Inalgebraic geometry, aprojective variety is analgebraic variety that is a closedsubvariety of aprojective space. That is, it is the zero-locus in${\displaystyle {\mathbb{P}}^n}$ of some finite family ofhomogeneous polynomials that generate aprime ideal, the defining ideal of the variety.

A projective variety is aprojective curve if its dimension is one; it is aprojective surface if its dimension is two; it is aprojective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a singlehomogeneous polynomial.

IfX is a projective variety defined by a homogeneous prime idealI, then thequotient ring

${\displaystyle k[x_0,\dots ,x_n]/I}$

is called thehomogeneous coordinate ring ofX. Basic invariants ofX such as thedegree and thedimension can be read off theHilbert polynomial of thisgraded ring.

Projective varieties arise in many ways. They arecomplete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, butChow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studyingline bundles ordivisors onX.

A salient feature of projective varieties are the finiteness constraints on sheaf cohomology. For smooth projective varieties,Serre duality can be viewed as an analog ofPoincaré duality. It also leads to theRiemann–Roch theorem for projective curves, i.e., projective varieties ofdimension 1. The theory of projective curves is particularly rich, including a classification by thegenus of the curve. The classification program for higher-dimensional projective varieties naturally leads to the construction of moduli of projective varieties.^[1]Hilbert schemes parametrize closed subschemes of${\displaystyle {\mathbb{P}}^n}$ with prescribed Hilbert polynomial. Hilbert schemes, of whichGrassmannians are special cases, are also projective schemes in their own right.Geometric invariant theory offers another approach. The classical approaches include theTeichmüller space andChow varieties.

A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials definingX havecomplex coefficients. Broadly, theGAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. For example, the theory ofholomorphic vector bundles (more generallycoherent analytic sheaves) onX coincide with that of algebraic vector bundles.Chow's theorem says that a subset of projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. The combination of analytic and algebraic methods for complex projective varieties lead to areas such asHodge theory.

Letk be an algebraically closed field. The basis of the definition of projective varieties is projective space${\displaystyle {\mathbb{P}}^n}$, which can be defined in different, but equivalent ways:

• as the set of all lines through the origin in${\displaystyle k^{n+1}}$ (i.e., all one-dimensional vector subspaces of${\displaystyle k^{n+1}}$)
• as the set of tuples${\displaystyle (x_0,\dots ,x_n)\in k^{n+1}}$, with${\displaystyle x_0,\dots ,x_n}$ not all zero, modulo the equivalence relation$${\displaystyle (x_0,\dots ,x_n)\sim \lambda (x_0,\dots ,x_n)}$$ for any${\displaystyle \lambda \in k\setminus \{0\}}$. The equivalence class of such a tuple is denoted by$${\displaystyle [x_0:\cdots :x_n].}$$ This equivalence class is the general point of projective space. The numbers${\displaystyle x_0,\dots ,x_n}$ are referred to as thehomogeneous coordinates of the point.

Aprojective variety is, by definition, a closed subvariety of${\displaystyle {\mathbb{P}}^n}$, where closed refers to theZariski topology.^[2] In general, closed subsets of the Zariski topology are defined to be the common zero-locus of a finite collection of homogeneous polynomial functions. Given a polynomial${\displaystyle f\in k[x_0,\dots ,x_n]}$, the condition

${\displaystyle f([x_0:\cdots :x_n])=0}$

does not make sense for arbitrary polynomials, but only iff ishomogeneous, i.e., the degrees of all themonomials (whose sum isf) are the same. In this case, the vanishing of

${\displaystyle f(\lambda x_0,\dots ,\lambda x_n)={\lambda }^{\mathrm{deg}f}f(x_0,\dots ,x_n)}$

is independent of the choice of${\displaystyle \lambda \ne 0}$.

Therefore, projective varieties arise from homogeneousprime idealsI of${\displaystyle k[x_0,\dots ,x_n]}$, and setting

${\displaystyle X=\left\{[x_0:\cdots :x_n]\in {\mathbb{P}}^n,f([x_0:\cdots :x_n])=0\text{ for all }f\in I\right\}.}$

Moreover, the projective varietyX is an algebraic variety, meaning that it is covered by open affine subvarieties and satisfies the separation axiom. Thus, the local study ofX (e.g., singularity) reduces to that of an affine variety. The explicit structure is as follows. The projective space${\displaystyle {\mathbb{P}}^n}$ is covered by the standard open affine charts

${\displaystyle U_i=\{[x_0:\cdots :x_n],x_i\ne 0\},}$

which themselves are affinen-spaces with the coordinate ring

${\displaystyle k\left[y_1^{(i)},\dots ,y_n^{(i)}\right],y_j^{(i)}=x_j/x_i.}$

Sayi = 0 for the notational simplicity and drop the superscript (0). Then${\displaystyle X\cap U_0}$ is a closed subvariety of${\displaystyle U_0\simeq {\mathbb{A}}^n}$ defined by the ideal of${\displaystyle k[y_1,\dots ,y_n]}$ generated by

${\displaystyle f(1,y_1,\dots ,y_n)}$

for allf inI. Thus,X is an algebraic variety covered by (n+1) open affine charts${\displaystyle X\cap U_i}$.

Note thatX is the closure of the affine variety${\displaystyle X\cap U_0}$ in${\displaystyle {\mathbb{P}}^n}$. Conversely, starting from some closed (affine) variety${\displaystyle V\subset U_0\simeq {\mathbb{A}}^n}$, the closure ofV in${\displaystyle {\mathbb{P}}^n}$ is the projective variety called theprojective completion ofV. If${\displaystyle I\subset k[y_1,\dots ,y_n]}$ definesV, then the defining ideal of this closure is the homogeneous ideal^[3] of${\displaystyle k[x_0,\dots ,x_n]}$ generated by

${\displaystyle x_0^{\mathrm{deg}(f)}f(x_1/x_0,\dots ,x_n/x_0)}$

for allf inI.

For example, ifV is an affine curve given by, say,${\displaystyle y^2=x^3+ax+b}$ in the affine plane, then its projective completion in the projective plane is given by${\displaystyle y^{2}z=x^3+ax{z}^2+b{z}^3.}$

For various applications, it is necessary to consider more general algebro-geometric objects than projective varieties, namely projective schemes. The first step towards projective schemes is to endow projective space with a scheme structure, in a way refining the above description of projective space as an algebraic variety, i.e.,${\displaystyle {\mathbb{P}}^n(k)}$ is a scheme which it is a union of (n + 1) copies of the affinen-spacekⁿ. More generally,^[4] projective space over a ringA is the union of theaffine schemes

${\displaystyle U_i=\mathrm{Spec}A[x_0/x_i,\dots ,x_n/x_i],0\le i\le n,}$

in such a way the variables match up as expected. The set ofclosed points of${\displaystyle {\mathbb{P}}_k^n}$, for algebraically closed fieldsk, is then the projective space${\displaystyle {\mathbb{P}}^n(k)}$ in the usual sense.

An equivalent but streamlined construction is given by theProj construction, which is an analog of thespectrum of a ring, denoted "Spec", which defines an affine scheme.^[5] For example, ifA is a ring, then

${\displaystyle {\mathbb{P}}_A^n=\mathrm{Proj}A[x_0,\dots ,x_n].}$

IfR is aquotient of${\displaystyle k[x_0,\dots ,x_n]}$ by a homogeneous idealI, then the canonical surjection induces theclosed immersion

${\displaystyle \mathrm{Proj}R\hookrightarrow {\mathbb{P}}_k^n.}$

Compared to projective varieties, the condition that the idealI be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand thetopological space${\displaystyle X=\mathrm{Proj}R}$ may have multipleirreducible components. Moreover, there may benilpotent functions onX.

Closed subschemes of${\displaystyle {\mathbb{P}}_k^n}$ correspond bijectively to the homogeneous idealsI of${\displaystyle k[x_0,\dots ,x_n]}$ that aresaturated; i.e.,${\displaystyle I:(x_0,\dots ,x_n)=I.}$^[6] This fact may be considered as a refined version ofprojective Nullstellensatz.

We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector spaceV overk, we let

${\displaystyle \mathbb{P}(V)=\mathrm{Proj}k[V]}$

where${\displaystyle k[V]=\mathrm{Sym}(V^{\ast })}$ is thesymmetric algebra of${\displaystyle V^{\ast }}$.^[7] It is theprojectivization ofV; i.e., it parametrizes lines inV. There is a canonical surjective map${\displaystyle \pi :V\setminus \{0\}\to \mathbb{P}(V)}$, which is defined using the chart described above.^[8] One important use of the construction is this (cf.,§ Duality and linear system). A divisorD on a projective varietyX corresponds to a line bundleL. One then set

${\displaystyle |D|=\mathbb{P}(\mathrm{\Gamma }(X,L))}$;

it is called thecomplete linear system ofD.

Projective space over anyschemeS can be defined as afiber product of schemes

${\displaystyle {\mathbb{P}}_S^n={\mathbb{P}}_{\mathbb{Z}}^n{\times }_{\mathrm{Spec}\mathbb{Z}}S.}$

If${\displaystyle \mathcal{O}(1)}$ is thetwisting sheaf of Serre on${\displaystyle {\mathbb{P}}_{\mathbb{Z}}^n}$, we let${\displaystyle \mathcal{O}(1)}$ denote thepullback of${\displaystyle \mathcal{O}(1)}$ to${\displaystyle {\mathbb{P}}_S^n}$; that is,${\displaystyle \mathcal{O}(1)=g^{\ast }(\mathcal{O}(1))}$ for the canonical map${\displaystyle g:{\mathbb{P}}_S^n\to {\mathbb{P}}_{\mathbb{Z}}^n.}$

A schemeX →S is calledprojective overS if it factors as a closed immersion

${\displaystyle X\to {\mathbb{P}}_S^n}$

followed by the projection toS.

A line bundle (or invertible sheaf)${\displaystyle \mathcal{L}}$ on a schemeX overS is said to bevery ample relative toS if there is animmersion (i.e., an open immersion followed by a closed immersion)

${\displaystyle i:X\to {\mathbb{P}}_S^n}$

for somen so that${\displaystyle \mathcal{O}(1)}$ pullbacks to${\displaystyle \mathcal{L}}$. Then aS-schemeX is projective if and only if it isproper and there exists a very ample sheaf onX relative toS. Indeed, ifX is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, ifX is projective, then the pullback of${\displaystyle \mathcal{O}(1)}$ under the closed immersion ofX into a projective space is very ample. That "projective" implies "proper" is deeper: themain theorem of elimination theory.

By definition, a variety iscomplete, if it isproper overk. Thevaluative criterion of properness expresses the intuition that in a proper variety, there are no points "missing".

There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:

• Asmooth curveC is projective if and only if it iscomplete. This is proved by identifyingC with the set ofdiscrete valuation rings of thefunction fieldk(C) overk. This set has a natural Zariski topology called theZariski–Riemann space.
• Chow's lemma states that for any complete varietyX, there is a projective varietyZ and abirational morphismZ →X.^[9] (Moreover, throughnormalization, one can assume this projective variety is normal.)

Some properties of a projective variety follow from completeness. For example,

${\displaystyle \mathrm{\Gamma }(X,{\mathcal{O}}_X)=k}$

for any projective varietyX overk.^[10] This fact is an algebraic analogue ofLiouville's theorem (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.

Quasi-projective varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includesaffine varieties. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globallyregular functions on a projective variety.

By definition, any homogeneous ideal in a polynomial ring yields a projective scheme (required to be prime ideal to give a variety). In this sense, examples of projective varieties abound. The following list mentions various classes of projective varieties which are noteworthy since they have been studied particularly intensely. The important class of complex projective varieties, i.e., the case${\displaystyle k=\mathbb{C}}$, is discussed further below.

The product of two projective spaces is projective. In fact, there is the explicit immersion (calledSegre embedding)

${\displaystyle \{\begin{array}{l}{\mathbb{P}}^n\times {\mathbb{P}}^m\to {\mathbb{P}}^{(n+1)(m+1)-1}\\ (x_i,y_j)\mapsto x_i{y}_j\end{array}}$

As a consequence, theproduct of projective varieties overk is again projective. ThePlücker embedding exhibits aGrassmannian as a projective variety.Flag varieties such as the quotient of thegeneral linear group${\displaystyle {\mathrm{G}\mathrm{L}}_n(k)}$ modulo the subgroup of uppertriangular matrices, are also projective, which is an important fact in the theory ofalgebraic groups.^[11]

As the prime idealP defining a projective varietyX is homogeneous, thehomogeneous coordinate ring

${\displaystyle R=k[x_0,\dots ,x_n]/P}$

is agraded ring, i.e., can be expressed as thedirect sum of its graded components:

${\displaystyle R=\underset{n\in \mathbb{N}}{⨁}{R}_n.}$

There exists a polynomialP such that${\displaystyle \dim R_n=P(n)}$ for all sufficiently largen; it is called theHilbert polynomial ofX. It is a numerical invariant encoding some extrinsic geometry ofX. The degree ofP is thedimensionr ofX and its leading coefficient timesr! is thedegree of the varietyX. Thearithmetic genus ofX is (−1)^r (P(0) − 1) whenX is smooth.

For example, the homogeneous coordinate ring of${\displaystyle {\mathbb{P}}^n}$ is${\displaystyle k[x_0,\dots ,x_n]}$ and its Hilbert polynomial is${\displaystyle P(z)=(\genfrac{}{}{0ex}{}{z+n}{n})}$; its arithmetic genus is zero.

If the homogeneous coordinate ringR is anintegrally closed domain, then the projective varietyX is said to beprojectively normal. Note, unlikenormality, projective normality depends onR, the embedding ofX into a projective space. The normalization of a projective variety is projective; in fact, it's the Proj of the integral closure of some homogeneous coordinate ring ofX.

Let${\displaystyle X\subset {\mathbb{P}}^N}$ be a projective variety. There are at least two equivalent ways to define the degree ofX relative to its embedding. The first way is to define it as the cardinality of the finite set

${\displaystyle \mathrm{\#}(X\cap H_1\cap \cdots \cap H_d)}$

whered is the dimension ofX andH_i's are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, ifX is a hypersurface, then the degree ofX is the degree of the homogeneous polynomial definingX. The "general positions" can be made precise, for example, byintersection theory; one requires that the intersection isproper and that the multiplicities of irreducible components are all one.

The other definition, which is mentioned in the previous section, is that the degree ofX is the leading coefficient of theHilbert polynomial ofX times (dimX)!. Geometrically, this definition means that the degree ofX is the multiplicity of the vertex of the affine cone overX.^[12]

Let${\displaystyle V_1,\dots ,V_r\subset {\mathbb{P}}^N}$ be closed subschemes of pure dimensions that intersect properly (they are in general position). Ifm_i denotes the multiplicity of an irreducible componentZ_i in the intersection (i.e.,intersection multiplicity), then the generalization ofBézout's theorem says:^[13]

${\displaystyle \sum _1^s{m}_i\mathrm{deg}{Z}_i=\prod _1^r\mathrm{deg}{V}_i.}$

The intersection multiplicitym_i can be defined as the coefficient ofZ_i in the intersection product${\displaystyle V_1\cdot \cdots \cdot V_r}$ in theChow ring of${\displaystyle {\mathbb{P}}^N}$.

In particular, if${\displaystyle H\subset {\mathbb{P}}^N}$ is a hypersurface not containingX, then

${\displaystyle \sum _1^s{m}_i\mathrm{deg}{Z}_i=\mathrm{deg}(X)\mathrm{deg}(H)}$

whereZ_i are the irreducible components of thescheme-theoretic intersection ofX andH with multiplicity (length of the local ring)m_i.

A complex projective variety can be viewed as acompact complex manifold; the degree of the variety (relative to the embedding) is then the volume of the variety as a manifold with respect to the metric inherited from the ambientcomplex projective space. A complex projective variety can be characterized as a minimizer of the volume (in a sense).

LetX be a projective variety andL a line bundle on it. Then the graded ring

${\displaystyle R(X,L)=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{H}^0(X,L^{\otimes n})}$

is called thering of sections ofL. IfL isample, then Proj of this ring isX. Moreover, ifX is normal andL is very ample, then${\displaystyle R(X,L)}$ is the integral closure of the homogeneous coordinate ring ofX determined byL; i.e.,${\displaystyle X\hookrightarrow {\mathbb{P}}^N}$ so that${\displaystyle {\mathcal{O}}_{{\mathbb{P}}^N}(1)}$ pulls-back toL.^[14]

For applications, it is useful to allow fordivisors (or${\displaystyle \mathbb{Q}}$-divisors) not just line bundles; assumingX is normal, the resulting ring is then called a generalized ring of sections. If${\displaystyle K_X}$ is acanonical divisor onX, then the generalized ring of sections

${\displaystyle R(X,K_X)}$

is called thecanonical ring ofX. If the canonical ring is finitely generated, then Proj of the ring is called thecanonical model ofX. The canonical ring or model can then be used to define theKodaira dimension ofX.

Projective schemes of dimension one are calledprojective curves. Much of the theory of projective curves is about smooth projective curves, since thesingularities of curves can be resolved bynormalization, which consists in taking locally theintegral closure of the ring of regular functions. Smooth projective curves are isomorphic if and only if theirfunction fields are isomorphic. The study of finite extensions of

${\displaystyle {\mathbb{F}}_p(t),}$

or equivalently smooth projective curves over${\displaystyle {\mathbb{F}}_p}$ is an important branch inalgebraic number theory.^[15]

A smooth projective curve of genus one is called anelliptic curve. As a consequence of theRiemann–Roch theorem, such a curve can be embedded as a closed subvariety in${\displaystyle {\mathbb{P}}^2}$. In general, any (smooth) projective curve can be embedded in${\displaystyle {\mathbb{P}}^3}$ (for a proof, seeSecant variety#Examples). Conversely, any smooth closed curve in${\displaystyle {\mathbb{P}}^2}$ of degree three has genus one by thegenus formula and is thus an elliptic curve.

A smooth complete curve of genus greater than or equal to two is called ahyperelliptic curve if there is a finite morphism${\displaystyle C\to {\mathbb{P}}^1}$ of degree two.^[16]

Every irreducible closed subset of${\displaystyle {\mathbb{P}}^n}$ of codimension one is ahypersurface; i.e., the zero set of some homogeneous irreducible polynomial.^[17]

Another important invariant of a projective varietyX is thePicard group${\displaystyle \mathrm{Pic}(X)}$ ofX, the set of isomorphism classes of line bundles onX. It is isomorphic to${\displaystyle H^1(X,{\mathcal{O}}_X^{\ast })}$ and therefore an intrinsic notion (independent of embedding). For example, the Picard group of${\displaystyle {\mathbb{P}}^n}$ is isomorphic to${\displaystyle \mathbb{Z}}$ via the degree map. The kernel of${\displaystyle \mathrm{deg}:\mathrm{Pic}(X)\to \mathbb{Z}}$ is not only an abstract abelian group, but there is a variety called theJacobian variety ofX, Jac(X), whose points equal this group. The Jacobian of a (smooth) curve plays an important role in the study of the curve. For example, the Jacobian of an elliptic curveE isE itself. For a curveX of genusg, Jac(X) has dimensiong.

Varieties, such as the Jacobian variety, which are complete and have a group structure are known asabelian varieties, in honor ofNiels Abel. In marked contrast toaffine algebraic groups such as${\displaystyle G{L}_n(k)}$, such groups are always commutative, whence the name. Moreover, they admit an ampleline bundle and are thus projective. On the other hand, anabelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties andK3 surfaces.

Let${\displaystyle E\subset {\mathbb{P}}^n}$ be a linear subspace; i.e.,${\displaystyle E=\{s_0=s_1=\cdots =s_r=0\}}$ for some linearly independent linear functionalss_i. Then theprojection fromE is the (well-defined) morphism

${\displaystyle \{\begin{array}{l}\varphi :{\mathbb{P}}^n-E\to {\mathbb{P}}^r\\ x\mapsto [s_0(x):\cdots :s_r(x)]\end{array}}$

The geometric description of this map is as follows:^[18]

• We view${\displaystyle {\mathbb{P}}^r\subset {\mathbb{P}}^n}$ so that it is disjoint fromE. Then, for any${\displaystyle x\in {\mathbb{P}}^n\setminus E}$,$${\displaystyle \varphi (x)=W_x\cap {\mathbb{P}}^r,}$$ where${\displaystyle W_x}$ denotes the smallest linear space containingE andx (called thejoin ofE andx.)
• ${\displaystyle {\varphi }^{-1}(\{y_i\ne 0\})=\{s_i\ne 0\},}$ where${\displaystyle y_i}$ are the homogeneous coordinates on${\displaystyle {\mathbb{P}}^r.}$
• For any closed subscheme${\displaystyle Z\subset {\mathbb{P}}^n}$ disjoint fromE, the restriction${\displaystyle \varphi :Z\to {\mathbb{P}}^r}$ is afinite morphism.^[19]

Projections can be used to cut down the dimension in which a projective variety is embedded, up tofinite morphisms. Start with some projective variety${\displaystyle X\subset {\mathbb{P}}^n.}$ If${\displaystyle n>\dim X,}$ the projection from a point not onX gives${\displaystyle \varphi :X\to {\mathbb{P}}^{n-1}.}$ Moreover,${\displaystyle \varphi }$ is a finite map to its image. Thus, iterating the procedure, one sees there is a finite map

${\displaystyle X\to {\mathbb{P}}^d,d=\dim X.}$

This result is the projective analog ofNoether's normalization lemma. (In fact, it yields a geometric proof of the normalization lemma.)

The same procedure can be used to show the following slightly more precise result: given a projective varietyX over a perfect field, there is a finite birational morphism fromX to a hypersurfaceH in${\displaystyle {\mathbb{P}}^{d+1}.}$^[20] In particular, ifX is normal, then it is the normalization ofH.

While a projectiven-space${\displaystyle {\mathbb{P}}^n}$ parameterizes the lines in an affinen-space, thedual of it parametrizes the hyperplanes on the projective space, as follows. Fix a fieldk. By${\displaystyle {\breve{\mathbb{P}}}_k^n}$, we mean a projectiven-space

${\displaystyle {\breve{\mathbb{P}}}_k^n=\mathrm{Proj}(k[u_0,\dots ,u_n])}$

equipped with the construction:

${\displaystyle f\mapsto H_f=\{{\alpha }_0{x}_0+\cdots +{\alpha }_n{x}_n=0\}}$, a hyperplane on${\displaystyle {\mathbb{P}}_L^n}$

where${\displaystyle f:\mathrm{Spec}L\to {\breve{\mathbb{P}}}_k^n}$ is anL-point of${\displaystyle {\breve{\mathbb{P}}}_k^n}$ for a field extensionL ofk and${\displaystyle {\alpha }_i=f^{\ast }(u_i)\in L.}$

For eachL, the construction is a bijection between the set ofL-points of${\displaystyle {\breve{\mathbb{P}}}_k^n}$ and the set of hyperplanes on${\displaystyle {\mathbb{P}}_L^n}$. Because of this, the dual projective space${\displaystyle {\breve{\mathbb{P}}}_k^n}$ is said to be themoduli space of hyperplanes on${\displaystyle {\mathbb{P}}_k^n}$.

A line in${\displaystyle {\breve{\mathbb{P}}}_k^n}$ is called apencil: it is a family of hyperplanes on${\displaystyle {\mathbb{P}}_k^n}$ parametrized by${\displaystyle {\mathbb{P}}_k^1}$.

IfV is a finite-dimensional vector space overk, then, for the same reason as above,${\displaystyle \mathbb{P}(V^{\ast })=\mathrm{Proj}(\mathrm{Sym}(V))}$ is the space of hyperplanes on${\displaystyle \mathbb{P}(V)}$. An important case is whenV consists of sections of a line bundle. Namely, letX be an algebraic variety,L a line bundle onX and${\displaystyle V\subset \mathrm{\Gamma }(X,L)}$ a vector subspace of finite positive dimension. Then there is a map:^[21]

${\displaystyle \{\begin{array}{l}{\phi }_V:X\setminus B\to \mathbb{P}(V^{\ast })\\ x\mapsto H_x=\{s\in V|s(x)=0\}\end{array}}$

determined by the linear systemV, whereB, called thebase locus, is theintersection of the divisors of zero of nonzero sections inV (seeLinear system of divisors#A map determined by a linear system for the construction of the map).

LetX be a projective scheme over a field (or, more generally over a Noetherian ringA).Cohomology of coherent sheaves${\displaystyle \mathcal{F}}$ onX satisfies the following important theorems due to Serre:

1. ${\displaystyle H^p(X,\mathcal{F})}$ is a finite-dimensionalk-vector space for anyp.
2. There exists an integer${\displaystyle n_0}$ (depending on${\displaystyle \mathcal{F}}$; see alsoCastelnuovo–Mumford regularity) such that$${\displaystyle H^p(X,\mathcal{F}(n))=0}$$ for all${\displaystyle n\ge n_0}$ andp > 0, where${\displaystyle \mathcal{F}(n)=\mathcal{F}\otimes \mathcal{O}(n)}$ is the twisting with a power of a very ample line bundle${\displaystyle \mathcal{O}(1).}$

These results are proven reducing to the case${\displaystyle X={\mathbb{P}}^n}$ using the isomorphism

${\displaystyle H^p(X,\mathcal{F})=H^p({\mathbb{P}}^r,\mathcal{F}),p\ge 0}$

where in the right-hand side${\displaystyle \mathcal{F}}$ is viewed as a sheaf on the projective space by extension by zero.^[22] The result then follows by a direct computation for${\displaystyle \mathcal{F}={\mathcal{O}}_{{\mathbb{P}}^r}(n),}$n any integer, and for arbitrary${\displaystyle \mathcal{F}}$ reduces to this case without much difficulty.^[23]

As a corollary to 1. above, iff is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image${\displaystyle R^p{f}_{\ast }\mathcal{F}}$ is coherent. The same result holds for proper morphismsf, as can be shown with the aid ofChow's lemma.

Sheaf cohomology groupsHⁱ on a noetherian topological space vanish fori strictly greater than the dimension of the space. Thus the quantity, called theEuler characteristic of${\displaystyle \mathcal{F}}$,

${\displaystyle \chi (\mathcal{F})=\sum _{i=0}^{\mathrm{\infty }}(-1{)}^i\dim H^i(X,\mathcal{F})}$

is a well-defined integer (forX projective). One can then show${\displaystyle \chi (\mathcal{F}(n))=P(n)}$ for some polynomialP over rational numbers.^[24] Applying this procedure to the structure sheaf${\displaystyle {\mathcal{O}}_X}$, one recovers the Hilbert polynomial ofX. In particular, ifX is irreducible and has dimensionr, the arithmetic genus ofX is given by

${\displaystyle (-1{)}^r(\chi ({\mathcal{O}}_X)-1),}$

which is manifestly intrinsic; i.e., independent of the embedding.

The arithmetic genus of a hypersurface of degreed is${\displaystyle (\genfrac{}{}{0ex}{}{d-1}{n})}$ in${\displaystyle {\mathbb{P}}^n}$. In particular, a smooth curve of degreed in${\displaystyle {\mathbb{P}}^2}$ has arithmetic genus${\displaystyle (d-1)(d-2)/2}$. This is thegenus formula.

LetX be a smooth projective variety where all of its irreducible components have dimensionn. In this situation, thecanonical sheaf ω_X, defined as the sheaf ofKähler differentials of top degree (i.e., algebraicn-forms), is a line bundle.

Serre duality states that for any locally free sheaf${\displaystyle \mathcal{F}}$ onX,

${\displaystyle H^i(X,\mathcal{F})\simeq H^{n-i}(X,{\mathcal{F}}^{\vee }\otimes {\omega }_X{)}^{\prime }}$

where the superscript prime refers to the dual space and${\displaystyle {\mathcal{F}}^{\vee }}$ is the dual sheaf of${\displaystyle \mathcal{F}}$.A generalization to projective, but not necessarily smooth schemes is known asVerdier duality.

For a (smooth projective) curveX,H² and higher vanish for dimensional reason and the space of the global sections of the structure sheaf is one-dimensional. Thus the arithmetic genus ofX is the dimension of${\displaystyle H^1(X,{\mathcal{O}}_X)}$. By definition, thegeometric genus ofX is the dimension ofH⁰(X,ω_X). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus ofX.

Serre duality is also a key ingredient in the proof of theRiemann–Roch theorem. SinceX is smooth, there is an isomorphism of groups

${\displaystyle \{\begin{array}{l}\mathrm{Cl}(X)\to \mathrm{Pic}(X)\\ D\mapsto \mathcal{O}(D)\end{array}}$

from the group of(Weil) divisors modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ω_X is called the canonical divisor and is denoted byK. Letl(D) be the dimension of${\displaystyle H^0(X,\mathcal{O}(D))}$. Then the Riemann–Roch theorem states: ifg is a genus ofX,

${\displaystyle l(D)-l(K-D)=\mathrm{deg}D+1-g,}$

for any divisorD onX. By the Serre duality, this is the same as:

${\displaystyle \chi (\mathcal{O}(D))=\mathrm{deg}D+1-g,}$

which can be readily proved.^[25] A generalization of the Riemann–Roch theorem to higher dimension is theHirzebruch–Riemann–Roch theorem, as well as the far-reachingGrothendieck–Riemann–Roch theorem.

Hilbert schemes parametrize all closed subvarieties of a projective schemeX in the sense that the points (in the functorial sense) ofH correspond to the closed subschemes ofX. As such, the Hilbert scheme is an example of amoduli space, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whoseHilbert polynomial equals a prescribed polynomialP.^[26] It is a deep theorem of Grothendieck that there is a scheme^[27]${\displaystyle H_X^P}$ overk such that, for anyk-schemeT, there is a bijection

${\displaystyle \{\text{morphisms }T\to H_X^P\}⟷\{\text{closed subschemes of }X{\times }_{k}T\text{ flat over }T,\text{ with Hilbert polynomial }P.\}}$

The closed subscheme of${\displaystyle X\times H_X^P}$ that corresponds to the identity map${\displaystyle H_X^P\to H_X^P}$ is called theuniversal family.

For${\displaystyle P(z)=(\genfrac{}{}{0ex}{}{z+r}{r})}$, the Hilbert scheme${\displaystyle H_{{\mathbb{P}}^n}^P}$ is called theGrassmannian ofr-planes in${\displaystyle {\mathbb{P}}^n}$ and, ifX is a projective scheme,${\displaystyle H_X^P}$ is called theFano scheme ofr-planes onX.^[28]

In this section, all algebraic varieties arecomplex algebraic varieties. A key feature of the theory of complex projective varieties is the combination of algebraic and analytic methods. The transition between these theories is provided by the following link: since any complex polynomial is also a holomorphic function, any complex varietyX yields a complexanalytic space, denoted${\displaystyle X(\mathbb{C})}$. Moreover, geometric properties ofX are reflected by the ones of${\displaystyle X(\mathbb{C})}$. For example, the latter is acomplex manifold if and only ifX is smooth; it is compact if and only ifX is proper over${\displaystyle \mathbb{C}}$.

Complex projective space is aKähler manifold. This implies that, for any projective algebraic varietyX,${\displaystyle X(\mathbb{C})}$ is a compact Kähler manifold. The converse is not in general true, but theKodaira embedding theorem gives a criterion for a Kähler manifold to be projective.

In low dimensions, there are the following results:

• (Riemann) Acompact Riemann surface (i.e., compact complex manifold of dimension one) is a projective variety. By theTorelli theorem, it is uniquely determined by its Jacobian.
• (Chow-Kodaira) A compactcomplex manifold of dimension two with two algebraically independentmeromorphic functions is a projective variety.^[29]

Chow's theorem provides a striking way to go the other way, from analytic to algebraic geometry. It states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that aholomorphic function satisfying certain growth condition is necessarily algebraic: "projective" provides this growth condition. One can deduce from the theorem the following:

• Meromorphic functions on the complex projective space are rational.
• If an algebraic map between algebraic varieties is an analyticisomorphism, then it is an (algebraic) isomorphism. (This part is a basic fact in complex analysis.) In particular, Chow's theorem implies that a holomorphic map between projective varieties is algebraic. (consider the graph of such a map.)
• Everyholomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.^[30]
• Every holomorphic line bundle on a projective variety is a line bundle of a divisor.^[31]

Chow's theorem can be shown via Serre'sGAGA principle. Its main theorem states:

LetX be a projective scheme over${\displaystyle \mathbb{C}}$. Then the functor associating the coherent sheaves onX to the coherent sheaves on the corresponding complex analytic spaceX^an is an equivalence of categories. Furthermore, the natural maps

${\displaystyle H^i(X,\mathcal{F})\to H^i(X^{\text{an}},\mathcal{F})}$

are isomorphisms for alli and all coherent sheaves${\displaystyle \mathcal{F}}$ onX.^[32]

The complex manifold associated to an abelian varietyA over${\displaystyle \mathbb{C}}$ is a compactcomplex Lie group. These can be shown to be of the form

${\displaystyle {\mathbb{C}}^g/L}$

and are also referred to ascomplex tori. Here,g is the dimension of the torus andL is a lattice (also referred to asperiod lattice).

According to theuniformization theorem already mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from anelliptic curve. In fact, theWeierstrass's elliptic function${\displaystyle \mathrm{\wp }}$ attached toL satisfies a certain differential equation and as a consequence it defines a closed immersion:^[33]

${\displaystyle \{\begin{array}{l}\mathbb{C}/L\to {\mathbb{P}}^2\\ L\mapsto (0:0:1)\\ z\mapsto (1:\mathrm{\wp }(z):{\mathrm{\wp }}^{\prime }(z))\end{array}}$

There is ap-adic analog, thep-adic uniformization theorem.

For higher dimensions, the notions of complex abelian varieties and complex tori differ: onlypolarized complex tori come from abelian varieties.

The fundamentalKodaira vanishing theorem states that for an ample line bundle${\displaystyle \mathcal{L}}$ on a smooth projective varietyX over a field of characteristic zero,

${\displaystyle H^i(X,\mathcal{L}\otimes {\omega }_X)=0}$

fori > 0, or, equivalently by Serre duality${\displaystyle H^i(X,{\mathcal{L}}^{-1})=0}$ fori <n.^[34] The first proof of this theorem used analytic methods of Kähler geometry, but a purely algebraic proof was found later. The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic. Kodaira's theorem is one of various vanishing theorems, which give criteria for higher sheaf cohomologies to vanish. Since the Euler characteristic of a sheaf (see above) is often more manageable than individual cohomology groups, this often has important consequences about the geometry of projective varieties.^[35]

• Multi-projective variety
• Weighted projective variety, a closed subvariety of aweighted projective space^[36]

• Algebraic geometry of projective spaces
• Adequate equivalence relation
• Hilbert scheme
• Lefschetz hyperplane theorem
• Minimal model program

• The Hilbert Scheme by Charles Siegel - a blog post
• Projective varieties Ch. 1

• Algebraic geometry
• Algebraic varieties
• Projective geometry

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