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'''[[Projective space]]''' plays a central role in [[algebraic geometry]]. The aim of this article is to define the notion in terms of abstract [[algebraic geometry]] and to describe some basic uses of projective space.

== Homogeneous polynomial ideals==

Let '''k''' be an [[algebraically closed]] [[field (mathematics)|field]], and ''V'' be a [[finite dimensional]] [[vector space]] over '''k'''. The [[symmetric algebra]] of the [[dual vector space]] ''V*'' is called the [[polynomial ring]] on ''V'' and denoted by '''k'''[''V'']. It is a naturally [[graded algebra]] by the degree of polynomials.

The projective [[Nullstellensatz]] states that, for any [[homogeneous ideal]] ''I'' that does not contain all polynomials of a certain degree (referred to as an [[irrelevant ideal]]), the common zero locus of all polynomials in ''I'' (or ''Nullstelle'') is non-trivial (i.e. the common zero locus contains more than the single element {0}), and, more precisely, the ideal of polynomials that vanish on that locus coincides with the [[radical of an ideal|radical]] of the ideal ''I''.

This last assertion is best summarized by the formula : for any relevant ideal ''I'',
:<math> \mathcal I (\mathcal V(I)) = \sqrt I .</math>

In particular, maximal homogeneous relevant ideals of '''k'''[''V''] are one-to-one with lines through the origin of ''V''.

== Construction of projectivized schemes==

Let ''V'' be a [[finite dimensional]] [[vector space]] over a [[field (mathematics)|field]] '''k'''. The [[scheme (mathematics)|scheme]] over '''k''' defined by [[Proj]]('''k'''[''V'']) is called '''projectivization''' of ''V''. The '''projective ''n''-space''' on '''k''' is the projectivization of the vector space <math>\mathbb A_k^{n+1}</math>.

The definition of the sheaf is done on the [[base (topology)|base of open sets]] of principal open sets ''D''(''P''), where ''P'' varies over the set of homogeneous polynomials, by setting the sections
:<math>\Gamma (D(P), \mathcal O_{\mathbb P (V)})</math>

to be the ring <math> (k[V]_P)_0</math>, the zero degree component of the ring obtained by [[localization of a ring|localization]] at ''P''. Its elements are therefore the rational functions with homogeneous numerator and some power of ''P'' as the denominator, with same degree as the numerator.

The situation is most clear at a non-vanishing [[linear form]] φ. The restriction of the structure sheaf to the open set ''D''(φ) is then canonically identified <ref>In coordinates this correspondence is given by <math>\frac{P (X_0, \ldots, X_n)} {X_0^{deg(P)}} \mapsto P(1,X_1,\ldots, X_n) </math></ref> with the [[affine scheme]] spec('''k'''[ker φ]). Since the ''D''(''φ'') form an [[open cover]] of ''X'' the projective schemes can be thought of as being obtained by the gluing via projectivization of isomorphic affine schemes.

It can be noted that the ring of global sections of this scheme is a field, which implies that the scheme is not affine. Any two open sets intersect non-trivially: ''ie'' the scheme is [[irreducible component|irreducible]]. When the field '''k''' is [[algebraically closed]], <math>\mathbb P(V) </math> is in fact an [[abstract variety]], that furthermore is complete. ''cf. '' [[Glossary of scheme theory]]

== Divisors and twisting sheaves ==
The Proj functor in fact gives more than a mere scheme: a sheaf in graded modules over the structure sheaf is defined in the process. The homogeneous components of this graded sheaf are denoted <math>\mathcal O (i)</math>, the [[Serre twisting sheaf|Serre twisting sheaves]]. All of these sheaves are in fact [[line bundle]]s. By the correspondence between [[Cartier divisor]]s and line bundles, the first twisting sheaf <math>\mathcal O(1)</math> is equivalent to hyperplane divisors.

Since the ring of polynomials is a [[unique factorization domain]], any [[prime ideal]] of [[height (ring theory)|height]] 1 is [[principal ideal|principal]], which shows that any Weil divisor is linearly equivalent to some power of a hyperplane divisor. This consideration proves that the Picard group of a projective space is free of rank 1. That is <math>\mathrm{Pic}\ \mathbf P^n_\mathbf k = \mathbb Z</math>, and the isomorphism is given by the degree of divisors.

===Classification of vector bundles===

The [[invertible sheaves]], or ''line bundles'', on the [[projective space]] <math>\mathbb{P}^n_k,\,</math> for ''k'' a [[field (mathematics)|field]], are '''exactly''' the twisting [[sheaf (mathematics)|sheaves]] <math>\mathcal{O}(m),\ m \in \mathbb{Z},</math> so the [[Picard group]] of <math>\mathbb{P}^n_k</math> is isomorphic to <math>\mathbb{Z}</math>. The isomorphism is given by the [[first Chern class]].

The space of local sections on an open set <math>U \subseteq \mathbb P (V)</math> of the line bundle <math>\mathcal O(k)</math> is the space of homogeneous degree ''k'' regular functions on the cone in ''V'' associated to ''U''. In particular, the space of global sections
:<math> \Gamma (\mathbb P, \mathcal O (m))</math>
vanishes if ''m < 0'', and consists of constants in '''k''' for ''m=0'' and of homogeneous polynomials of degree ''m'' for ''m > 0''.

The [[Birkhoff-Grothendieck theorem]] states that on the projective line, any vector bundle splits in a unique way as a direct sum of the line bundles.

===Important line bundles ===
The [[tautological bundle]], which appears for instance as the [[exceptional divisor]] of the [[blowing up]] of a [[Singular point of an algebraic variety|smooth point]] is the sheaf <math>\mathcal O (-1)</math>. The [[canonical bundle]]
:<math>\mathcal K(\mathbb{P}^n_k),\,</math> is <math>\mathcal O(-(n+1))</math>.
This fact derives from a fundamental geometric statement on projective spaces: the [[Euler exact sequence]].

The negativity of the canonical line bundle makes projective spaces prime examples of [[Fano variety|Fano varieties]], equivalently, their anticanonical line bundle is ample (in fact very ample). Their index (''cf.'' [[Fano variety|Fano varieties]]) is given by <math>\mathrm {Ind} (\mathbb P^n) = n+1</math>, and, by a theorem of Kobayashi-Ochiai, projective spaces are ''characterized'' amongst Fano varieties by the property
:<math>\mathrm {Ind} (X) = \mathrm{dim} X +1</math>.

== Morphisms to projective schemes ==

As affine spaces can be embedded in projective spaces, all [[affine varieties]] can be embedded in projective spaces too.

Any choice of a finite system of nonsimultaneously vanishing global sections of a '''globally generated [[line bundle]]''' defines a [[morphism (algebraic geometry)|morphism]] to a projective space. A line bundle whose base can be embedded in a projective space by such a morphism is called [[very ample line bundle|very ample]].

The group of symmetries of the projective space <math>\mathbb P^n_{\mathbf k}</math> is the group of projectivized linear automorphisms <math>\mathrm {PGL}_{n+1}(\mathbf k)</math>. The choice of a morphism to a projective space <math>j : X \to \mathbf P^n</math> ''modulo'' the action of this group is in fact ''equivalent'' to the choice of a '''globally generating''' ''n''-dimensional [[linear system of divisors]] on a '''[[line bundle]]''' on ''X''. The choice of a projective embedding of ''X'', ''modulo'' projective transformations is likewise equivalent to the choice of a [[very ample line bundle]] on ''X''.

A morphism to a projective space <math>j : X \to \mathbf P^n</math> defines a globally generated line bundle by <math>j^* \mathcal O (1)</math> and a linear system
:<math>j^* (\Gamma (\mathbf P^n, \mathcal O(1))) \subset \Gamma (X, j^*\mathcal O(1) ).</math>

If the range of the morphism <math>j</math> is not contained in a hyperplane divisor, then the pull-back is an injection and the [[linear system of divisors]]
:<math>j^* (\Gamma (\mathbf P^n, \mathcal O(1)))</math> is a linear system of dimension ''n''.

===An example: the Veronese embeddings===

The Veronese embeddings are embeddings <math>\mathbb P^n \to \mathbb P^N</math> for <math>N=\binom{n+d}{d} -1</math>

See the [http://mathoverflow.net/questions/60324/lefschetz-hyperplane-section-theorem answer] on [[MathOverflow]] for an application of the Veronese embedding to the calculation of cohomology groups of smooth projective [[hypersurface]]s (smooth divisors).

==Curves in projective spaces==

As Fano varieties, the projective spaces are [[ruled variety|ruled varieties]]. The intersection theory of curves in the projective plane yields the [[Bézout theorem]].

== See also ==
=== General algebraic geometry ===
*[[Scheme (mathematics)]]
*[[Projective variety]]
*[[Proj construction]]

===General projective geometry===
*[[Projective space]]
*[[Projective geometry]]
*[[Homogeneous polynomial]]

==References==
{{Reflist}}
*{{cite book
| author = [[Robin Hartshorne]]
| year = 1977
| title = [[Hartshorne's Algebraic Geometry|Algebraic Geometry]]
| publisher = [[Springer Science+Business Media|Springer-Verlag]]
| isbn = 0-387-90244-9
}}
* [http://www.math.u-psud.fr/~laszlo/M2/2010-2011/TD2.pdf Exercise sheet] (in French) on projective spaces, on the [http://www.math.u-psud.fr/~laszlo/ page] of Yves Laszlo.

{{DEFAULTSORT:Algebraic Geometry Of Projective Spaces}}
[[Category:Algebraic geometry]]
[[Category:Projective geometry]]
[[Category:Algebraic varieties]]
[[Category:Geometry of divisors]]

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