Editing Projective variety (section)

== Examples and basic invariants ==

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 <math>k=\Complex</math>, is discussed further below.

The product of two projective spaces is projective. In fact, there is the explicit immersion (called [[Segre embedding]])

:<math>\begin{cases}
\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{cases}</math>

As a consequence, the [[fiber product of schemes|product]] of projective varieties over ''k'' is again projective. The [[Plücker embedding]] exhibits a [[Grassmannian]] as a projective variety. [[Generalized flag variety|Flag varieties]] such as the quotient of the [[general linear group]] <math>\mathrm{GL}_n(k)</math> modulo the subgroup of upper [[triangular matrices]], are also projective, which is an important fact in the theory of [[algebraic group]]s.<ref>{{citation|author=Humphreys|first=James|title=Linear algebraic groups|publisher=Springer|year=1981}}, Theorem 21.3</ref>

=== Homogeneous coordinate ring and Hilbert polynomial ===
{{main|Hilbert series and Hilbert polynomial}}
As the prime ideal ''P'' defining a projective variety ''X'' is homogeneous, the [[homogeneous coordinate ring]]

:<math>R = k[x_0, \dots, x_n] / P</math>

is a [[graded ring]], i.e., can be expressed as the [[direct sum]] of its graded components:

:<math>R = \bigoplus_{n \in \N} R_n.</math>

There exists a polynomial ''P'' such that <math>\dim R_n = P(n)</math> for all sufficiently large ''n''; it is called the [[Hilbert polynomial]] of ''X''. It is a numerical invariant encoding some extrinsic geometry of ''X''. The degree of ''P'' is the [[dimension of an algebraic variety|dimension]] ''r'' of ''X'' and its leading coefficient times '''r!''' is the [[degree of an algebraic variety|degree]] of the variety ''X''. The [[arithmetic genus]] of ''X'' is (−1)<sup>''r''</sup> (''P''(0) − 1) when ''X'' is smooth.

For example, the homogeneous coordinate ring of <math>\mathbb{P}^n</math> is <math>k[x_0, \ldots, x_n]</math> and its Hilbert polynomial is <math>P(z) = \binom{z+n}{n}</math>; its arithmetic genus is zero.

If the homogeneous coordinate ring ''R'' is an [[integrally closed domain]], then the projective variety ''X'' is said to be [[projectively normal]]. Note, unlike [[normal variety|normality]], projective normality depends on ''R'', the embedding of ''X'' 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 of ''X''.

=== Degree ===
{{main|Degree of an algebraic variety|Hilbert series and Hilbert polynomial}}
Let <math>X \subset \mathbb{P}^N</math> be a projective variety. There are at least two equivalent ways to define the degree of ''X'' relative to its embedding. The first way is to define it as the cardinality of the finite set

:<math>\# (X \cap H_1 \cap \cdots \cap H_d)</math>

where ''d'' is the dimension of ''X'' and ''H''<sub>''i''</sub>'s are hyperplanes in "general positions". This definition corresponds to an intuitive idea of a degree. Indeed, if ''X'' is a hypersurface, then the degree of ''X'' is the degree of the homogeneous polynomial defining ''X''. The "general positions" can be made precise, for example, by [[intersection theory]]; one requires that the intersection is [[proper intersection|proper]] and that the multiplicities of irreducible components are all one.

The other definition, which is mentioned in the previous section, is that the degree of ''X'' is the leading coefficient of the [[Hilbert polynomial]] of ''X'' times (dim ''X'')!. Geometrically, this definition means that the degree of ''X'' is the multiplicity of the vertex of the affine cone over ''X''.<ref>{{harvnb|Hartshorne|1977|loc=Ch. V, Exercise 3.4. (e).}}</ref>

Let <math>V_1, \dots, V_r \subset \mathbb{P}^N</math> be closed subschemes of pure dimensions that intersect properly (they are in general position). If ''m<sub>i</sub>'' denotes the multiplicity of an irreducible component ''Z<sub>i</sub>'' in the intersection (i.e., [[intersection multiplicity]]), then the generalization of [[Bézout's theorem]] says:<ref>{{harvnb|Fulton|1998|loc=Proposition 8.4.}}</ref>

:<math>\sum_1^s m_i \deg Z_i = \prod_1^r \deg V_i.</math>

The intersection multiplicity ''m<sub>i</sub>'' can be defined as the coefficient of ''Z<sub>i</sub>'' in the intersection product <math>V_1 \cdot \cdots \cdot V_r</math> in the [[Chow ring]] of <math>\mathbb{P}^N</math>.

In particular, if <math>H \subset \mathbb{P}^N</math> is a hypersurface not containing ''X'', then

:<math>\sum_1^s m_i \deg Z_i = \deg(X) \deg(H)</math>

where ''Z<sub>i</sub>'' are the irreducible components of the [[scheme-theoretic intersection]] of ''X'' and ''H'' with multiplicity (length of the local ring) ''m<sub>i</sub>''.

A complex projective variety can be viewed as a [[compact 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 ambient [[complex projective space]]. A complex projective variety can be characterized as a minimizer of the volume (in a sense).

=== The ring of sections ===
Let ''X'' be a projective variety and ''L'' a line bundle on it. Then the graded ring

:<math>R(X, L) = \bigoplus_{n=0}^{\infty} H^0(X, L^{\otimes n})</math>

is called the [[ring of sections]] of ''L''. If ''L'' is [[ample line bundle|ample]], then Proj of this ring is ''X''. Moreover, if ''X'' is normal and ''L'' is very ample, then <math>R(X,L)</math> is the integral closure of the homogeneous coordinate ring of ''X'' determined by ''L''; i.e., <math>X \hookrightarrow \mathbb{P}^N</math> so that <math>\mathcal{O}_{\mathbb{P}^N}(1)</math> pulls-back to ''L''.<ref>{{harvnb|Hartshorne|1977|loc=Ch. II, Exercise 5.14. (a)}}</ref>

For applications, it is useful to allow for [[divisor (algebraic geometry)|divisor]]s (or <math>\Q</math>-divisors) not just line bundles; assuming ''X'' is normal, the resulting ring is then called a generalized ring of sections. If <math>K_X</math> is a [[canonical divisor]] on ''X'', then the generalized ring of sections

:<math>R(X, K_X)</math>

is called the [[canonical ring]] of ''X''. If the canonical ring is finitely generated, then Proj of the ring is called the [[canonical model]] of ''X''. The canonical ring or model can then be used to define the [[Kodaira dimension]] of ''X''.

=== Projective curves ===
{{Further|Algebraic curve}}
Projective schemes of dimension one are called ''projective curves''. Much of the theory of projective curves is about smooth projective curves, since the [[Singular point of an algebraic variety|singularities]] of curves can be resolved by [[Normalization of an algebraic variety|normalization]], which consists in taking locally the [[integral closure]] of the ring of regular functions. Smooth projective curves are isomorphic if and only if their [[Function field of an algebraic variety|function fields]] are isomorphic. The study of finite extensions of

:<math>\mathbb F_p(t),</math>

or equivalently smooth projective curves over <math>\mathbb F_p</math> is an important branch in [[algebraic number theory]].<ref>{{citation|author=Rosen|first=Michael|title=Number theory in Function Fields|year=2002|publisher=Springer}}</ref>

A smooth projective curve of genus one is called an [[elliptic curve]]. As a consequence of the [[Riemann–Roch theorem]], such a curve can be embedded as a closed subvariety in <math>\mathbb{P}^2</math>. In general, any (smooth) projective curve can be embedded in <math>\mathbb{P}^3</math> (for a proof, see [[Secant variety#Examples]]). Conversely, any smooth closed curve in <math>\mathbb{P}^2</math> of degree three has genus one by the [[genus formula]] and is thus an elliptic curve.

A smooth complete curve of genus greater than or equal to two is called a [[hyperelliptic curve]] if there is a finite morphism <math>C \to \mathbb{P}^1</math> of degree two.<ref>{{harvnb|Hartshorne|1977|loc=Ch IV, Exercise 1.7.}}</ref>

=== Projective hypersurfaces ===

Every irreducible closed subset of <math>\mathbb{P}^n</math> of codimension one is a [[hypersurface]]; i.e., the zero set of some homogeneous irreducible polynomial.<ref>{{harvnb|Hartshorne|1977|loc=Ch I, Exercise 2.8}}; this is because the homogeneous coordinate ring of <math>\mathbb{P}^n</math> is a [[unique factorization domain]] and in a UFD every prime ideal of height 1 is principal.</ref>

=== Abelian varieties ===
Another important invariant of a projective variety ''X'' is the [[Picard group]] <math>\operatorname{Pic}(X)</math> of ''X'', the set of isomorphism classes of line bundles on ''X''. It is isomorphic to <math>H^1(X, \mathcal O_X^*)</math> and therefore an intrinsic notion (independent of embedding). For example, the Picard group of <math>\mathbb{P}^n</math> is isomorphic to <math>\Z</math> via the degree map. The kernel of <math>\deg: \operatorname{Pic}(X) \to \Z</math> is not only an abstract abelian group, but there is a variety called the [[Jacobian variety]] of ''X'', 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 curve ''E'' is ''E'' itself. For a curve ''X'' of genus ''g'', Jac(''X'') has dimension ''g''.

Varieties, such as the Jacobian variety, which are complete and have a group structure are known as [[abelian variety|abelian varieties]], in honor of [[Niels Abel]]. In marked contrast to [[affine algebraic group]]s such as <math>GL_n(k)</math>, such groups are always commutative, whence the name. Moreover, they admit an ample [[line bundle]] and are thus projective. On the other hand, an [[abelian scheme]] may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and [[K3 surface]]s.