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A closed subscheme (cf. [[Scheme|Scheme]]) of a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753301.png" />. In homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753302.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753303.png" />, a projective scheme is given by a system of homogeneous algebraic equations
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A closed sub-[[Scheme|scheme]] of a projective space $ \mathbf{P}_{\mathbb{k}}^{n} $. In homogeneous coordinates $ x_{0},\ldots,x_{n} $ on $ \mathbf{P}_{\mathbb{k}}^{n} $, a projective scheme is given by a system of homogeneous algebraic equations:
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$$
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{f_{1}}(x_{0},\ldots,x_{n}) = 0, \quad \ldots \quad {f_{r}}(x_{0},\ldots,x_{n}) = 0.
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753304.png" /></td> </tr></table>
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Every projective scheme is complete (compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an [[Ample sheaf|ample]], [[Invertible sheaf|invertible]] sheaf on it. There are also other criteria of projectivity.
  
Every projective scheme is complete (compact in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753305.png" />); conversely, a complete scheme is projective if there is an ample invertible sheaf (cf. [[Ample sheaf|Ample sheaf]]; [[Invertible sheaf|Invertible sheaf]]) on it. There are also other criteria of projectivity.
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A generalization of the concept of a projective scheme is a '''projective morphism'''. A morphism $ f: X \to Y $ of schemes is called '''projective''' (and $ X $ is called a '''projective scheme''' over $ Y $) if and only if $ X $ is a closed sub-scheme of the projective fiber bundle $ {\mathbf{P}_{Y}}(\mathcal{E}) $, where $ \mathcal{E} $ is a locally free $ \mathcal{O}_{Y} $-module. A composition of projective morphisms is projective. The projectivity of a morphism is also preserved under a [[Base change|base change]]; in particular, the fibers of a projective morphism are projective schemes (but not conversely). If a scheme $ X $ is projective and $ X \to Z $ is a finite surjective morphism, then $ Z $ is also projective.
  
A generalization of the concept of a projective scheme is a projective morphism. A morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753306.png" /> of schemes is called projective (and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753307.png" /> is called a scheme projective over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753308.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p0753309.png" /> is a closed subscheme of the projective fibre bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533011.png" /> is a locally free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533012.png" />-module. A composite of projective morphisms is projective. The projectivity of a morphism is preserved also under a [[Base change|base change]]; in particular, the fibres of a projective morphism are projective schemes (but not conversely). If a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533013.png" /> is projective and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533014.png" /> is a finite surjective morphism, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533015.png" /> is also projective.
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Any projective scheme (over $ Y $) can be obtained using the construction of the [[Projective spectrum of a ring|projective spectrum]]. Restricting to the case of an affine base, $ Y = \operatorname{Spec}(R) $, suppose that $ \displaystyle A = \bigoplus_{i \geq 0} A_{i} $ is a graded $ R $-algebra with the $ R $-module $ A_{1} $ being of finite type and generating the algebra $ A $, and suppose that $ \operatorname{Proj}(A) $ is the set of homogeneous prime ideals $ \mathfrak{p} \subseteq A $ not containing $ A_{1} $. Equipped with the natural topology and a structure sheaf, the set $ \operatorname{Proj}(A) $ is a projective $ Y $-scheme; moreover, any projective $ Y $-scheme has such a form.
 
 
Any projective scheme (over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533016.png" />) can be obtained using the construction of the projective spectrum (cf. [[Projective spectrum of a ring|Projective spectrum of a ring]]). Restricting to the case of an affine base, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533017.png" />, suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533018.png" /> is a graded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533019.png" />-algebra with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533020.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533021.png" /> being of finite type and generating the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533022.png" />, and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533023.png" /> is the set of homogeneous prime ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533024.png" /> not containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533025.png" />. Equipped with the natural topology and a structure sheaf, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533026.png" /> is a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533027.png" />-scheme; moreover, any projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533028.png" />-scheme has such a form.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Algebraic geometry" , '''1. Complex projective varieties''' , Springer (1976) {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR></table>
 
  
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<table>
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<TR><TD valign="top">[1]</TD><TD valign="top">
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D. Mumford, “Algebraic geometry”, '''1. Complex projective varieties''', Springer (1976). {{MR|0453732}} {{ZBL|0356.14002}} </TD></TR>
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</table>
  
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====Comments====
  
====Comments====
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Given a vector bundle $ E $ over $ Y $ (or, equivalently, a locally free $ \mathcal{O}_{Y} $-module $ \mathcal{E} $), the associated projective bundle, or projective fiber bundle, has as fiber over $ y \in Y $ the projective space $ \mathbf{P}(E_{Y}) $ of all $ 1 $-dimensional sub-spaces of the vector space $ E_{Y} $.
Given a vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533029.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533030.png" /> (or, equivalently, a locally free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533031.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533032.png" />), the associated projective bundle, or projective fibre bundle, has as fibre over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533033.png" /> the projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533034.png" /> of all one-dimensional subspaces of the vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075330/p07533035.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) pp. 91 {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD><TD valign="top">
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R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
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</table>

Latest revision as of 08:05, 14 December 2016

A closed sub-scheme of a projective space $ \mathbf{P}_{\mathbb{k}}^{n} $. In homogeneous coordinates $ x_{0},\ldots,x_{n} $ on $ \mathbf{P}_{\mathbb{k}}^{n} $, a projective scheme is given by a system of homogeneous algebraic equations: $$ {f_{1}}(x_{0},\ldots,x_{n}) = 0, \quad \ldots \quad {f_{r}}(x_{0},\ldots,x_{n}) = 0. $$

Every projective scheme is complete (compact in the case $ \mathbb{k} = \mathbb{C} $). Conversely, a complete scheme is projective if there is an ample, invertible sheaf on it. There are also other criteria of projectivity.

A generalization of the concept of a projective scheme is a projective morphism. A morphism $ f: X \to Y $ of schemes is called projective (and $ X $ is called a projective scheme over $ Y $) if and only if $ X $ is a closed sub-scheme of the projective fiber bundle $ {\mathbf{P}_{Y}}(\mathcal{E}) $, where $ \mathcal{E} $ is a locally free $ \mathcal{O}_{Y} $-module. A composition of projective morphisms is projective. The projectivity of a morphism is also preserved under a base change; in particular, the fibers of a projective morphism are projective schemes (but not conversely). If a scheme $ X $ is projective and $ X \to Z $ is a finite surjective morphism, then $ Z $ is also projective.

Any projective scheme (over $ Y $) can be obtained using the construction of the projective spectrum. Restricting to the case of an affine base, $ Y = \operatorname{Spec}(R) $, suppose that $ \displaystyle A = \bigoplus_{i \geq 0} A_{i} $ is a graded $ R $-algebra with the $ R $-module $ A_{1} $ being of finite type and generating the algebra $ A $, and suppose that $ \operatorname{Proj}(A) $ is the set of homogeneous prime ideals $ \mathfrak{p} \subseteq A $ not containing $ A_{1} $. Equipped with the natural topology and a structure sheaf, the set $ \operatorname{Proj}(A) $ is a projective $ Y $-scheme; moreover, any projective $ Y $-scheme has such a form.

References

[1] D. Mumford, “Algebraic geometry”, 1. Complex projective varieties, Springer (1976). MR0453732 Zbl 0356.14002

Comments

Given a vector bundle $ E $ over $ Y $ (or, equivalently, a locally free $ \mathcal{O}_{Y} $-module $ \mathcal{E} $), the associated projective bundle, or projective fiber bundle, has as fiber over $ y \in Y $ the projective space $ \mathbf{P}(E_{Y}) $ of all $ 1 $-dimensional sub-spaces of the vector space $ E_{Y} $.

References

[a1] R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. MR0463157 Zbl 0367.14001
How to Cite This Entry:
Projective scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Projective_scheme&oldid=23935
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article