="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">sharply 3-transitive</a>. The computational aspect of this is the <a href="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></div>

<p>The projective line is a fundamental example of an <a href="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="Algebraic_curve#Singularities" title="Algebraic curve">non-singular</a> curve of <a href="Genus_(mathematics)" title="Genus (mathematics)">genus</a> 0. If <i>K</i> is <a href="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="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="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="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="Rational_function" title="Rational function">rational functions</a> over <i>K</i>, in a single indeterminate <i>T</i>. The <a href="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="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="Birational_geometry" title="Birational geometry">birational geometry</a>, this means that there will be a <a href="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="Meromorphic_function" title="Meromorphic function">meromorphic functions</a> of <a href="Complex_analysis" title="Complex analysis">complex analysis</a>, and indeed in the case of <a href="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="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="Ramification_(mathematics)" title="Ramification (mathematics)">ramification</a>.

</p><p>Many curves, for example <a href="Hyperelliptic_curve" title="Hyperelliptic curve">hyperelliptic curves</a>, may be presented abstractly, as <a href="Ramified_cover" class="mw-redirect" title="Ramified cover">ramified covers</a> of the projective line. According to the <a href="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="Birational_equivalence" class="mw-redirect" title="Birational equivalence">birationally equivalent</a> to a projective line (see <a href="Rational_variety" title="Rational variety">rational variety</a>); its <a href="Genus_(mathematics)" title="Genus (mathematics)">genus</a> is 0. A <a href="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>

<p>See <i><a href="Twisted_cubic" title="Twisted cubic">Twisted cubic</a></i> for the first interesting case.

</p>

<ul><li><a href="Algebraic_curve" title="Algebraic curve">Algebraic curve</a></li>

<li><a href="Cross-ratio" title="Cross-ratio">Cross-ratio</a></li>

<li><a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a></li>

<li><a href="Projective_line_over_a_ring" title="Projective line over a ring">Projective line over a ring</a></li>

<li><a href="Projectively_extended_real_line" title="Projectively extended real line">Projectively extended real line</a></li>

<li><a href="Projective_range" title="Projective range">Projective range</a></li>

<li><a href="Wheel_theory" title="Wheel theory">Wheel theory</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<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>

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</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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780387977164</bdi></cite>.</span>

</li>

</ol></div></div>

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</style><div id="Topics_in_algebraic_curves148" style="font-size:114%;margin:0 4em">Topics in <a href="Algebraic_curve" title="Algebraic curve">algebraic curves</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Five_points_determine_a_conic" title="Five points determine a conic">Five points determine a conic</a></li>

<li><a href="Rational_normal_curve" title="Rational normal curve">Rational normal curve</a></li>

<li><a href="Riemann_sphere" title="Riemann sphere">Riemann sphere</a></li>

<li><a href="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="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="Elliptic_function" title="Elliptic function">Elliptic function</a></li>

<li><a href="Elliptic_integral" title="Elliptic integral">Elliptic integral</a></li>

<li><a href="Fundamental_pair_of_periods" title="Fundamental pair of periods">Fundamental pair of periods</a></li>

<li><a href="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="Counting_points_on_elliptic_curves" title="Counting points on elliptic curves">Counting points on elliptic curves</a></li>

<li><a href="Division_polynomials" title="Division polynomials">Division polynomials</a></li>

<li><a href="Hasse's_theorem_on_elliptic_curves" title="Hasse's theorem on elliptic curves">Hasse's theorem on elliptic curves</a></li>

<li><a href="Mazur's_torsion_theorem" class="mw-redirect" title="Mazur's torsion theorem">Mazur's torsion theorem</a></li>

<li><a href="Modular_elliptic_curve" title="Modular elliptic curve">Modular elliptic curve</a></li>

<li><a href="Modularity_theorem" title="Modularity theorem">Modularity theorem</a></li>

<li><a href="Mordell%E2%80%93Weil_theorem" title="Mordell–Weil theorem">Mordell–Weil theorem</a></li>

<li><a href="Nagell%E2%80%93Lutz_theorem" title="Nagell–Lutz theorem">Nagell–Lutz theorem</a></li>

<li><a href="Supersingular_elliptic_curve" title="Supersingular elliptic curve">Supersingular elliptic curve</a></li>

<li><a href="Schoof's_algorithm" title="Schoof's algorithm">Schoof's algorithm</a></li>

<li><a href="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="Elliptic_curve_cryptography" class="mw-redirect" title="Elliptic curve cryptography">Elliptic curve cryptography</a></li>

<li><a href="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="De_Franchis_theorem" title="De Franchis theorem">De Franchis theorem</a></li>

<li><a href="Faltings's_theorem" title="Faltings's theorem">Faltings's theorem</a></li>

<li><a href="Hurwitz's_automorphisms_theorem" title="Hurwitz's automorphisms theorem">Hurwitz's automorphisms theorem</a></li>

<li><a href="Hurwitz_surface" title="Hurwitz surface">Hurwitz surface</a></li>

<li><a href="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="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="AF%2BBG_theorem" title="AF+BG theorem">AF+BG theorem</a></li>

<li><a href="B%C3%A9zout's_theorem" title="Bézout's theorem">Bézout's theorem</a></li>

<li><a href="Bitangent" title="Bitangent">Bitangent</a></li>

<li><a href="Cayley%E2%80%93Bacharach_theorem" title="Cayley–Bacharach theorem">Cayley–Bacharach theorem</a></li>

<li><a href="Conic_section" title="Conic section">Conic section</a></li>

<li><a href="Cramer's_paradox" title="Cramer's paradox">Cramer's paradox</a></li>

<li><a href="Cubic_plane_curve" title="Cubic plane curve">Cubic plane curve</a></li>

<li><a href="Fermat_curve" title="Fermat curve">Fermat curve</a></li>

<li><a href="Genus%E2%80%93degree_formula" title="Genus–degree formula">Genus–degree formula</a></li>

<li><a href="Hilbert's_sixteenth_problem" title="Hilbert's sixteenth problem">Hilbert's sixteenth problem</a></li>

<li><a href="Nagata's_conjecture_on_curves" title="Nagata's conjecture on curves">Nagata's conjecture on curves</a></li>

<li><a href="Pl%C3%BCcker_formula" title="Plücker formula">Plücker formula</a></li>

<li><a href="Quartic_plane_curve" title="Quartic plane curve">Quartic plane curve</a></li>

<li><a href="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="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="Belyi's_theorem" title="Belyi's theorem">Belyi's theorem</a></li>

<li><a href="Bring's_curve" title="Bring's curve">Bring's curve</a></li>

<li><a href="Bolza_surface" title="Bolza surface">Bolza surface</a></li>

<li><a href="Compact_Riemann_surface" class="mw-redirect" title="Compact Riemann surface">Compact Riemann surface</a></li>

<li><a href="Dessin_d'enfant" title="Dessin d'enfant">Dessin d'enfant</a></li>

<li><a href="Differential_of_the_first_kind" title="Differential of the first kind">Differential of the first kind</a></li>

<li><a href="Klein_quartic" title="Klein quartic">Klein quartic</a></li>

<li><a href="Riemann's_existence_theorem" title="Riemann's existence theorem">Riemann's existence theorem</a></li>

<li><a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li>

<li><a href="Teichm%C3%BCller_space" title="Teichmüller space">Teichmüller space</a></li>

<li><a href="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="Dual_curve" title="Dual curve">Dual curve</a></li>

<li><a href="Polar_curve" title="Polar curve">Polar curve</a></li>

<li><a href="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="Abel%E2%80%93Jacobi_map" title="Abel–Jacobi map">Abel–Jacobi map</a></li>

<li><a href="Brill%E2%80%93Noether_theory" title="Brill–Noether theory">Brill–Noether theory</a></li>

<li><a href="Clifford's_theorem_on_special_divisors" title="Clifford's theorem on special divisors">Clifford's theorem on special divisors</a></li>

<li><a href="Gonality_of_an_algebraic_curve" title="Gonality of an algebraic curve">Gonality of an algebraic curve</a></li>

<li><a href="Jacobian_variety" title="Jacobian variety">Jacobian variety</a></li>

<li><a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a></li>

<li><a href="Weierstrass_point" title="Weierstrass point">Weierstrass point</a></li>

<li><a href="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="ELSV_formula" title="ELSV formula">ELSV formula</a></li>

<li><a href="Gromov%E2%80%93Witten_invariant" title="Gromov–Witten invariant">Gromov–Witten invariant</a></li>

<li><a href="Hodge_bundle" title="Hodge bundle">Hodge bundle</a></li>

<li><a href="Moduli_of_algebraic_curves" title="Moduli of algebraic curves">Moduli of algebraic curves</a></li>

<li><a href="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="Hasse%E2%80%93Witt_matrix" title="Hasse–Witt matrix">Hasse–Witt matrix</a></li>

<li><a href="Riemann%E2%80%93Hurwitz_formula" title="Riemann–Hurwitz formula">Riemann–Hurwitz formula</a></li>

<li><a href="Prym_variety" title="Prym variety">Prym variety</a></li>

<li><a href="Weber's_theorem_(Algebraic_curves)" class="mw-redirect" 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="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="Ak_singularity" title="Ak singularity"><i>A<sub>k</sub></i> singularity</a></li>

<li><a href="Acnode" title="Acnode">Acnode</a></li>

<li><a href="Crunode" title="Crunode">Crunode</a></li>

<li><a href="Delta_invariant" class="mw-redirect" title="Delta invariant">Delta invariant</a></li>

<li><a href="Tacnode" title="Tacnode">Tacnode</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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="Birkhoff%E2%80%93Grothendieck_theorem" title="Birkhoff–Grothendieck theorem">Birkhoff–Grothendieck theorem</a></li>

<li><a href="Stable_vector_bundle" title="Stable vector bundle">Stable vector bundle</a></li>

<li><a href="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><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-17" href="https://en.wikipedia.org/wiki/?title=Projective_line&oldid=1301044799">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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