The concept of aProjective space plays a central role inalgebraic geometry. This article aims to define the notion in terms of abstractalgebraic geometry and to describe some basic uses of projective spaces.

Letk be analgebraically closedfield, andV be afinite-dimensionalvector space overk. Thesymmetric algebra of thedual vector spaceV* is called thepolynomial ring onV and denoted byk[V]. It is a naturallygraded algebra by the degree of polynomials.

The projectiveNullstellensatz states that, for anyhomogeneous idealI that does not contain all polynomials of a certain degree (referred to as anirrelevant ideal), the common zero locus of all polynomials inI (orNullstelle) 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 theradical of the idealI.

This last assertion is best summarized by the formula: for any relevant idealI,

${\displaystyle \mathcal{I}(\mathcal{V}(I))=\sqrt{I}.}$

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

LetV be afinite-dimensionalvector space over afieldk. Thescheme overk defined byProj(k[V]) is calledprojectivization ofV. Theprojectiven-space onk is the projectivization of the vector space${\displaystyle {\mathbb{A}}_k^{n+1}}$.

The definition of the sheaf is done on thebase of open sets of principal open sets D(P), whereP varies over the set of homogeneous polynomials, by setting the sections

${\displaystyle \mathrm{\Gamma }(D(P),{\mathcal{O}}_{\mathbb{P}(V)})}$

to be the ring${\displaystyle (k[V{]}_P{)}_0}$, the zero degree component of the ring obtained bylocalization atP. Its elements are therefore the rational functions with homogeneous numerator and some power ofP as the denominator, with same degree as the numerator.

The situation is most clear at a non-vanishinglinear form φ. The restriction of the structure sheaf to the open setD(φ) is then canonically identified^{[note 1]} with theaffine scheme spec(k[ker φ]). Since theD(φ) form anopen cover ofX 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 isirreducible. When the fieldk isalgebraically closed,${\displaystyle \mathbb{P}(V)}$ is in fact anabstract variety, that furthermore iscomplete.cf.Glossary of scheme theory

The Proj construction 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${\displaystyle \mathcal{O}(i)}$, theSerre twisting sheaves. All of these sheaves are in factline bundles. By the correspondence betweenCartier divisors and line bundles, the first twisting sheaf${\displaystyle \mathcal{O}(1)}$ is equivalent to hyperplane divisors.

Since the ring of polynomials is aunique factorization domain, anyprime ideal ofheight 1 isprincipal, which shows that any Weil divisor is linearly equivalent to some power of a hyperplane divisor. This consideration proves that thePicard group of a projective space is free of rank 1. That is${\displaystyle \mathrm{Pic}{\mathbf{P}}_{\mathbf{k}}^n=\mathbb{Z}}$, and the isomorphism is given by the degree of divisors.

Theinvertible sheaves, orline bundles, on theprojective space${\displaystyle {\mathbb{P}}_k^n,}$ fork afield, areexactly the twistingsheaves${\displaystyle \mathcal{O}(m),m\in \mathbb{Z},}$ so thePicard group of${\displaystyle {\mathbb{P}}_k^n}$ is isomorphic to${\displaystyle \mathbb{Z}}$. The isomorphism is given by thefirst Chern class.

The space of local sections on an open set${\displaystyle U\subseteq \mathbb{P}(V)}$ of the line bundle${\displaystyle \mathcal{O}(k)}$ is the space of homogeneous degreek regular functions on the cone inV associated toU. In particular, the space of global sections

${\displaystyle \mathrm{\Gamma }(\mathbb{P},\mathcal{O}(m))}$

vanishes ifm < 0, and consists of constants ink form=0 and of homogeneous polynomials of degreem form > 0. (Hence has dimension$(\genfrac{}{}{0ex}{}{m+n}{m})=(\genfrac{}{}{0ex}{}{m+n}{n})$).

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

Thetautological bundle, which appears for instance as theexceptional divisor of theblowing up of asmooth point is the sheaf${\displaystyle \mathcal{O}(-1)}$. Thecanonical bundle

${\displaystyle \mathcal{K}({\mathbb{P}}_k^n),}$ is${\displaystyle \mathcal{O}(-(n+1))}$.

This fact derives from a fundamental geometric statement on projective spaces: theEuler sequence.

The negativity of the canonical line bundle makes projective spaces prime examples ofFano varieties, equivalently, their anticanonical line bundle isample (in fact very ample). Their index (cf.Fano varieties) is given by${\displaystyle \mathrm{Ind}({\mathbb{P}}^n)=n+1}$, and, by a theorem of Kobayashi-Ochiai, projective spaces arecharacterized amongst Fano varieties by the property

${\displaystyle \mathrm{Ind}(X)=\dim X+1.}$

As affine spaces can be embedded in projective spaces, allaffine varieties can be embedded in projective spaces too.

Any choice of a finite system of nonsimultaneously vanishing global sections of aglobally generatedline bundle defines amorphism to a projective space. A line bundle whose base can be embedded in a projective space by such a morphism is calledvery ample.

The group of symmetries of the projective space${\displaystyle {\mathbb{P}}_{\mathbf{k}}^n}$ is thegroup of projectivized linear automorphisms${\displaystyle {\mathrm{P}\mathrm{G}\mathrm{L}}_{n+1}(\mathbf{k})}$. The choice of a morphism to a projective space${\displaystyle j:X\to {\mathbf{P}}^n}$modulo the action of this group is in factequivalent to the choice of aglobally generatingn-dimensionallinear system of divisors on a line bundle onX. The choice of a projective embedding ofX,modulo projective transformations is likewise equivalent to the choice of a very ample line bundle onX.

A morphism to a projective space${\displaystyle j:X\to {\mathbf{P}}^n}$ defines a globally generated line bundle by${\displaystyle j^{\ast }\mathcal{O}(1)}$ and a linear system

${\displaystyle j^{\ast }(\mathrm{\Gamma }({\mathbf{P}}^n,\mathcal{O}(1)))\subset \mathrm{\Gamma }(X,j^{\ast }\mathcal{O}(1)).}$

If the range of the morphism${\displaystyle j}$ is not contained in a hyperplane divisor, then thepull-back is an injection and thelinear system of divisors

${\displaystyle j^{\ast }(\mathrm{\Gamma }({\mathbf{P}}^n,\mathcal{O}(1)))}$ is a linear system of dimensionn.

TheVeronese embeddings are embeddings${\displaystyle {\mathbb{P}}^n\to {\mathbb{P}}^N}$ for$N=(\genfrac{}{}{0ex}{}{n+d}{d})-1.$

See theanswer onMathOverflow for an application of the Veronese embedding to the calculation of cohomology groups of smooth projectivehypersurfaces (smooth divisors).

As Fano varieties, the projective spaces areruled varieties. Theintersection theory of curves in the projective plane yields theBézout theorem.

• Scheme (mathematics)
• Projective variety
• Proj construction

• Projective space
• Projective geometry
• Homogeneous polynomial

• Robin Hartshorne (1977).Algebraic Geometry.Springer-Verlag.ISBN 0-387-90244-9.

• Algebraic geometry
• Projective geometry
• Algebraic varieties
• Geometry of divisors
• Space (mathematics)

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