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Ample line bundle

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</style><div role="note" class="hatnote navigation-not-searchable">"Ample" redirects here. For the company, see <a href="Ample_(company)" title="Ample (company)">Ample (company)</a>. For a general definition of the word, see the Wiktionary entry <a href="https://en.wiktionary.org/wiki/ample" class="extiw external" title="wikt:ample">ample</a>.</div>

In mathematics, a distinctive feature of <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> is that some <a href="Line_bundle" title="Line bundle">line bundles</a> on a <a href="Projective_variety" title="Projective variety">projective variety</a> can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">sections</a>. Understanding the ample line bundles on a given variety <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> amounts to understanding the different ways of mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> into <a href="Projective_space" title="Projective space">projective spaces</a>. In view of the correspondence between line bundles and <a href="Divisor_(algebraic_geometry)" title="Divisor (algebraic geometry)">divisors</a> (built from <a href="Codimension" title="Codimension">codimension</a>-1 subvarieties), there is an equivalent notion of an ample divisor.

In more detail, a line bundle is called basepoint-free if it has enough sections to give a <a href="Morphism_of_algebraic_varieties" title="Morphism of algebraic varieties">morphism</a> to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a <a href="Line_bundle" title="Line bundle">line bundle</a> on a complete variety <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is very ample if it has enough sections to give a <a href="Closed_immersion" title="Closed immersion">closed immersion</a> (or "embedding") of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> into a projective space. A line bundle is ample if some positive power is very ample.

An ample line bundle on a projective variety <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has positive degree on every <a href="Algebraic_curve" title="Algebraic curve">curve</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness.

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

<div class="mw-heading mw-heading3"><h3 id="Pullback_of_a_line_bundle_and_hyperplane_divisors">Pullback of a line bundle and hyperplane divisors</h3></div>

Given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation>

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</math><img src="./07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"> of <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a>, a <a href="Vector_bundle" title="Vector bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon E\to Y}">

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<annotation encoding="application/x-tex">{\displaystyle p\colon E\to Y}</annotation>

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</math><img src="./818c561a30ff5f3969403fe6c9d4b50420e625b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.456ex; height:2.509ex;" alt="{\displaystyle p\colon E\to Y}" loading="lazy"> (or more generally a <a href="Coherent_sheaf" title="Coherent sheaf">coherent sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">) has a <a href="Pullback_bundle" title="Pullback bundle">pullback</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}E=\{(x,e)\in X\times E,\;f(x)=p(e)\}}">

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<annotation encoding="application/x-tex">{\displaystyle f^{*}E=\{(x,e)\in X\times E,\;f(x)=p(e)\}}</annotation>

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</math><img src="./501d99b6fb6f0977e1a18eff7a0b4153be79ab46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.524ex; height:2.843ex;" alt="{\displaystyle f^{*}E=\{(x,e)\in X\times E,\;f(x)=p(e)\}}" loading="lazy"> where the projection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p'\colon f^{*}E\to X}">

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<annotation encoding="application/x-tex">{\displaystyle p'\colon f^{*}E\to X}</annotation>

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</math><img src="./ba48674c1b5f382ee89edebcc8ced5e733f87bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:12.722ex; height:2.843ex;" alt="{\displaystyle p'\colon f^{*}E\to X}" loading="lazy"> is the projection on the first coordinate (see <a href="Sheaf_of_modules#Operations" title="Sheaf of modules">Sheaf of modules#Operations</a>). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}E}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f^{*}E}</annotation>

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</math><img src="./6900d52f70c54b42932a5c6acd2c65833400a9eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.15ex; height:2.676ex;" alt="{\displaystyle f^{*}E}" loading="lazy"> at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy"> is the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

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</math><img src="./4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)\in Y}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)\in Y}</annotation>

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</math><img src="./959a48d7bbac6f7fc2304c25f4997e6224d582cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.032ex; height:2.843ex;" alt="{\displaystyle f(x)\in Y}" loading="lazy">.)

The notions described in this article are related to this construction in the case of a morphism to projective space

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to \mathbb {P} ^{n},}</annotation>

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</math><img src="./204138b1b425d58d6ad4ea405d61761ca183c58c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.192ex; height:2.676ex;" alt="{\displaystyle f\colon X\to \mathbb {P} ^{n},}" loading="lazy"></dd></dl>

with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\mathcal {O}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle E={\mathcal {O}}(1)}</annotation>

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</math><img src="./c19810b795e5a2c0c7fd9e0f45c2c9324f396b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.696ex; height:2.843ex;" alt="{\displaystyle E={\mathcal {O}}(1)}" loading="lazy"> the <a href="Coherent_sheaf#Examples_of_vector_bundles" title="Coherent sheaf">line bundle on projective space</a> whose global sections are the <a href="Homogeneous_polynomial" title="Homogeneous polynomial">homogeneous polynomials</a> of degree 1 (that is, linear functions) in variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0},\ldots ,x_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{0},\ldots ,x_{n}}</annotation>

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</math><img src="./67b6164003ad568582d3e60e82053f49ba6577c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle x_{0},\ldots ,x_{n}}" loading="lazy">. The line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

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</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> can also be described as the line bundle associated to a <a href="Hyperplane" title="Hyperplane">hyperplane</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

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</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy"> (because the zero set of a section of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

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</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> is a hyperplane). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a closed immersion, for example, it follows that the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}O(1)}">

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<annotation encoding="application/x-tex">{\displaystyle f^{*}O(1)}</annotation>

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</math><img src="./27b2120521efc68b9d2030185dc6aff75c5ee95d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.12ex; height:2.843ex;" alt="{\displaystyle f^{*}O(1)}" loading="lazy"> is the line bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> associated to a hyperplane section (the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with a hyperplane in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

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</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy">).

<div class="mw-heading mw-heading3"><h3 id="Basepoint-free_line_bundles">Basepoint-free line bundles</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> be a scheme over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> (for example, an algebraic variety) with a line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. (A line bundle may also be called an <a href="Invertible_sheaf" title="Invertible sheaf">invertible sheaf</a>.) Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},...,a_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},...,a_{n}}</annotation>

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</math><img src="./a88f789a7215734e9d8e3fa49c7fd762043b9d30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.902ex; height:2.009ex;" alt="{\displaystyle a_{0},...,a_{n}}" loading="lazy"> be elements of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">-<a href="Vector_space" title="Vector space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(X,L)}">

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<annotation encoding="application/x-tex">{\displaystyle H^{0}(X,L)}</annotation>

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</math><img src="./eacdd324708dfcfa9852bdf5628af69596935026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.564ex; height:3.176ex;" alt="{\displaystyle H^{0}(X,L)}" loading="lazy"> of <a href="Global_section" class="mw-redirect" title="Global section">global sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. The zero set of each section is a closed subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">; let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

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</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> be the open subset of points at which at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},\ldots ,a_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},\ldots ,a_{n}}</annotation>

</semantics>

</math><img src="./403d004cd76aacd38b6432fdc449ddbc226075f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.911ex; height:2.009ex;" alt="{\displaystyle a_{0},\ldots ,a_{n}}" loading="lazy"> is not zero. Then these sections define a morphism

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon U\to \mathbb {P} _{k}^{n},\ x\mapsto [a_{0}(x),\ldots ,a_{n}(x)].}</annotation>

</semantics>

</math><img src="./bbde057484014c09ca864cfc0ba7cdf12a404887.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.035ex; height:2.843ex;" alt="{\displaystyle f\colon U\to \mathbb {P} _{k}^{n},\ x\mapsto [a_{0}(x),\ldots ,a_{n}(x)].}" loading="lazy"></dd></dl>

In more detail: for each point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, the fiber of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> is a 1-dimensional vector space over the <a href="Residue_field" title="Residue field">residue field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k(x)}</annotation>

</semantics>

</math><img src="./6e58c3a9efef5b14a9ba8f2bc27baf8bac2c3f25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.35ex; height:2.843ex;" alt="{\displaystyle k(x)}" loading="lazy">. Choosing a basis for this fiber makes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}(x),\ldots ,a_{n}(x)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{0}(x),\ldots ,a_{n}(x)}</annotation>

</semantics>

</math><img src="./f8fb0ec05e5f88c37390777dbcba70755e776745.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.189ex; height:2.843ex;" alt="{\displaystyle a_{0}(x),\ldots ,a_{n}(x)}" loading="lazy"> into a sequence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>

</semantics>

</math><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"> numbers, not all zero, and hence a point in projective space. Changing the choice of basis scales all the numbers by the same nonzero constant, and so the point in projective space is independent of the choice.

Moreover, this morphism has the property that the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> is isomorphic to the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}{\mathcal {O}}(1)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}{\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./7c280277187561f6861d00c873ef4b216f99392a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.196ex; height:2.843ex;" alt="{\displaystyle f^{*}{\mathcal {O}}(1)}" loading="lazy">.<a href="#cite_note-1">[1]</a>

The base locus of a line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> on a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the intersection of the zero sets of all global sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. A line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is called basepoint-free if its base locus is empty. That is, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> there is a global section of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> which is nonzero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is <a href="Proper_morphism" title="Proper morphism">proper</a> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, then the vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(X,L)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{0}(X,L)}</annotation>

</semantics>

</math><img src="./eacdd324708dfcfa9852bdf5628af69596935026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.564ex; height:3.176ex;" alt="{\displaystyle H^{0}(X,L)}" loading="lazy"> of global sections has finite dimension; the dimension is called <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{0}(X,L)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h^{0}(X,L)}</annotation>

</semantics>

</math><img src="./80c266fe54371268d28917dc15f6ea33519be3a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.799ex; height:3.176ex;" alt="{\displaystyle h^{0}(X,L)}" loading="lazy">.<a href="#cite_note-2">[2]</a> So a basepoint-free line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> determines a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X\to \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./9e779738a67d892bc6ac3cc30b581d0f45902a5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.545ex; height:2.676ex;" alt="{\displaystyle f\colon X\to \mathbb {P} ^{n}}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=h^{0}(X,L)-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=h^{0}(X,L)-1}</annotation>

</semantics>

</math><img src="./c27f210603444d9525b2290b0448748f91d53d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.295ex; height:3.176ex;" alt="{\displaystyle n=h^{0}(X,L)-1}" loading="lazy">, given by choosing a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(X,L)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{0}(X,L)}</annotation>

</semantics>

</math><img src="./eacdd324708dfcfa9852bdf5628af69596935026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.564ex; height:3.176ex;" alt="{\displaystyle H^{0}(X,L)}" loading="lazy">. Without making a choice, this can be described as the morphism

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X\to \mathbb {P} (H^{0}(X,L))}</annotation>

</semantics>

</math><img src="./d3a8ccb8767acacdc9025a04d54939a8cacc2ace.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.7ex; height:3.176ex;" alt="{\displaystyle f\colon X\to \mathbb {P} (H^{0}(X,L))}" loading="lazy"></dd></dl>

from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to the space of hyperplanes in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(X,L)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{0}(X,L)}</annotation>

</semantics>

</math><img src="./eacdd324708dfcfa9852bdf5628af69596935026.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.564ex; height:3.176ex;" alt="{\displaystyle H^{0}(X,L)}" loading="lazy">, canonically associated to the basepoint-free line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">. This morphism has the property that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is the pullback <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}{\mathcal {O}}(1)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}{\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./7c280277187561f6861d00c873ef4b216f99392a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.196ex; height:2.843ex;" alt="{\displaystyle f^{*}{\mathcal {O}}(1)}" loading="lazy">.

Conversely, for any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> from a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> to a projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">, the pullback line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}{\mathcal {O}}(1)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}{\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./7c280277187561f6861d00c873ef4b216f99392a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.196ex; height:2.843ex;" alt="{\displaystyle f^{*}{\mathcal {O}}(1)}" loading="lazy"> is basepoint-free. Indeed, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> is basepoint-free on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy">, because for every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy"> there is a hyperplane not containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">. Therefore, for every point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, there is a section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy"> that is not zero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">, and the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s}</annotation>

</semantics>

</math><img src="./01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> is a global section of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}{\mathcal {O}}(1)}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}{\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./7c280277187561f6861d00c873ef4b216f99392a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.196ex; height:2.843ex;" alt="{\displaystyle f^{*}{\mathcal {O}}(1)}" loading="lazy"> that is not zero at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">. In short, basepoint-free line bundles are exactly those that can be expressed as the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> by some morphism to a projective space.

<div class="mw-heading mw-heading3"><h3 id="Nef,_globally_generated,_semi-ample">Nef, globally generated, semi-ample</h3></div>

The <a href="Divisor_(algebraic_geometry)#Divisors_on_a_Riemann_surface" title="Divisor (algebraic geometry)">degree</a> of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(C,L)\neq 0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{0}(C,L)\neq 0}</annotation>

</semantics>

</math><img src="./b471b876cb67b0041b3c7d0abebba7207ccc2f23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.611ex; height:3.176ex;" alt="{\displaystyle H^{0}(C,L)\neq 0}" loading="lazy"> has nonnegative degree (because sections of L over C, as opposed to rational sections, have no poles).<a href="#cite_note-3">[3]</a> In particular, every basepoint-free line bundle on a curve has nonnegative degree. As a result, a basepoint-free line bundle L on any proper scheme X over a field is <a href="Nef_line_bundle" title="Nef line bundle">nef</a>, meaning that L has nonnegative degree on every (irreducible) curve in X.<a href="#cite_note-4">[4]</a>

More generally, a sheaf F of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{X}}</annotation>

</semantics>

</math><img src="./a816172129d917db5f1bf8e9c091513579890806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.406ex; height:2.509ex;" alt="{\displaystyle O_{X}}" loading="lazy">-modules on a scheme X is said to be globally generated if there is a set I of global sections <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}\in H^{0}(X,F)}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{i}\in H^{0}(X,F)}</annotation>

</semantics>

</math><img src="./3d09c295849a4449e700ed53fb69ed9b345b3dc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.453ex; height:3.176ex;" alt="{\displaystyle s_{i}\in H^{0}(X,F)}" loading="lazy"> such that the corresponding morphism

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{i\in I}O_{X}\to F}</annotation>

</semantics>

</math><img src="./a5044879168069bd5965835bc47131d77346f4ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.658ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{i\in I}O_{X}\to F}" loading="lazy"></dd></dl>

of sheaves is surjective.<a href="#cite_note-FOOTNOTEtag_01AM-5">[5]</a> A line bundle is globally generated if and only if it is basepoint-free.

For example, every <a href="Quasi-coherent_sheaf" class="mw-redirect" title="Quasi-coherent sheaf">quasi-coherent sheaf</a> on an <a href="Affine_scheme" class="mw-redirect" title="Affine scheme">affine scheme</a> is globally generated.<a href="#cite_note-6">[6]</a> Analogously, in <a href="Complex_geometry" title="Complex geometry">complex geometry</a>, <a href="Cartan's_theorem_A" class="mw-redirect" title="Cartan's theorem A">Cartan's theorem A</a> says that every coherent sheaf on a <a href="Stein_manifold" title="Stein manifold">Stein manifold</a> is globally generated.

A line bundle L on a proper scheme over a field is semi-ample if there is a positive integer r such that the <a href="Tensor_product_of_line_bundles" class="mw-redirect" title="Tensor product of line bundles">tensor power</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\otimes r}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L^{\otimes r}}</annotation>

</semantics>

</math><img src="./a079b55b942aff3a14be3059eda3086a8a739ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.835ex; height:2.509ex;" alt="{\displaystyle L^{\otimes r}}" loading="lazy"> is basepoint-free. A semi-ample line bundle is nef (by the corresponding fact for basepoint-free line bundles).<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading3"><h3 id="Very_ample_line_bundles">Very ample line bundles</h3></div>

A line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> on a proper scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> over a field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is said to be very ample if it is basepoint-free and the associated morphism

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon X\to \mathbb {P} _{k}^{n}}</annotation>

</semantics>

</math><img src="./eb3d24088d2f79f6b1623976d1a2ed03d752428c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.545ex; height:2.843ex;" alt="{\displaystyle f\colon X\to \mathbb {P} _{k}^{n}}" loading="lazy"></dd></dl>

is an immersion. Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=h^{0}(X,L)-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=h^{0}(X,L)-1}</annotation>

</semantics>

</math><img src="./c27f210603444d9525b2290b0448748f91d53d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.295ex; height:3.176ex;" alt="{\displaystyle n=h^{0}(X,L)-1}" loading="lazy">. Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is very ample if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> can be embedded into a projective space of some dimension over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> in such a way that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> is the restriction of the line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

</semantics>

</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.<a href="#cite_note-8">[8]</a> The latter definition is used to define very ampleness for a line bundle on a proper scheme over any <a href="Commutative_ring" title="Commutative ring">commutative ring</a>.<a href="#cite_note-FOOTNOTEtag_02NP-9">[9]</a>

The name "very ample" was introduced by <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> in 1961.<a href="#cite_note-10">[10]</a> Various names had been used earlier in the context of <a href="Linear_systems_of_divisors" class="mw-redirect" title="Linear systems of divisors">linear systems of divisors</a>.

For a very ample line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> on a proper scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> over a field with associated morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, the degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> on a curve <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C}</annotation>

</semantics>

</math><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is the <a href="Degree_of_an_algebraic_variety" title="Degree of an algebraic variety">degree</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(C)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(C)}</annotation>

</semantics>

</math><img src="./6b2b925cba46183c6b54011f186728ce596e92ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.854ex; height:2.843ex;" alt="{\displaystyle f(C)}" loading="lazy"> as a curve in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy">. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> has positive degree on every curve in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> (because every subvariety of projective space has positive degree).<a href="#cite_note-11">[11]</a>

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

<div class="mw-heading mw-heading3"><h3 id="Ample_invertible_sheaves_on_quasi-compact_schemes">Ample invertible sheaves on quasi-compact schemes</h3></div>

Ample line bundles are used most often on proper schemes, but they can be defined in much wider generality.

Let X be a scheme, and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> be an invertible sheaf on X. For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}_{x}}</annotation>

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</math><img src="./44c94ffb93cc0ffd1686deeb415e446405e91e16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.955ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {m}}_{x}}" loading="lazy"> denote the ideal sheaf of the reduced subscheme supported only at x. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \Gamma (X,{\mathcal {L}})}">

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<annotation encoding="application/x-tex">{\displaystyle s\in \Gamma (X,{\mathcal {L}})}</annotation>

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</math><img src="./16fd747f5740db2d5cbfb9a15039beaf1dca53e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.811ex; height:2.843ex;" alt="{\displaystyle s\in \Gamma (X,{\mathcal {L}})}" loading="lazy">, define

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<annotation encoding="application/x-tex">{\displaystyle X_{s}=\{x\in X\colon s_{x}\not \in {\mathfrak {m}}_{x}{\mathcal {L}}_{x}\}.}</annotation>

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Equivalently, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa (x)}">

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<annotation encoding="application/x-tex">{\displaystyle \kappa (x)}</annotation>

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</math><img src="./236a014894dc9f0248a8b9f55f2a6e1162f9e834.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.478ex; height:2.843ex;" alt="{\displaystyle \kappa (x)}" loading="lazy"> denotes the residue field at x (considered as a skyscraper sheaf supported at x), then

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<annotation encoding="application/x-tex">{\displaystyle X_{s}=\{x\in X\colon {\bar {s}}_{x}\neq 0\in \kappa (x)\otimes {\mathcal {L}}_{x}\},}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {s}}_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle {\bar {s}}_{x}}</annotation>

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</math><img src="./2d27d8a536aa4232f797c2e81c3c959f41d9da2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.464ex; height:2.343ex;" alt="{\displaystyle {\bar {s}}_{x}}" loading="lazy"> is the image of s in the tensor product.

Fix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \Gamma (X,{\mathcal {L}})}">

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<annotation encoding="application/x-tex">{\displaystyle s\in \Gamma (X,{\mathcal {L}})}</annotation>

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</math><img src="./16fd747f5740db2d5cbfb9a15039beaf1dca53e7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.811ex; height:2.843ex;" alt="{\displaystyle s\in \Gamma (X,{\mathcal {L}})}" loading="lazy">. For every s, the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}|_{X_{s}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}|_{X_{s}}}</annotation>

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</math><img src="./a2a5815bf720a21d4421b9d97bf23d426fb3182b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.634ex; height:3.176ex;" alt="{\displaystyle {\mathcal {L}}|_{X_{s}}}" loading="lazy"> is a free <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

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</math><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy">-module trivialized by the restriction of s, meaning the multiplication-by-s morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X_{s}}\to {\mathcal {L}}|_{X_{s}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X_{s}}\to {\mathcal {L}}|_{X_{s}}}</annotation>

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</math><img src="./ea8a94e0503c5c5079def21cda550c0b588f0581.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.481ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}_{X_{s}}\to {\mathcal {L}}|_{X_{s}}}" loading="lazy"> is an isomorphism. The set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation>

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</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy"> is always open, and the inclusion morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}\to X}">

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<annotation encoding="application/x-tex">{\displaystyle X_{s}\to X}</annotation>

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</math><img src="./5c8f19337e91093b3c53ccbeb4051fc98111a9a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.522ex; height:2.509ex;" alt="{\displaystyle X_{s}\to X}" loading="lazy"> is an affine morphism. Despite this, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation>

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</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy"> need not be an affine scheme. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1\in \Gamma (X,{\mathcal {O}}_{X})}">

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<annotation encoding="application/x-tex">{\displaystyle s=1\in \Gamma (X,{\mathcal {O}}_{X})}</annotation>

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</math><img src="./620d740511696140aa413a0a44383ea93c427a94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.95ex; height:2.843ex;" alt="{\displaystyle s=1\in \Gamma (X,{\mathcal {O}}_{X})}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}=X}">

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<annotation encoding="application/x-tex">{\displaystyle X_{s}=X}</annotation>

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</math><img src="./67b49e591c8b5f725b23a60b87b7d8eb21b5a93a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.006ex; height:2.509ex;" alt="{\displaystyle X_{s}=X}" loading="lazy"> is open in itself and affine over itself but generally not affine.

Assume X is quasi-compact. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> is ample if, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X}</annotation>

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</math><img src="./3e580967f68f36743e894aa7944f032dda6ea01d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.15ex; height:2.176ex;" alt="{\displaystyle x\in X}" loading="lazy">, there exists an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 1}">

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<annotation encoding="application/x-tex">{\displaystyle n\geq 1}</annotation>

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</math><img src="./d8ce9ce38d06f6bf5a3fe063118c09c2b6202bfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 1}" loading="lazy"> and an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}">

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<annotation encoding="application/x-tex">{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}</annotation>

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</math><img src="./dcbda647dcb2beddc43c9f6156b869c975a99139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:3.009ex;" alt="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle x\in X_{s}}</annotation>

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</math><img src="./1a62b39af7c6a9042b8f5317f903ed4896f6cf6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.098ex; height:2.509ex;" alt="{\displaystyle x\in X_{s}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}">

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<annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation>

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</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy"> is an affine scheme.<a href="#cite_note-FOOTNOTEtag_01PS-12">[12]</a> For example, the trivial line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

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</math><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy"> is ample if and only if X is <a href="Quasi-affine_morphism" class="mw-redirect" title="Quasi-affine morphism">quasi-affine</a>.<a href="#cite_note-FOOTNOTEtag_01QE-13">[13]</a>

In general, it is not true that every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}">

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</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy"> is affine. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbf {P} ^{2}\setminus \{O\}}">

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<annotation encoding="application/x-tex">{\displaystyle X=\mathbf {P} ^{2}\setminus \{O\}}</annotation>

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</math><img src="./08836b9eaa7cbeb3d5f31ef2dbf5b0ba2fb8ec67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.252ex; height:3.176ex;" alt="{\displaystyle X=\mathbf {P} ^{2}\setminus \{O\}}" loading="lazy"> for some point O, and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> is the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}</annotation>

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</math><img src="./2ca571c617af98f809b5d4770aba65f66bf6261f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.177ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}" loading="lazy"> to X, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}</annotation>

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</math><img src="./2ca571c617af98f809b5d4770aba65f66bf6261f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.177ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}" loading="lazy"> have the same global sections, and the non-vanishing locus of a section of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> is affine if and only if the corresponding section of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}</annotation>

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</math><img src="./2ca571c617af98f809b5d4770aba65f66bf6261f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.177ex; height:3.009ex;" alt="{\displaystyle {\mathcal {O}}_{\mathbf {P} ^{2}}(1)}" loading="lazy"> contains O.

It is necessary to allow powers of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

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</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> in the definition. In fact, for every N, it is possible that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{s}}">

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</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy"> is non-affine for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}">

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<annotation encoding="application/x-tex">{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}</annotation>

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</math><img src="./dcbda647dcb2beddc43c9f6156b869c975a99139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:3.009ex;" alt="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\leq N}">

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</math><img src="./26619ac273b9b4a895a2f2bd8542bbf7fe40a52f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.557ex; height:2.343ex;" alt="{\displaystyle n\leq N}" loading="lazy">. Indeed, suppose Z is a finite set of points in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} ^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {P} ^{2}}</annotation>

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</math><img src="./287baf61ec5cf3c23c03fc497016127997d54612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.881ex; height:2.676ex;" alt="{\displaystyle \mathbf {P} ^{2}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbf {P} ^{2}\setminus Z}">

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<annotation encoding="application/x-tex">{\displaystyle X=\mathbf {P} ^{2}\setminus Z}</annotation>

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</math><img src="./8c667575988258a982995a571b40b31b275633a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.835ex; height:3.176ex;" alt="{\displaystyle X=\mathbf {P} ^{2}\setminus Z}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}={\mathcal {O}}_{\mathbf {P} ^{2}}(1)|_{X}}">

</mrow>

<msub>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}={\mathcal {O}}_{\mathbf {P} ^{2}}(1)|_{X}}</annotation>

</semantics>

</math><img src="./f483ed83d0599dc92647b7704094d3b03b94a19e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.159ex; height:3.009ex;" alt="{\displaystyle {\mathcal {L}}={\mathcal {O}}_{\mathbf {P} ^{2}}(1)|_{X}}" loading="lazy">. The vanishing loci of the sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{\otimes N}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{\otimes N}}</annotation>

</semantics>

</math><img src="./82d6c52be9ebc881f19c6d9a03d84068cd094b39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.574ex; height:2.676ex;" alt="{\displaystyle {\mathcal {L}}^{\otimes N}}" loading="lazy"> are plane curves of degree N. By taking Z to be a sufficiently large set of points in general position, we may ensure that no plane curve of degree N (and hence any lower degree) contains all the points of Z. In particular their non-vanishing loci are all non-affine.

Define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle S=\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {L}}^{\otimes n})}">

</mrow>

</munder>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle S=\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {L}}^{\otimes n})}</annotation>

</semantics>

</math><img src="./446c2dc60012581e270005d59356d3304e1077c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:21.262ex; height:3.343ex;" alt="{\displaystyle \textstyle S=\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {L}}^{\otimes n})}" loading="lazy">. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\colon X\to \operatorname {Spec} \mathbf {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\colon X\to \operatorname {Spec} \mathbf {Z} }</annotation>

</semantics>

</math><img src="./07058865a6abc08d87dae4b69a1648e79398ed33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:14.558ex; height:2.509ex;" alt="{\displaystyle p\colon X\to \operatorname {Spec} \mathbf {Z} }" loading="lazy"> denote the structural morphism. There is a natural isomorphism between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy">-algebra homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle p^{*}({\tilde {S}})\to \bigoplus _{n\geq 0}{\mathcal {L}}^{\otimes n}}">

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mrow>

</munder>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle p^{*}({\tilde {S}})\to \bigoplus _{n\geq 0}{\mathcal {L}}^{\otimes n}}</annotation>

</semantics>

</math><img src="./3f72d3cc7836406d4513ee26712f0a0ddd54728f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; margin-left: -0.089ex; width:19.698ex; height:3.509ex;" alt="{\displaystyle \textstyle p^{*}({\tilde {S}})\to \bigoplus _{n\geq 0}{\mathcal {L}}^{\otimes n}}" loading="lazy"> and endomorphisms of the graded ring S. The identity endomorphism of S corresponds to a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

</semantics>

</math><img src="./a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy">. Applying the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Proj} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Proj} }</annotation>

</semantics>

</math><img src="./6bb0e47c5502f6b647f71ab37b94436b7173ea71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.369ex; height:2.509ex;" alt="{\displaystyle \operatorname {Proj} }" loading="lazy"> functor produces a morphism from an open subscheme of X, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\varepsilon )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\varepsilon )}</annotation>

</semantics>

</math><img src="./4be8478cc3b02cde7d704deba1bb28662c74a018.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.72ex; height:2.843ex;" alt="{\displaystyle G(\varepsilon )}" loading="lazy">, to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Proj} S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Proj} S}</annotation>

</semantics>

</math><img src="./549b2dd0cfbc6e437bcb77919051e203293e08b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.255ex; height:2.509ex;" alt="{\displaystyle \operatorname {Proj} S}" loading="lazy">.

The basic characterization of ample invertible sheaves states that if X is a quasi-compact quasi-separated scheme and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

</semantics>

</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> is an invertible sheaf on X, then the following assertions are equivalent:<a href="#cite_note-14">[14]</a>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

</semantics>

</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> is ample.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation>

</semantics>

</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}</annotation>

</semantics>

</math><img src="./dcbda647dcb2beddc43c9f6156b869c975a99139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:3.009ex;" alt="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>

</semantics>

</math><img src="./ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}" loading="lazy">, form a basis for the topology of X.</li>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{s}}</annotation>

</semantics>

</math><img src="./cc1b40229a003b8b719a29ac70c15fe4f704a777.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.928ex; height:2.509ex;" alt="{\displaystyle X_{s}}" loading="lazy"> with the property of being affine, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}</annotation>

</semantics>

</math><img src="./dcbda647dcb2beddc43c9f6156b869c975a99139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.308ex; height:3.009ex;" alt="{\displaystyle s\in \Gamma (X,{\mathcal {L}}^{\otimes n})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq 0}</annotation>

</semantics>

</math><img src="./ce8a1b7b3bc3c790054d93629fc3b08cd1da1fd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 0}" loading="lazy">, form a basis for the topology of X.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\varepsilon )=X}</annotation>

</semantics>

</math><img src="./8bb91dd7a67f22d6d7555a84ae7f1e14216391cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.798ex; height:2.843ex;" alt="{\displaystyle G(\varepsilon )=X}" loading="lazy"> and the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\varepsilon )\to \operatorname {Proj} S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\varepsilon )\to \operatorname {Proj} S}</annotation>

</semantics>

</math><img src="./20cdd07aec018f59f4dec87f87000caf4ec85495.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.589ex; height:2.843ex;" alt="{\displaystyle G(\varepsilon )\to \operatorname {Proj} S}" loading="lazy"> is a dominant open immersion.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\varepsilon )=X}</annotation>

</semantics>

</math><img src="./8bb91dd7a67f22d6d7555a84ae7f1e14216391cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.798ex; height:2.843ex;" alt="{\displaystyle G(\varepsilon )=X}" loading="lazy"> and the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\varepsilon )\to \operatorname {Proj} S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G(\varepsilon )\to \operatorname {Proj} S}</annotation>

</semantics>

</math><img src="./20cdd07aec018f59f4dec87f87000caf4ec85495.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.589ex; height:2.843ex;" alt="{\displaystyle G(\varepsilon )\to \operatorname {Proj} S}" loading="lazy"> is a homeomorphism of the underlying topological space of X with its image.</li>

<li>For every quasi-coherent sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> on X, the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {F}}}">

</mrow>

</munder>

</mrow>

<msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./ef9a95b60d50d1e36acb8626daf83fe32af0c612.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.031ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {F}}}" loading="lazy"> is surjective.</li>

<li>For every quasi-coherent sheaf of ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}}</annotation>

</semantics>

</math><img src="./f8fb0b896b1b2a45546779ecafc567f4f1688714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.95ex; height:2.343ex;" alt="{\displaystyle {\mathcal {J}}}" loading="lazy"> on X, the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}}">

</mrow>

</munder>

</mrow>

<msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}}</annotation>

</semantics>

</math><img src="./8e3b1628dcd5ab9aa314b5d38d144d675a0f54ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.077ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}}" loading="lazy"> is surjective.</li>

<li>For every quasi-coherent sheaf of ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}}</annotation>

</semantics>

</math><img src="./f8fb0b896b1b2a45546779ecafc567f4f1688714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.95ex; height:2.343ex;" alt="{\displaystyle {\mathcal {J}}}" loading="lazy"> on X, the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}}">

</mrow>

</munder>

</mrow>

<msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}}</annotation>

</semantics>

</math><img src="./8e3b1628dcd5ab9aa314b5d38d144d675a0f54ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.077ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}}" loading="lazy"> is surjective.</li>

<li>For every quasi-coherent sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> of finite type on X, there exists an integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{0}}</annotation>

</semantics>

</math><img src="./63584d203ecb012a7bcb90f422408bbfe4018956.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.449ex; height:2.009ex;" alt="{\displaystyle n_{0}}" loading="lazy"> such that for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq n_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n\geq n_{0}}</annotation>

</semantics>

</math><img src="./fff23d9c13da1b2a7ec6f7fab8fa413b129c6b6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.942ex; height:2.343ex;" alt="{\displaystyle n\geq n_{0}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\otimes {\mathcal {L}}^{\otimes n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\otimes {\mathcal {L}}^{\otimes n}}</annotation>

</semantics>

</math><img src="./62983079f9501e476e0674122bc6bb31ea6531f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.868ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}\otimes {\mathcal {L}}^{\otimes n}}" loading="lazy"> is generated by its global sections.</li>

<li>For every quasi-coherent sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> of finite type on X, there exists integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

</semantics>

</math><img src="./27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>

</semantics>

</math><img src="./27b3af208b148139eefc03f0f80fa94c38c5af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> is isomorphic to a quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}}</annotation>

</semantics>

</math><img src="./f69e69f9005ab794557a7ac6763033e776bdc7d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.981ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}}" loading="lazy">.</li>

<li>For every quasi-coherent sheaf of ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}}</annotation>

</semantics>

</math><img src="./f8fb0b896b1b2a45546779ecafc567f4f1688714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.95ex; height:2.343ex;" alt="{\displaystyle {\mathcal {J}}}" loading="lazy"> of finite type on X, there exists integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

</semantics>

</math><img src="./27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>

</semantics>

</math><img src="./27b3af208b148139eefc03f0f80fa94c38c5af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {J}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {J}}}</annotation>

</semantics>

</math><img src="./f8fb0b896b1b2a45546779ecafc567f4f1688714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.95ex; height:2.343ex;" alt="{\displaystyle {\mathcal {J}}}" loading="lazy"> is isomorphic to a quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}}</annotation>

</semantics>

</math><img src="./f69e69f9005ab794557a7ac6763033e776bdc7d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.981ex; height:3.343ex;" alt="{\displaystyle {\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}}" loading="lazy">.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="On_proper_schemes">On proper schemes</h3></div>

When X is separated and finite type over an affine scheme, an invertible sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}}</annotation>

</semantics>

</math><img src="./9027196ecb178d598958555ea01c43157d83597c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.604ex; height:2.176ex;" alt="{\displaystyle {\mathcal {L}}}" loading="lazy"> is ample if and only if there exists a positive integer r such that the tensor power <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {L}}^{\otimes r}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {L}}^{\otimes r}}</annotation>

</semantics>

</math><img src="./a7ced80c8d4195fb1f5438316b39d8787e8fc576.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.856ex; height:2.509ex;" alt="{\displaystyle {\mathcal {L}}^{\otimes r}}" loading="lazy"> is very ample.<a href="#cite_note-15">[15]</a><a href="#cite_note-FOOTNOTEtag_01VU-16">[16]</a> In particular, a proper scheme over R has an ample line bundle if and only if it is projective over R. Often, this characterization is taken as the definition of ampleness.

The rest of this article will concentrate on ampleness on proper schemes over a field, as this is the most important case. An ample line bundle on a proper scheme X over a field has positive degree on every curve in X, by the corresponding statement for very ample line bundles.

A <a href="Cartier_divisor" class="mw-redirect" title="Cartier divisor">Cartier divisor</a> D on a proper scheme X over a field k is said to be ample if the corresponding line bundle O(D) is ample. (For example, if X is smooth over k, then a Cartier divisor can be identified with a finite <a href="Linear_combination" title="Linear combination">linear combination</a> of closed codimension-1 subvarieties of X with integer coefficients.)

Weakening the notion of "very ample" to "ample" gives a flexible concept with a wide variety of different characterizations. A first point is that tensoring high powers of an ample line bundle with any coherent sheaf whatsoever gives a sheaf with many global sections. More precisely, a line bundle L on a proper scheme X over a field (or more generally over a <a href="Noetherian_ring" title="Noetherian ring">Noetherian ring</a>) is ample if and only if for every coherent sheaf F on X, there is an integer s such that the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\otimes L^{\otimes r}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\otimes L^{\otimes r}}</annotation>

</semantics>

</math><img src="./ab28b051b7a12c87cf1e2d2a71359c6c25b70585.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.416ex; height:2.676ex;" alt="{\displaystyle F\otimes L^{\otimes r}}" loading="lazy"> is globally generated for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\geq s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\geq s}</annotation>

</semantics>

</math><img src="./3e2e2dd3b20bdb59786819e76201acd83c49e968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.238ex; height:2.176ex;" alt="{\displaystyle r\geq s}" loading="lazy">. Here s may depend on F.<a href="#cite_note-17">[17]</a><a href="#cite_note-CSG-18">[18]</a>

Another characterization of ampleness, known as the <a href="Henri_Cartan" title="Henri Cartan">Cartan</a>–<a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre</a>–<a href="Grothendieck" class="mw-redirect" title="Grothendieck">Grothendieck</a> theorem, is in terms of <a href="Coherent_sheaf_cohomology" title="Coherent sheaf cohomology">coherent sheaf cohomology</a>. Namely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{i}(X,F\otimes L^{\otimes r})=0}</annotation>

</semantics>

</math><img src="./872dcc78b92fa46190260978dcf1ddfa60e3060a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.404ex; height:3.176ex;" alt="{\displaystyle H^{i}(X,F\otimes L^{\otimes r})=0}" loading="lazy"></dd></dl>

for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i>0}</annotation>

</semantics>

</math><img src="./1f49f2878fd68a89c3da37eb537198e887cf0293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i>0}" loading="lazy"> and all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\geq s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\geq s}</annotation>

</semantics>

</math><img src="./3e2e2dd3b20bdb59786819e76201acd83c49e968.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.238ex; height:2.176ex;" alt="{\displaystyle r\geq s}" loading="lazy">.<a href="#cite_note-19">[19]</a><a href="#cite_note-CSG-18">[18]</a> In particular, high powers of an ample line bundle kill cohomology in positive degrees. This implication is called the Serre vanishing theorem, proved by <a href="Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a> in his 1955 paper <a href="Faisceaux_alg%C3%A9briques_coh%C3%A9rents" class="mw-redirect" title="Faisceaux algébriques cohérents">Faisceaux algébriques cohérents</a>.

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

<ul><li>The trivial line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O_{X}}</annotation>

</semantics>

</math><img src="./a816172129d917db5f1bf8e9c091513579890806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.406ex; height:2.509ex;" alt="{\displaystyle O_{X}}" loading="lazy"> on a projective variety X of positive dimension is basepoint-free but not ample. More generally, for any morphism f from a projective variety X to some projective space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{n}}</annotation>

</semantics>

</math><img src="./7e712abd7ff167cfb12ffa7c8c9bf7b0029cf192.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.639ex; height:2.343ex;" alt="{\displaystyle \mathbb {P} ^{n}}" loading="lazy"> over a field, the pullback line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=f^{*}O(1)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=f^{*}O(1)}</annotation>

</semantics>

</math><img src="./d4c288a34606ca9222dd300ec16bd1a679435644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.801ex; height:2.843ex;" alt="{\displaystyle L=f^{*}O(1)}" loading="lazy"> is always basepoint-free, whereas L is ample if and only if the morphism f is <a href="Finite_morphism" title="Finite morphism">finite</a> (that is, all fibers of f have dimension 0 or are empty).<a href="#cite_note-finite-20">[20]</a></li>

<li>For an integer d, the space of sections of the line bundle O(d) over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}">

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}</annotation>

</semantics>

</math><img src="./cbbfe3cec9a0f3e68e2deacbaeb7cb042d6092ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.839ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} _{\mathbb {C} }^{1}}" loading="lazy"> is the <a href="Complex_number" title="Complex number">complex</a> vector space of homogeneous polynomials of degree d in variables x,y. In particular, this space is zero for d < 0. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\geq 0}</annotation>

</semantics>

</math><img src="./74c3a2cca0c610ded6e0f188788933fb11f07df4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 0}" loading="lazy">, the morphism to projective space given by O(d) is</li></ul>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{1}\to \mathbb {P} ^{d}}</annotation>

</semantics>

</math><img src="./1ae697e2bd278d51d37c3abf4ba2eb6cb4c58ccc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.601ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{1}\to \mathbb {P} ^{d}}" loading="lazy"></dd></dl></dd>

<dd>by

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [x,y]\mapsto [x^{d},x^{d-1}y,\ldots ,y^{d}].}</annotation>

</semantics>

</math><img src="./2e377c5f0a56e7e4c4a97cb970555e00ee812b29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.931ex; height:3.176ex;" alt="{\displaystyle [x,y]\mapsto [x^{d},x^{d-1}y,\ldots ,y^{d}].}" loading="lazy"></dd></dl></dd>

<dd>This is a closed immersion for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\geq 1}</annotation>

</semantics>

</math><img src="./33476a779aa3e8bc64413c16f777ef15e9d78df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 1}" loading="lazy">, with image a <a href="Rational_normal_curve" title="Rational normal curve">rational normal curve</a> of degree d in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{d}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{d}}</annotation>

</semantics>

</math><img src="./813b404adcc8c1a0adcc863400b996d88d2434c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.512ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{d}}" loading="lazy">. Therefore, O(d) is basepoint-free if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\geq 0}</annotation>

</semantics>

</math><img src="./74c3a2cca0c610ded6e0f188788933fb11f07df4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 0}" loading="lazy">, and very ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\geq 1}</annotation>

</semantics>

</math><img src="./33476a779aa3e8bc64413c16f777ef15e9d78df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 1}" loading="lazy">. It follows that O(d) is ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\geq 1}</annotation>

</semantics>

</math><img src="./33476a779aa3e8bc64413c16f777ef15e9d78df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 1}" loading="lazy">.</dd></dl>

<ul><li>For an example where "ample" and "very ample" are different, let X be a smooth projective curve of <a href="Genus_(mathematics)" title="Genus (mathematics)">genus</a> 1 (an <a href="Elliptic_curve" title="Elliptic curve">elliptic curve</a>) over C, and let p be a complex point of X. Let O(p) be the associated line bundle of degree 1 on X. Then the complex vector space of global sections of O(p) has dimension 1, spanned by a section that vanishes at p.<a href="#cite_note-21">[21]</a> So the base locus of O(p) is equal to p. On the other hand, O(2p) is basepoint-free, and O(dp) is very ample for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\geq 3}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d\geq 3}</annotation>

</semantics>

</math><img src="./7ebd85f457285fae5a6938211a05b373a4263d6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.477ex; height:2.343ex;" alt="{\displaystyle d\geq 3}" loading="lazy"> (giving an embedding of X as an elliptic curve of degree d in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{d-1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{d-1}}</annotation>

</semantics>

</math><img src="./043f6b4845976d00b8e283d5f70b89b236135322.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.613ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{d-1}}" loading="lazy">). Therefore, O(p) is ample but not very ample. Also, O(2p) is ample and basepoint-free but not very ample; the associated morphism to projective space is a <a href="Branched_covering" title="Branched covering">ramified</a> double cover <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\to \mathbb {P} ^{1}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\to \mathbb {P} ^{1}}</annotation>

</semantics>

</math><img src="./b3d39420f9822c297255f167265ddde74c6f9fd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.069ex; height:2.676ex;" alt="{\displaystyle X\to \mathbb {P} ^{1}}" loading="lazy">.</li>

<li>On curves of higher genus, there are ample line bundles L for which every global section is zero. (But high multiples of L have many sections, by definition.) For example, let X be a smooth plane quartic curve (of degree 4 in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{2}}</annotation>

</semantics>

</math><img src="./02e892a2d48ff09d2f95b0e8b8eb419d42e179c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{2}}" loading="lazy">) over C, and let p and q be distinct complex points of X. Then the line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=O(2p-q)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L=O(2p-q)}</annotation>

</semantics>

</math><img src="./e58cc108f364bda02f8b48169d6b1ab30b73aedf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.506ex; height:2.843ex;" alt="{\displaystyle L=O(2p-q)}" loading="lazy"> is ample but has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(X,L)=0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{0}(X,L)=0}</annotation>

</semantics>

</math><img src="./4390801011731329f4ba949d9c03a20c0fa5b66d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.825ex; height:3.176ex;" alt="{\displaystyle H^{0}(X,L)=0}" loading="lazy">.<a href="#cite_note-22">[22]</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Criteria_for_ampleness_of_line_bundles">Criteria for ampleness of line bundles</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Intersection_theory">Intersection theory</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Intersection_theory#Intersection_theory_in_algebraic_geometry" title="Intersection theory">intersection theory § Intersection theory in algebraic geometry</a></div>

To determine whether a given line bundle on a projective variety X is ample, the following numerical criteria (in terms of intersection numbers) are often the most useful. It is equivalent to ask when a Cartier divisor D on X is ample, meaning that the associated line bundle O(D) is ample. The intersection number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\cdot C}">

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<annotation encoding="application/x-tex">{\displaystyle D\cdot C}</annotation>

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</math><img src="./b2f5137f9de9421b32adc528cf1b001adabde910.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.37ex; height:2.176ex;" alt="{\displaystyle D\cdot C}" loading="lazy"> can be defined as the degree of the line bundle O(D) restricted to C. In the other direction, for a line bundle L on a projective variety, the <a href="Cartier_divisor" class="mw-redirect" title="Cartier divisor">first Chern class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(L)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(L)}</annotation>

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</math><img src="./9e87f59f473579aeb2a6dce99f8fbedb0cd81beb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.453ex; height:2.843ex;" alt="{\displaystyle c_{1}(L)}" loading="lazy"> means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section of L.

On a <a href="Smooth_scheme" title="Smooth scheme">smooth</a> projective curve X over an <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> k, a line bundle L is very ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{0}(X,L\otimes O(-x-y))=h^{0}(X,L)-2}">

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<annotation encoding="application/x-tex">{\displaystyle h^{0}(X,L\otimes O(-x-y))=h^{0}(X,L)-2}</annotation>

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</math><img src="./9da3565c439e087ede236707058782797977fcc9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.257ex; height:3.176ex;" alt="{\displaystyle h^{0}(X,L\otimes O(-x-y))=h^{0}(X,L)-2}" loading="lazy"> for all k-<a href="Rational_point" title="Rational point">rational points</a> x,y in X.<a href="#cite_note-23">[23]</a> Let g be the genus of X. By the <a href="Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a>, every line bundle of degree at least 2g + 1 satisfies this condition and hence is very ample. As a result, a line bundle on a curve is ample if and only if it has positive degree.<a href="#cite_note-24">[24]</a>

For example, the <a href="Canonical_bundle" title="Canonical bundle">canonical bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{X}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{X}}</annotation>

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</math><img src="./3485fd0e87deb55b2544efb4320a21496f22bcea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.605ex; height:2.509ex;" alt="{\displaystyle K_{X}}" loading="lazy"> of a curve X has degree 2g − 2, and so it is ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}">

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<annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation>

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</math><img src="./b84b2394f64ae9927e09ee51b848be213edecafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}" loading="lazy">. The curves with ample canonical bundle form an important class; for example, over the complex numbers, these are the curves with a metric of negative <a href="Sectional_curvature" title="Sectional curvature">curvature</a>. The canonical bundle is very ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\geq 2}">

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<annotation encoding="application/x-tex">{\displaystyle g\geq 2}</annotation>

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</math><img src="./b84b2394f64ae9927e09ee51b848be213edecafe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.377ex; height:2.509ex;" alt="{\displaystyle g\geq 2}" loading="lazy"> and the curve is not <a href="Hyperelliptic_curve" title="Hyperelliptic curve">hyperelliptic</a>.<a href="#cite_note-25">[25]</a>

The Nakai–Moishezon criterion (named for Yoshikazu Nakai (1963) and <a href="Boris_Moishezon" title="Boris Moishezon">Boris Moishezon</a> (1964)) states that a line bundle L on a proper scheme X over a field is ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{Y}c_{1}(L)^{{\text{dim}}(Y)}>0}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{Y}c_{1}(L)^{{\text{dim}}(Y)}>0}</annotation>

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</math><img src="./9f20dcc0c850ced8e14e909252fd1e1086706a14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.386ex; height:5.676ex;" alt="{\displaystyle \int _{Y}c_{1}(L)^{{\text{dim}}(Y)}>0}" loading="lazy"> for every (<a href="Irreducible_component" title="Irreducible component">irreducible</a>) closed subvariety Y of X (Y is not allowed to be a point).<a href="#cite_note-26">[26]</a> In terms of divisors, a Cartier divisor D is ample if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{{\text{dim}}(Y)}\cdot Y>0}">

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<annotation encoding="application/x-tex">{\displaystyle D^{{\text{dim}}(Y)}\cdot Y>0}</annotation>

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</math><img src="./c77f4bd3cfca72e2d3e3a7d93a823744354fcc9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.143ex; height:2.843ex;" alt="{\displaystyle D^{{\text{dim}}(Y)}\cdot Y>0}" loading="lazy"> for every (nonzero-dimensional) subvariety Y of X. For X a curve, this says that a divisor is ample if and only if it has positive degree. For X a surface, the criterion says that a divisor D is ample if and only if its <a href="Self-intersection_number" class="mw-redirect" title="Self-intersection number">self-intersection number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle D^{2}}</annotation>

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</math><img src="./7a1e7d554748b105413b259fdfb8c9587acb9f1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.979ex; height:2.676ex;" alt="{\displaystyle D^{2}}" loading="lazy"> is positive and every curve C on X has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\cdot C>0}">

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<annotation encoding="application/x-tex">{\displaystyle D\cdot C>0}</annotation>

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</math><img src="./7df1a7f1579a053bc0fc0ff0637411d040cd81d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.631ex; height:2.176ex;" alt="{\displaystyle D\cdot C>0}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Kleiman's_criterion">Kleiman's criterion</h3></div>

To state Kleiman's criterion (1966), let X be a projective scheme over a field. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{1}(X)}</annotation>

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</math><img src="./246f28dcc9d09fb9f8467319ad6e2e8555d59c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle N_{1}(X)}" loading="lazy"> be the <a href="Real_number" title="Real number">real</a> vector space of 1-cycles (real linear combinations of curves in X) modulo numerical equivalence, meaning that two 1-cycles A and B are equal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{1}(X)}</annotation>

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</math><img src="./246f28dcc9d09fb9f8467319ad6e2e8555d59c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle N_{1}(X)}" loading="lazy"> if and only if every line bundle has the same degree on A and on B. By the <a href="N%C3%A9ron%E2%80%93Severi_group" title="Néron–Severi group">Néron–Severi theorem</a>, the real vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{1}(X)}</annotation>

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</math><img src="./246f28dcc9d09fb9f8467319ad6e2e8555d59c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle N_{1}(X)}" loading="lazy"> has finite dimension. Kleiman's criterion states that a line bundle L on X is ample if and only if L has positive degree on every nonzero element C of the <a href="Closure_(topology)" title="Closure (topology)">closure</a> of the <a href="Cone_of_curves" title="Cone of curves">cone of curves</a> NE(X) in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N_{1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle N_{1}(X)}</annotation>

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</math><img src="./246f28dcc9d09fb9f8467319ad6e2e8555d59c08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.71ex; height:2.843ex;" alt="{\displaystyle N_{1}(X)}" loading="lazy">. (This is slightly stronger than saying that L has positive degree on every curve.) Equivalently, a line bundle is ample if and only if its class in the <a href="Dual_vector_space" class="mw-redirect" title="Dual vector space">dual vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle N^{1}(X)}</annotation>

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</math><img src="./302a6599a45c990eed23ece9e6fb6745a0907a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.967ex; height:3.176ex;" alt="{\displaystyle N^{1}(X)}" loading="lazy"> is in the interior of the <a href="Nef_line_bundle" title="Nef line bundle">nef cone</a>.<a href="#cite_note-27">[27]</a>

Kleiman's criterion fails in general for proper (rather than projective) schemes X over a field, although it holds if X is smooth or more generally Q-factorial.<a href="#cite_note-28">[28]</a>

A line bundle on a projective variety is called strictly nef if it has positive degree on every curve. <a href="#CITEREFNagata1959">Nagata (1959)</a> and <a href="David_Mumford" title="David Mumford">David Mumford</a> constructed line bundles on smooth projective surfaces that are strictly nef but not ample. This shows that the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(L)^{2}>0}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(L)^{2}>0}</annotation>

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</math><img src="./4d57452e4a517919bac76d76a7173bceda5e622a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.768ex; height:3.176ex;" alt="{\displaystyle c_{1}(L)^{2}>0}" loading="lazy"> cannot be omitted in the Nakai–Moishezon criterion, and it is necessary to use the closure of NE(X) rather than NE(X) in Kleiman's criterion.<a href="#cite_note-29">[29]</a> Every nef line bundle on a surface has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(L)^{2}\geq 0}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(L)^{2}\geq 0}</annotation>

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</math><img src="./fd60e35248a7e7a96021bea2d828a1abba66cf43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.768ex; height:3.176ex;" alt="{\displaystyle c_{1}(L)^{2}\geq 0}" loading="lazy">, and Nagata and Mumford's examples have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1}(L)^{2}=0}">

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<annotation encoding="application/x-tex">{\displaystyle c_{1}(L)^{2}=0}</annotation>

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</math><img src="./566d78f226e6d5f1950cefaaac744839a0fe0cb5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.768ex; height:3.176ex;" alt="{\displaystyle c_{1}(L)^{2}=0}" loading="lazy">.

<a href="C._S._Seshadri" title="C. S. Seshadri">C. S. Seshadri</a> showed that a line bundle L on a proper scheme over an algebraically closed field is ample if and only if there is a positive real number ε such that deg(L|C) ≥ εm(C) for all (irreducible) curves C in X, where m(C) is the maximum of the multiplicities at the points of C.<a href="#cite_note-30">[30]</a>

Several characterizations of ampleness hold more generally for line bundles on a proper <a href="Algebraic_space" title="Algebraic space">algebraic space</a> over a field k. In particular, the Nakai-Moishezon criterion is valid in that generality.<a href="#cite_note-31">[31]</a> The Cartan-Serre-Grothendieck criterion holds even more generally, for a proper algebraic space over a Noetherian ring R.<a href="#cite_note-FOOTNOTEtag_0D38-32">[32]</a> (If a proper algebraic space over R has an ample line bundle, then it is in fact a projective scheme over R.) Kleiman's criterion fails for proper algebraic spaces X over a field, even if X is smooth.<a href="#cite_note-33">[33]</a>

<div class="mw-heading mw-heading3"><h3 id="Openness_of_ampleness">Openness of ampleness</h3></div>

On a projective scheme X over a field, Kleiman's criterion implies that ampleness is an open condition on the class of an R-divisor (an R-linear combination of Cartier divisors) in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{1}(X)}">

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<annotation encoding="application/x-tex">{\displaystyle N^{1}(X)}</annotation>

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</math><img src="./302a6599a45c990eed23ece9e6fb6745a0907a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.967ex; height:3.176ex;" alt="{\displaystyle N^{1}(X)}" loading="lazy">, with its topology based on the topology of the real numbers. (An R-divisor is defined to be ample if it can be written as a positive linear combination of ample Cartier divisors.<a href="#cite_note-34">[34]</a>) An elementary special case is: for an ample divisor H and any divisor E, there is a positive real number b such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H+aE}">

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<annotation encoding="application/x-tex">{\displaystyle H+aE}</annotation>

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</math><img src="./25b65027d526da693a2afe16cba705b739492fb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.909ex; height:2.343ex;" alt="{\displaystyle H+aE}" loading="lazy"> is ample for all real numbers a of absolute value less than b. In terms of divisors with integer coefficients (or line bundles), this means that nH + E is ample for all sufficiently large positive integers n.

Ampleness is also an open condition in a quite different sense, when the variety or line bundle is varied in an algebraic family. Namely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}">

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</math><img src="./07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"> be a proper morphism of schemes, and let L be a line bundle on X. Then the set of points y in Y such that L is ample on the <a href="Scheme-theoretic_fiber" class="mw-redirect" title="Scheme-theoretic fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{y}}">

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</math><img src="./00ba221e3f5c92da8c1bd62dcad2083e7493160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.974ex; height:2.843ex;" alt="{\displaystyle X_{y}}" loading="lazy"> is open (in the <a href="Zariski_topology" title="Zariski topology">Zariski topology</a>). More strongly, if L is ample on one fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{y}}">

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</math><img src="./00ba221e3f5c92da8c1bd62dcad2083e7493160d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.974ex; height:2.843ex;" alt="{\displaystyle X_{y}}" loading="lazy">, then there is an affine open neighborhood U of y such that L is ample on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(U)}</annotation>

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</math><img src="./b8467d2b1831e3e4ef040a899742a9d6db350f90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.245ex; height:3.176ex;" alt="{\displaystyle f^{-1}(U)}" loading="lazy"> over U.<a href="#cite_note-35">[35]</a>

<div class="mw-heading mw-heading3"><h3 id="Kleiman's_other_characterizations_of_ampleness">Kleiman's other characterizations of ampleness</h3></div>

Kleiman also proved the following characterizations of ampleness, which can be viewed as intermediate steps between the definition of ampleness and numerical criteria. Namely, for a line bundle L on a proper scheme X over a field, the following are equivalent:<a href="#cite_note-36">[36]</a>

<ul><li>L is ample.</li>

<li>For every (irreducible) subvariety <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset X}">

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</math><img src="./22ed7bcf8cec0dba504a57e067b74f9f85f1ea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle Y\subset X}" loading="lazy"> of positive dimension, there is a positive integer r and a section <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in H^{0}(Y,{\mathcal {L}}^{\otimes r})}">

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<annotation encoding="application/x-tex">{\displaystyle s\in H^{0}(Y,{\mathcal {L}}^{\otimes r})}</annotation>

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</math><img src="./ddf64127d11d2508208b32548cbf95fdecb13187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.561ex; height:3.176ex;" alt="{\displaystyle s\in H^{0}(Y,{\mathcal {L}}^{\otimes r})}" loading="lazy"> which is not identically zero but vanishes at some point of Y.</li>

<li>For every (irreducible) subvariety <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\subset X}">

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<annotation encoding="application/x-tex">{\displaystyle Y\subset X}</annotation>

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</math><img src="./22ed7bcf8cec0dba504a57e067b74f9f85f1ea0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.852ex; height:2.176ex;" alt="{\displaystyle Y\subset X}" loading="lazy"> of positive dimension, the <a href="Holomorphic_Euler_characteristic" class="mw-redirect" title="Holomorphic Euler characteristic">holomorphic Euler characteristics</a> of powers of L on Y go to infinity:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle \chi (Y,{\mathcal {L}}^{\otimes r})\to \infty }</annotation>

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</math><img src="./90758ab97af21b8768bdccac8b372af0452619be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.866ex; height:3.009ex;" alt="{\displaystyle \chi (Y,{\mathcal {L}}^{\otimes r})\to \infty }" loading="lazy"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\to \infty }">

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<annotation encoding="application/x-tex">{\displaystyle r\to \infty }</annotation>

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</math><img src="./dcd3a85ea2e3d6b4027434e502cace4177d7a3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.986ex; height:1.843ex;" alt="{\displaystyle r\to \infty }" loading="lazy">.</dd></dl></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Ample_vector_bundles">Ample vector bundles</h3></div>

<a href="Robin_Hartshorne" title="Robin Hartshorne">Robin Hartshorne</a> defined a <a href="Algebraic_vector_bundle" class="mw-redirect" title="Algebraic vector bundle">vector bundle</a> F on a projective scheme X over a field to be ample if the line bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)}">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)}</annotation>

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</math><img src="./5aeb15c854068604d35a2dd82a925899fafd3690.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.822ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)}" loading="lazy"> on the space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (F)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (F)}</annotation>

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</math><img src="./a7c6b467b58c28096beaf3c89de29e4aea1993ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.97ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (F)}" loading="lazy"> of hyperplanes in F is ample.<a href="#cite_note-37">[37]</a>

Several properties of ample line bundles extend to ample vector bundles. For example, a vector bundle F is ample if and only if high symmetric powers of F kill the cohomology <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{i}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle H^{i}}</annotation>

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</math><img src="./615a92cf445eebd5e7401387c3eef1ae09b0af14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.903ex; height:2.676ex;" alt="{\displaystyle H^{i}}" loading="lazy"> of coherent sheaves for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i>0}">

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<annotation encoding="application/x-tex">{\displaystyle i>0}</annotation>

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</math><img src="./1f49f2878fd68a89c3da37eb537198e887cf0293.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i>0}" loading="lazy">.<a href="#cite_note-38">[38]</a> Also, the <a href="Chern_class" title="Chern class">Chern class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{r}(F)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{r}(F)}</annotation>

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</math><img src="./f4f8f0de5551927bdf58f33ac9c862a7d102a015.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.531ex; height:2.843ex;" alt="{\displaystyle c_{r}(F)}" loading="lazy"> of an ample vector bundle has positive degree on every r-dimensional subvariety of X, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq r\leq {\text{rank}}(F)}">

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1\leq r\leq {\text{rank}}(F)}</annotation>

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</math><img src="./b93ddc73c0dda28301a6b35fb844e7f8aaed8e60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.552ex; height:2.843ex;" alt="{\displaystyle 1\leq r\leq {\text{rank}}(F)}" loading="lazy">.<a href="#cite_note-39">[39]</a>

<div class="mw-heading mw-heading3"><h3 id="Big_line_bundles">Big line bundles</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Iitaka_dimension" title="Iitaka dimension">Iitaka dimension</a></div>

A useful weakening of ampleness, notably in <a href="Birational_geometry" title="Birational geometry">birational geometry</a>, is the notion of a big line bundle. A line bundle L on a projective variety X of dimension n over a field is said to be big if there is a positive real number a and a positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle j_{0}}</annotation>

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</math><img src="./a7829ecaa6c8e08ce2a635f4d0232b6ac68a81f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:2.039ex; height:2.509ex;" alt="{\displaystyle j_{0}}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{0}(X,L^{\otimes j})\geq aj^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle h^{0}(X,L^{\otimes j})\geq aj^{n}}</annotation>

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</math><img src="./029591a66d7c8ce2c813ac30980e9c6f1d4221aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.492ex; height:3.176ex;" alt="{\displaystyle h^{0}(X,L^{\otimes j})\geq aj^{n}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\geq j_{0}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle j\geq j_{0}}</annotation>

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</math><img src="./0133d447c13f667a7f972f836b92661c94189de0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:6.096ex; height:2.509ex;" alt="{\displaystyle j\geq j_{0}}" loading="lazy">. This is the maximum possible growth rate for the spaces of sections of powers of L, in the sense that for every line bundle L on X there is a positive number b with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h^{0}(X,L^{\otimes j})\leq bj^{n}}">

<msup>

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</msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle h^{0}(X,L^{\otimes j})\leq bj^{n}}</annotation>

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</math><img src="./7de6b1cc894607740edb5de427f19f9e2466844f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.26ex; height:3.176ex;" alt="{\displaystyle h^{0}(X,L^{\otimes j})\leq bj^{n}}" loading="lazy"> for all j > 0.<a href="#cite_note-40">[40]</a>

There are several other characterizations of big line bundles. First, a line bundle is big if and only if there is a positive integer r such that the <a href="Rational_map" class="mw-redirect" title="Rational map">rational map</a> from X to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (H^{0}(X,L^{\otimes r}))}">

</mrow>

<msup>

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (H^{0}(X,L^{\otimes r}))}</annotation>

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</math><img src="./01ce7d3150016727493fa4aaf88fb9bb40ae1709.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.046ex; height:3.176ex;" alt="{\displaystyle \mathbb {P} (H^{0}(X,L^{\otimes r}))}" loading="lazy"> given by the sections of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\otimes r}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle L^{\otimes r}}</annotation>

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</math><img src="./a079b55b942aff3a14be3059eda3086a8a739ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.835ex; height:2.509ex;" alt="{\displaystyle L^{\otimes r}}" loading="lazy"> is <a href="Birational" class="mw-redirect" title="Birational">birational</a> onto its image.<a href="#cite_note-41">[41]</a> Also, a line bundle L is big if and only if it has a positive tensor power which is the tensor product of an ample line bundle A and an effective line bundle B (meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{0}(X,B)\neq 0}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle H^{0}(X,B)\neq 0}</annotation>

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</math><img src="./9786cd0b3a6652df8fbd1c6c9ba461a7c3881ca3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.006ex; height:3.176ex;" alt="{\displaystyle H^{0}(X,B)\neq 0}" loading="lazy">).<a href="#cite_note-42">[42]</a> Finally, a line bundle is big if and only if its class in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N^{1}(X)}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle N^{1}(X)}</annotation>

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</math><img src="./302a6599a45c990eed23ece9e6fb6745a0907a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.967ex; height:3.176ex;" alt="{\displaystyle N^{1}(X)}" loading="lazy"> is in the interior of the cone of effective divisors.<a href="#cite_note-43">[43]</a>

Bigness can be viewed as a birationally invariant analog of ampleness. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to Y}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f\colon X\to Y}</annotation>

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</math><img src="./07b9ff205beb51e7899846aeae788ae5e5546a3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.68ex; height:2.509ex;" alt="{\displaystyle f\colon X\to Y}" loading="lazy"> is a dominant rational map between smooth projective varieties of the same dimension, then the pullback of a big line bundle on Y is big on X. (At first sight, the pullback is only a line bundle on the open subset of X where f is a morphism, but this extends uniquely to a line bundle on all of X.) For ample line bundles, one can only say that the pullback of an ample line bundle by a finite morphism is ample.<a href="#cite_note-finite-20">[20]</a>

Example: Let X be the <a href="Blowing_up" title="Blowing up">blow-up</a> of the projective plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{2}}</annotation>

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</math><img src="./02e892a2d48ff09d2f95b0e8b8eb419d42e179c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{2}}" loading="lazy"> at a point over the complex numbers. Let H be the pullback to X of a line on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} ^{2}}">

<msup>

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} ^{2}}</annotation>

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</math><img src="./02e892a2d48ff09d2f95b0e8b8eb419d42e179c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.475ex; height:2.676ex;" alt="{\displaystyle \mathbb {P} ^{2}}" loading="lazy">, and let E be the exceptional curve of the blow-up <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon X\to \mathbb {P} ^{2}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \pi \colon X\to \mathbb {P} ^{2}}</annotation>

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</math><img src="./4274a8580a9872d208c5164c6732c01f8d108cd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.435ex; height:2.676ex;" alt="{\displaystyle \pi \colon X\to \mathbb {P} ^{2}}" loading="lazy">. Then the divisor H + E is big but not ample (or even nef) on X, because

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<annotation encoding="application/x-tex">{\displaystyle (H+E)\cdot E=E^{2}=-1<0.}</annotation>

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</math><img src="./9a8893fde3bc2b3c364928f48e41682d33cdfbbb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.867ex; height:3.176ex;" alt="{\displaystyle (H+E)\cdot E=E^{2}=-1<0.}" loading="lazy"></dd></dl>

This negativity also implies that the base locus of H + E (or of any positive multiple) contains the curve E. In fact, this base locus is equal to E.

<div class="mw-heading mw-heading2"><h2 id="Relative_ampleness">Relative ampleness</h2></div>

Given a <a href="Quasi-compact_morphism" title="Quasi-compact morphism">quasi-compact morphism</a> of schemes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to S}">

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<annotation encoding="application/x-tex">{\displaystyle f:X\to S}</annotation>

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</math><img src="./a21a941b35354b73f5a62849067deac7ee9794dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.309ex; height:2.509ex;" alt="{\displaystyle f:X\to S}" loading="lazy">, an invertible sheaf L on X is said to be ample relative to f or f-ample if the following equivalent conditions are met:<a href="#cite_note-FOOTNOTEtag_01VG-44">[44]</a><a href="#cite_note-FOOTNOTEGrothendieckDieudonné1961Proposition_4.6.3-45">[45]</a>

<ol><li>For each open affine subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\subset S}">

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<annotation encoding="application/x-tex">{\displaystyle U\subset S}</annotation>

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</math><img src="./f0ddfd77a2b02dfa9aed180392e1f37c4414ac96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.38ex; height:2.176ex;" alt="{\displaystyle U\subset S}" loading="lazy">, the restriction of L to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle f^{-1}(U)}</annotation>

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</math><img src="./b8467d2b1831e3e4ef040a899742a9d6db350f90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.245ex; height:3.176ex;" alt="{\displaystyle f^{-1}(U)}" loading="lazy"> is <a href="Ample_invertible_sheaf" class="mw-redirect" title="Ample invertible sheaf">ample</a> (in the usual sense).</li>

<li>f is <a href="Quasi-separated_morphism" title="Quasi-separated morphism">quasi-separated</a> and there is an open immersion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\hookrightarrow \operatorname {Proj} _{S}({\mathcal {R}}),\,{\mathcal {R}}:=f_{*}\left(\bigoplus _{0}^{\infty }L^{\otimes n}\right)}">

<msub>

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<mrow>

<mrow>

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</munderover>

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<annotation encoding="application/x-tex">{\displaystyle X\hookrightarrow \operatorname {Proj} _{S}({\mathcal {R}}),\,{\mathcal {R}}:=f_{*}\left(\bigoplus _{0}^{\infty }L^{\otimes n}\right)}</annotation>

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</math><img src="./104369def1962e76a5dcbae4c63ec99f51c79d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.704ex; height:7.509ex;" alt="{\displaystyle X\hookrightarrow \operatorname {Proj} _{S}({\mathcal {R}}),\,{\mathcal {R}}:=f_{*}\left(\bigoplus _{0}^{\infty }L^{\otimes n}\right)}" loading="lazy"> induced by the <a href="Adjunction_map" class="mw-redirect" title="Adjunction map">adjunction map</a>:

<msup>

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<annotation encoding="application/x-tex">{\displaystyle f^{*}{\mathcal {R}}\to \bigoplus _{0}^{\infty }L^{\otimes n}}</annotation>

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</math><img src="./5354f0871d79c0569af50f9fbccfb26444cc9548.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:15.937ex; height:6.843ex;" alt="{\displaystyle f^{*}{\mathcal {R}}\to \bigoplus _{0}^{\infty }L^{\otimes n}}" loading="lazy">.</dd></dl></li>

<li>The condition 2. without "open".</li></ol>

The condition 2 says (roughly) that X can be openly compactified to a <a href="Projective_scheme" class="mw-redirect" title="Projective scheme">projective scheme</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(1)=L}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(1)=L}</annotation>

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</math><img src="./36c08375dcdad2f772acb59a5795dd4f2f2b4345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.503ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}(1)=L}" loading="lazy"> (not just to a proper scheme).

<div class="mw-heading mw-heading3"><h3 id="General_algebraic_geometry">General algebraic geometry</h3></div>

<ul><li><a href="Algebraic_geometry_of_projective_spaces" title="Algebraic geometry of projective spaces">Algebraic geometry of projective spaces</a></li>

<li><a href="Fano_variety" title="Fano variety">Fano variety</a>: a variety whose canonical bundle is anti-ample</li>

<li><a href="Matsusaka's_big_theorem" title="Matsusaka's big theorem">Matsusaka's big theorem</a></li>

<li><a href="Divisorial_scheme" title="Divisorial scheme">Divisorial scheme</a>: a scheme admitting an ample family of line bundles</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Ampleness_in_complex_geometry">Ampleness in complex geometry</h3></div>

<ul><li><a href="Holomorphic_vector_bundle" title="Holomorphic vector bundle">Holomorphic vector bundle</a></li>

<li><a href="Kodaira_embedding_theorem" title="Kodaira embedding theorem">Kodaira embedding theorem</a>: on a compact complex manifold, ampleness and positivity coincide.</li>

<li><a href="Kodaira_vanishing_theorem" title="Kodaira vanishing theorem">Kodaira vanishing theorem</a></li>

<li><a href="Lefschetz_hyperplane_theorem" title="Lefschetz hyperplane theorem">Lefschetz hyperplane theorem</a>: an ample divisor in a complex projective variety X is topologically similar to X.</li></ul>

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

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> Hartshorne (1977), Theorem II.7.1.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Hartshorne (1977), Theorem III.5.2; (<a href="#CITEREFtag_02O6">tag 02O6</a>).

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> Hartshorne (1977), Lemma IV.1.2.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> Lazarsfeld (2004), Example 1.4.5.

</li>

<li id="cite_note-FOOTNOTEtag_01AM-5"><a href="#cite_ref-FOOTNOTEtag_01AM_5-0">^</a> <a href="#CITEREFtag_01AM">tag 01AM</a>.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> Hartshorne (1977), Example II.5.16.2.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> Lazarsfeld (2004), Definition 2.1.26.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Hartshorne (1977), section II.5.

</li>

<li id="cite_note-FOOTNOTEtag_02NP-9"><a href="#cite_ref-FOOTNOTEtag_02NP_9-0">^</a> <a href="#CITEREFtag_02NP">tag 02NP</a>.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> Grothendieck, EGA II, Definition 4.2.2.

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> Hartshorne (1977), Proposition I.7.6 and Example IV.3.3.2.

</li>

<li id="cite_note-FOOTNOTEtag_01PS-12"><a href="#cite_ref-FOOTNOTEtag_01PS_12-0">^</a> <a href="#CITEREFtag_01PS">tag 01PS</a>.

</li>

<li id="cite_note-FOOTNOTEtag_01QE-13"><a href="#cite_ref-FOOTNOTEtag_01QE_13-0">^</a> <a href="#CITEREFtag_01QE">tag 01QE</a>.

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> EGA II, Théorème 4.5.2 and Proposition 4.5.5.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> EGA II, Proposition 4.5.10.

</li>

<li id="cite_note-FOOTNOTEtag_01VU-16"><a href="#cite_ref-FOOTNOTEtag_01VU_16-0">^</a> <a href="#CITEREFtag_01VU">tag 01VU</a>.

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> Hartshorne (1977), Theorem II.7.6

</li>

<li id="cite_note-CSG-18">^ <a href="#cite_ref-CSG_18-0">a</a> <a href="#cite_ref-CSG_18-1">b</a> Lazarsfeld (2004), Theorem 1.2.6.

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> Hartshorne (1977), Proposition III.5.3

</li>

<li id="cite_note-finite-20">^ <a href="#cite_ref-finite_20-0">a</a> <a href="#cite_ref-finite_20-1">b</a> Lazarsfeld (2004), Theorem 1.2.13.

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> Hartshorne (1977), Example II.7.6.3.

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> Hartshorne (1977), Exercise IV.3.2(b).

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> Hartshorne (1977), Proposition IV.3.1.

</li>

<li id="cite_note-24"><a href="#cite_ref-24">^</a> Hartshorne (1977), Corollary IV.3.3.

</li>

<li id="cite_note-25"><a href="#cite_ref-25">^</a> Hartshorne (1977), Proposition IV.5.2.

</li>

<li id="cite_note-26"><a href="#cite_ref-26">^</a> Lazarsfeld (2004), Theorem 1.2.23, Remark 1.2.29; Kleiman (1966), Theorem III.1.

</li>

<li id="cite_note-27"><a href="#cite_ref-27">^</a> Lazarsfeld (2004), Theorems 1.4.23 and 1.4.29; Kleiman (1966), Theorem IV.1.

</li>

<li id="cite_note-28"><a href="#cite_ref-28">^</a> Fujino (2005), Corollary 3.3; Lazarsfeld (2004), Remark 1.4.24.

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> Lazarsfeld (2004), Example 1.5.2.

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> Lazarsfeld (2004), Theorem 1.4.13; Hartshorne (1970), Theorem I.7.1.

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> Kollár (1990), Theorem 3.11.

</li>

<li id="cite_note-FOOTNOTEtag_0D38-32"><a href="#cite_ref-FOOTNOTEtag_0D38_32-0">^</a> <a href="#CITEREFtag_0D38">tag 0D38</a>.

</li>

<li id="cite_note-33"><a href="#cite_ref-33">^</a> Kollár (1996), Chapter VI, Appendix, Exercise 2.19.3.

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> Lazarsfeld (2004), Definition 1.3.11.

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> Lazarsfeld (2004), Theorem 1.2.17 and its proof.

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> Lazarsfeld (2004), Example 1.2.32; Kleiman (1966), Theorem III.1.

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> Lazarsfeld (2004), Definition 6.1.1.

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> Lazarsfeld (2004), Theorem 6.1.10.

</li>

<li id="cite_note-39"><a href="#cite_ref-39">^</a> Lazarsfeld (2004), Theorem 8.2.2.

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> Lazarsfeld (2004), Corollary 2.1.38.

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> Lazarsfeld (2004), section 2.2.A.

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> Lazarsfeld (2004), Corollary 2.2.7.

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> Lazarsfeld (2004), Theorem 2.2.26.

</li>

<li id="cite_note-FOOTNOTEtag_01VG-44"><a href="#cite_ref-FOOTNOTEtag_01VG_44-0">^</a> <a href="#CITEREFtag_01VG">tag 01VG</a>.

</li>

<li id="cite_note-FOOTNOTEGrothendieckDieudonné1961Proposition_4.6.3-45"><a href="#cite_ref-FOOTNOTEGrothendieckDieudonné1961Proposition_4.6.3_45-0">^</a> <a href="#CITEREFGrothendieckDieudonné1961">Grothendieck & Dieudonné 1961</a>, Proposition 4.6.3.

</li>

</ol></div>

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

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<li><cite id="CITEREFHartshorne1970" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1970), Ample Subvarieties of Algebraic Varieties, Lecture Notes in Mathematics, vol. 156, Berlin, Heidelberg: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBFb0067839">10.1007/BFb0067839</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-05184-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0282977">0282977</a></cite></li>

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<li><cite id="CITEREFKleiman1966" class="citation cs2"><a href="Steven_Kleiman" title="Steven Kleiman">Kleiman, Steven L.</a> (1966), "Toward a numerical theory of ampleness", <a href="Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, Second Series, 84 (3): 293–344, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1970447">10.2307/1970447</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1970447">1970447</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0206009">0206009</a></cite></li>

<li><cite id="CITEREFKollár1990" class="citation cs2"><a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a> (1990), "Projectivity of complete moduli", Journal of Differential Geometry, 32, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214445046">10.4310/jdg/1214445046</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1064874">1064874</a></cite></li>

<li><cite id="CITEREFKollár1996" class="citation cs2"><a href="J%C3%A1nos_Koll%C3%A1r" title="János Kollár">Kollár, János</a> (1996), Rational curves on algebraic varieties, Berlin, Heidelberg: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-662-03276-3">10.1007/978-3-662-03276-3</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-08219-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1440180">1440180</a></cite></li>

<li><cite id="CITEREFLazarsfeld2004" class="citation cs2"><a href="Robert_Lazarsfeld" title="Robert Lazarsfeld">Lazarsfeld, Robert</a> (2004), Positivity in algebraic geometry (2 vols.), Berlin: Springer-Verlag, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-18808-4">10.1007/978-3-642-18808-4</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-22533-1</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2095471">2095471</a></cite></li>

<li><cite id="CITEREFNagata1959" class="citation cs2"><a href="Masayoshi_Nagata" title="Masayoshi Nagata">Nagata, Masayoshi</a> (1959), "On the 14th problem of Hilbert", <a href="American_Journal_of_Mathematics" title="American Journal of Mathematics">American Journal of Mathematics</a>, 81 (3): 766–772, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2372927">10.2307/2372927</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2372927">2372927</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0154867">0154867</a></cite></li>

<li><cite id="CITEREFtag_01VG" class="citation web cs2"><a rel="nofollow" class="external text" href="https://stacks.math.columbia.edu/tag/01VG">"Section 29.37 (01VG): Relatively ample sheaves—The Stacks project"</a></cite></li>

<li><cite id="CITEREFtag_01AM" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/01AM">Stacks Project, Tag 01AM</a></cite></li>

<li><cite id="CITEREFtag_01PS" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/01PS">Stacks Project, Tag 01PS</a></cite></li>

<li><cite id="CITEREFtag_01QE" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/01QE">Stacks Project, Tag 01QE</a></cite></li>

<li><cite id="CITEREFtag_01VU" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/01VU">Stacks Project, Tag 01VU</a></cite></li>

<li><cite id="CITEREFtag_02NP" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/02NP">Stacks Project, Tag 02NP</a></cite></li>

<li><cite id="CITEREFtag_02O6" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/02O6">Stacks Project, Tag 02O6</a></cite></li>

<li><cite id="CITEREFtag_0D38" class="citation cs2"><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/tag/0D38">Stacks Project, Tag 0D38</a></cite></li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://stacks.math.columbia.edu/">The Stacks Project</a></li></ul></div><div><div class="zim-footer">

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