Projective variety

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An elliptic curve is a smooth projective curve of genus one.

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of Pn. A Zariski open subvariety of a projective variety is called a quasi-projective variety.

If X is a projective variety defined by a homogeneous prime ideal I, then the quotient ring

k[x_0, \ldots, x_n]/I

is called the homogeneous coordinate ring of X. The ring comes with the Hilbert polynomial P, an important invariant (depending on embedding) of X. The degree of P is the topological dimension r of X and its leading coefficient times r! is the degree of the variety X. The arithmetic genus of X is (−1)r (P(0) − 1) when X is smooth. For example, the homogeneous coordinate ring of Pn is k[x_0, \ldots, x_n] and its Hilbert polynomial is P(z) = \binom{z+n}{n}; its arithmetic genus is zero.

Another important invariant of a projective variety X is the Picard group \operatorname{Pic}(X) of X, the set of isomorphism classes of line bundles on X. It is isomorphic to H^1(X, {\mathcal{O}_X}^*). It is an intrinsic notion (independent of embedding). For example, the Picard group of Pn is isomorphic to Z via the degree map. The kernel of \operatorname{deg}: \operatorname{Pic}(X) \to \mathbf{Z} is called the Jacobian variety of X. The Jacobian of a (smooth) curve plays an important role in the study of the curve.

The classification program, classical and modern, naturally leads to the construction of moduli of projective varieties.[1] A Hilbert scheme, which is a projective scheme, is used to parametrize closed subschemes of Pn with the prescribed Hilbert polynomial. For example, a Grassmannian \mathbb{G}(k, n) is a Hilbert scheme with the specific Hilbert polynomial. The geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.

For complex projective varieties, there is a marriage of algebraic and complex-analytic approaches. Chow's theorem says that a subset of the projective space is the zero-locus of a family of holomorphic functions if and only if it is the zero-locus of homogeneous polynomials. (A corollary of this is that a "compact" complex space admits at most one variety structure.) The GAGA says that the theory of holomorphic vector bundles (more generally coherent analytic sheaves) on X coincide with that of algebraic vector bundles.

Examples[edit]

\mathbf{P}^n \times \mathbf{P}^m \to \mathbf{P}^{(n+1)(m+1)-1}, (x_i, y_j) \mapsto x_iy_j (lexicographical order).
It follows from this that the fibered product of projective varieties is also projective.
  • Every irreducible closed subset of Pn of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial.[2]
  • The arithmetic genus of a hypersurface of degree d is \binom{d-1}{n} in \mathbf{P}^n. In particular, a smooth curve of degree d in P2 has arithmetic genus (d-1)(d-2)/2. This is the genus formula.
  • A smooth curve is projective if and only if it is complete. This is because of the following consideration. If F is the function field of a smooth projective curve C (called the algebraic function field), then C may be identified with the set of discrete valuation rings of F over k and this set has a natural Zariski topology called the Zariski–Riemann space. See also algebraic curve for more specific examples of curves.
  • A smooth complete curve of genus one is called an elliptic curve. By an argument with the Riemann-Roch theorem, one can show that such a curve can be embedded as a closed subvariety in P2. (In general, any (smooth complete) curve can be embedded in P3.) Conversely, any smooth closed curve in P2 of degree three has genus one by the genus formula and is thus an elliptic curve. An elliptic curve is isomorphic to its own Jacobian and thus an abelian variety.
  • A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism C \to \mathbf{P}^1 of degree two.[3]
  • An abelian variety (i.e., a complete group variety) admits an ample line bundle and thus projective. On the other hand, an abelian scheme may not be projective. Examples of abelian varieties are elliptic curves, Jacobian varieties and K3 surfaces.
  • Some (but not all) complex tori are projective. A complex torus is of the form \mathbb{C}^g/L (period lattice construction) as a complex Lie group where L is a lattice and g is the complex dimension of the torus. Suppose g=1. Let \wp be the Weierstrass's elliptic function. The function satisfies a certain differential equation and as a consequence it defines a closed immersion:
    \mathbb{C}/L \to \mathbf{P}^2, L \mapsto (0:0:1), z \mapsto (1 : \wp(z) : \wp'(z))
for some lattice L.[4] Thus, \mathbb{C}/L is an elliptic curve. The uniformization theorem implies that every complex elliptic curve arises in this way. The case g > 1 is more complicated; it is a matter of polarization. (cf. Lefschetz's embedding theorem.) By the p-adic uniformization, the case g = 1 has a p-adic analog.

Variety and scheme structure[edit]

Variety structure[edit]

Let k be an algebraically closed field. Given a homogeneous prime ideal P of k[x_0, ..., x_n], let X be a subset of Pn(k) consisting of all roots of polynomials in P.[6] Here we show X admits a structure of variety by showing locally it is an affine variety.[7] Let

R = k[x_0, ..., x_n]/P

i.e., R is the homogeneous coordinate ring of X. The localization of R with respect to nonzero homogeneous elements is graded; let k(X) denote its zeroth piece. It is the function field of X. Explicitly, k(X) consists of zero and f/g, with f, g homogeneous of the same degree, inside the field of fractions of R. For each x in X, let \mathcal{O}_x \subset k(X) be the subring consisting of zero and f/g with g(x) ≠ 0; it is a local ring.

Now define the sheaf \mathcal{O}_X on X by: for each Zariski open subset U,

\mathcal{O}_X(U) = \bigcap_{x \in U} \mathcal{O}_x.

The stalk of \mathcal{O}_X at x in X is then \mathcal{O}_x.[8] We have thus constructed the locally ringed space (X, \mathcal{O}_X). We shall show it is locally an affine variety. For that, it is enough to show:

(U_i, \mathcal{O}_X|{U_i}), \quad U_i=\{ (x_0:x_1:\cdots:x_n) \in X | x_i \ne 0 \}

are affine varieties. For simplicity, we consider only the case i = 0. Let P′ be the kernel of

k[y_1, \dots, y_n] \to k(X), \quad y_i \mapsto x_i/x_0

and let X′ be the zero-locus of P′ in kn. X′ is an affine variety. We then verify

\phi: U_0 \to X', \quad (1:x_1:...:x_n) \mapsto (x_1, \dots, x_n)

is an isomorphism of ringed spaces. More specifically, we check:

(i) φ is a homeomorphism (by looking at closed subsets.)
(ii) \phi^{\#}: \mathcal{O}_{\phi(x)} \overset{\sim}\to \mathcal{O}_x, \, s \mapsto s \circ \phi (by noticing \phi^{\#}: k(X') \overset{\sim}\to k(X).)

Projective schemes[edit]

The discussion in the preceding section applies in particular to the projective space Pn(k); i.e., it is a union of (n + 1) copies of the affine n-space kn in such a way ringed space structures are compatible. This motivates the following definition:[9] for any ring A we let \mathbf{P}^n_A be the scheme that is the union of

U_i = \operatorname{Spec} A[x_1/x_i, \dots, x_n/x_i], \quad 0 \le i \le n,

in such a way the variables match up as expected. The set of closed points of \mathbf{P}^n_k is then the projective space Pn(k) in the usual sense.

An equivalent but streamlined construction is given by the Proj construction, which is an analog of the spectrum of a ring, denoted "Spec", which defines an affine scheme. For example, if A is a ring, then

\mathbf{P}^n_A = \operatorname{Proj}A[x_0, \ldots, x_n].

If R is a quotient of k[x_0, \ldots, x_n] by a homogeneous ideal, then the canonical surjection induces the closed immersion

\operatorname{Proj} R \to \mathbf{P}^n_k.

In fact, one has the following: every closed subscheme of \mathbf{P}^n_k corresponds bijectively to a homogeneous ideal I of k[x_0, \ldots, x_n] that is saturated; i.e., I : (x_0, \dots, x_n) = I.[10] This fact may be considered as a refined version of projective Nullstellensatz.

We can give a coordinate-free analog of the above. Namely, given a finite-dimensional vector space V over k, we let

\mathbf{P}(V) = \operatorname{Proj} k[V]

where k[V] = \operatorname{Sym}(V^*) is the symmetric algebra of V^*.[11] It is the projectivization of V; i.e., it parametrizes lines in V. There is a canonical surjective map \pi: V - 0 \to \mathbf{P}(V), which is defined using the chart described above.[12] One important use of the construction is this (for more of this see below). A divisor D on a projective variety X corresponds to a line bundle L. One then set

|D| = \mathbf{P}(\Gamma(X, L));

it is called the complete linear system of D.

For any noetherian scheme S, we let

\mathbf{P}^n_S = \mathbf{P}_\mathbf{Z}^n \times_{\operatorname{Spec}\mathbf{Z}} S.

If \mathcal{O}(1) is the twisting sheaf of Serre on \mathbf{P}_\mathbf{Z}^n, we let \mathcal{O}(1) denote the pullback of \mathcal{O}(1) to \mathbf{P}^n_S; that is, \mathcal{O}(1) = g^*(\mathcal{O}(1)) for the canonical map g: \mathbf{P}^n_{S} \to \mathbf{P}^n_{\mathbf{Z}}.

A scheme XS is called projective over S if it factors as a closed immersion

X \to \mathbf{P}^n_S

followed by the projection to S.

In general, a line bundle (or invertible sheaf) \mathcal{L} on a scheme X over S is said to be very ample relative to S if there is an immersion

i: X \to \mathbf{P}^n_S

for some n so that \mathcal{O}(1) pullbacks to \mathcal{L}. (An immersion is an open immersion followed by a closed immersion.) Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if X is projective, then the pullback of \mathcal{O}(1) under the closed immersion of X into a projective space is very ample. That "projective" implies "proper" is more difficult: the main theorem of elimination theory.

A complete variety (i.e., proper over k) is close to being a projective variety: Chow's lemma states that if X is a complete variety, there is a projective variety Z and a birational morphism ZX. (Moreover, through normalization, one can assume this projective variety is normal.) Some properties of a projective variety follow from completeness. For example, if X is a projective variety over k, then \Gamma(X, \mathcal{O}_X) = k.[13]

In general, a line bundle is called ample if some power of it is very ample. Thus, a variety is projective if and only if it is complete and it admis an ample line bundle. An issue of an embedding of a variety into a projective space is discussed in greater details in the following section.

Line bundle and divisors[edit]

Main article: Ample line bundle

We begin with a preliminary on a morphism into a projective space.[14] Let X be a scheme over a ring A. Suppose there is a morphism

\phi: X \to \mathbf{P}^n_A = \operatorname{Proj} A[x_1, \dots, x_n].

Then, along this map, \mathcal{O}(1) pulls-back to a line bundle L on X. L is then generated by the global sections \phi^*(x_i). Conversely, suppose L is generated by global sections s_0, ..., s_n. They define a morphism \phi: X \to \mathbf{P}^n_A as follows: let X_i and U_i be the complements of s_i = 0 in X and x_i = 0 in  \mathbf{P}^n_A (i.e., U_i are the standard open affine chart described above.) Let \phi: X_i \to U_i be given by x_i/x_j \mapsto s_i/s_j. It is then immediate that L is isomorphic to \phi^*(\mathcal{O}(1)) and s_i = \phi^*(x_i). Furthermore, \phi is a closed immersion if and only if X_i are affine and \Gamma(U_i, \mathcal{O}_{\mathbf{P}^n_A}) \to \Gamma(X_i, \mathcal{O}_{X_i}) are surjective.[15]

Let \mathcal{M}_X be the sheaf on X associated with U \mapsto the total ring of fractions of \Gamma(U, \mathcal{O}_X). A global section of \mathcal{M}_X^*/\mathcal{O}_X^* (* means multiplicative group) is called a Cartier divisor on X. The notion actually adds nothing new: there is the canonical bijection

D \mapsto \mathcal{L}(D)

from the set of all Cartier divisors on X to the set of all line bundles on X.

Coherent sheaves[edit]

Main article: coherent sheaf

Let X be a projective scheme over a field (possibly finite) k with very ample line bundle \mathcal{O}(1). Let \mathcal{F} be a coherent sheaf on it. Let i: X \to \mathbf{P}^r_A be the closed immersion. Then the cohomology of X can be computed from that of the projective space:

H^p(X, \mathcal{F}) = H^p(\mathbf{P}^r_A, \mathcal{F}), p \ge 0

where in the right-hand side \mathcal{F} is viewed as a sheaf on the projective space by extension by zero.[16] One can then deduce the following results due to Serre: let \mathcal{F}(n) = \mathcal{F} \otimes \mathcal{O}(n)

  • (a) H^p(X, \mathcal{F}) is a finite-dimensional k-vector space for any p.
  • (b) There exists an n_0 (depending on \mathcal{F}) such that
    H^p(X, \mathcal{F}(n)) = 0
for all n \ge n_0 and p.

Indeed, we can assume X is the projective space by the early discussion. Then this can be seen by a direct computation for \mathcal{F} = \mathcal{O}_{\mathbf{P}^r}(n), n any integer, and the general case reduces to this case without much difficulty.[17]

An analogous statement is true for X over a noetherian ring by the same argument. As a corollay to (a) bis, if f is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image R^p f_* \mathcal{F} is coherent. This is a special case of a more general case: f proper. But the general case follows from the projective case with the aid of Chow's lemma.

It is a feature of sheaf cohomology on a noetherian topological space that Hi vanishes for i strictly greater than the dimension of the space. Thus, in view of the above discussion, the quantity

\chi(\mathcal{F}) = \sum_{i=0}^\infty (-1)^i \operatorname{dim} H^i(X, \mathcal{F})

is a well-defined integer. It is called the Euler characteristic of \mathcal{F}. Then H^i(X, \mathcal{F}(n)) all vanish for sufficiently large n. One can then show \chi(\mathcal{F}(n)) = P(n) for some polynomial P over rational numbers.[18] Applying this procedure to the structure sheaf \mathcal{O}_X, one recovers the Hilbert polynomial of X. In particular, if X is irreducible and has dimension r, the arithmetic genus of X is given by

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

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

Smooth projective varieties[edit]

Let X be a smooth projective variety where all of its irreducible components have dimension n. Then one has the following version of the Serre duality: for any locally free sheaf \mathcal{F} on X,

H^i(X, \mathcal{F}) \simeq H^{n-i}(X, \mathcal{F}^\vee \otimes \omega_X)'

where the superscript prime refers to the dual space, ωX is the canonical sheaf and \mathcal{F}^\vee is the dual sheaf of \mathcal{F}.

Now, assume X is a curve (but still projective and nonsingular). Then 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 of X is the dimension of H^1(X, \mathcal{O}_X). By definition, the geometric genus of X is the dimension of H0(X, ωX). It thus follows from the Serre duality that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of X.

The Serre duality is also a key ingredient in the proof of the Riemann–Roch theorem. Since X is smooth, there is an isomorphism of groups

\operatorname{Cl}(X) \to \operatorname{Pic}(X), D \mapsto \mathcal{O}(D)

from the group of (Weil) divisors modulo principal divisors to the group of isomorphism classes of line bundles. A divisor corresponding to ωX is called the canonical divisor and is denoted by K. Let l(D) be the dimension of H^0(X, \mathcal{O}(D)). Then the Riemann–Roch theorem states: if g is a genus of X,

l(D) -l(K - D) = \operatorname{deg} D + 1 - g

for any divisor D on X. By the Serre duality, this is the same as:

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

which can be readily proved.[19]

For complex smooth projecive varieties, one of fundamental results is the Kodaira vanishing theorem, which states the following:[20]

Let X be a projective nonsingular variety of dimension n over C and \mathcal{L} an ample line bundle. Then
  1. H^i(X, \mathcal{L}\otimes \omega_X) = 0 for i > 0,
  2. H^i(X, \mathcal{L}^{-1}) = 0 for i < n.

The Kodaira vanishing in general fails for a smooth projective variety in positive characteristic.

Hilbert schemes[edit]

Main article: Hilbert scheme

(In this section, schemes mean locally noetherian schemes.)

Suppose we want to parametrize all closed subvarieties of a projective scheme X. The idea is to construct a scheme H so that each "point" (in the functorial sense) of H corresponds to a closed subscheme of X. (To make the construction to work, one needs to allow for a non-variety.) Such a scheme is called a Hilbert scheme. It is a deep theorem of Grothendieck that a Hilbert scheme exists at all. Let S be a scheme. One version of the theorem states that,[21] given a projective scheme X over S and a polynomial P, there exists a projective scheme H_X^P over S such that, for any S-scheme T,

to give a T-point of H_X^P; i.e., a morphism T \to H^P_X is the same as to give a closed subscheme of X \times_S T flat over T with Hilbert polynomial P.

The closed subscheme of X \times_S H_X^P that corresponds to the identity map H_X^P \to H_X^P is called the universal family.

Examples:

  • If P(z) = \binom{z+k}{k}, then H_{\mathbf{P}^n_S}^P is called the Grassmannian of k-planes in \mathbf{P}^n_S and, if X is a projective scheme over S, H_X^P is called the Fano scheme of k-planes on X.[22]

Complex projective varieties[edit]

In this section, all algebraic varieties are complex algebraic varieties.

One of the fundamental results here is Chow's theorem, which states that every analytic subvariety of a complex projective space is algebraic. The theorem may be interpreted to saying that a holomorphic 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 analytic isomorphism, 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.)
  • Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.[23]
  • Every holomorphic line bundle on a projective variety is a line bundle of a divisor.[24]

Chow's theorem is an instance of GAGA. Its main theorem due to Serre states:

Let X be a projective scheme over C. Then the functor associating the coherent sheaves on X to the coherent sheaves on the corresponding complex analytic space Xan is an equivalence of categories. Furthermore, the natural maps
H^i(X, \mathcal{F}) \to H^i(X^\text{an}, \mathcal{F})
are isomorphisms for all i and all coherent sheaves \mathcal{F} on X.[25]

In particular, the theorem gives a proof of Chow's theorem.

See also[edit]

Notes[edit]

  1. ^ Kollár Moduli, Ch I.
  2. ^ Hartshorne 1977, Ch I, Exercise 2.8; this is because the homogeneous coordinate ring of Pn is a unique factorization domain and in a UFD every prime ideal of height 1 is principal.
  3. ^ Hartshorne & 1977 Ch IV, Exercise 1.7.
  4. ^ Mumford 1970, pg. 36
  5. ^ Hartshorne, Appendix B. Theorem 3.4.
  6. ^ The definition makes sense since f(x_0, ..., x_n) = 0 if and only if f(\lambda x_0, ..., \lambda x_n) = 0 for any nonzero λ in k.
  7. ^ The construction follows Mumford's red book.
  8. ^ This follows from: for the complement Xf of homogeneous f = 0, \mathcal{O}_X(X_f) is the zeroth piece of the localization Rf. The proof uses the projective Nullstellensatz; cf. (Mumford 1999, pg. 20)
  9. ^ Mumford 1999, pg. 82
  10. ^ Mumford 1999, pg. 111
  11. ^ This definition differs from Eisenbud–Harris 2000, III.2.3 but is consistent with the other parts of Wikipedia.
  12. ^ cf. the proof of Hartshorne 1977, Ch II, Theorem 7.1
  13. ^ Hartshorne 1977, Ch II. Exercise 4.5
  14. ^ Hartshorne 1977, Ch II, Theorem 7.1
  15. ^ Hartshorne 1977, Ch II, Proposition 7.2
  16. ^ This is not difficult:(Hartshorne 1977, Ch III. Lemma 2.10) consider a flasque resolution of \mathcal{F} and its zero-extension to the whole projective space.
  17. ^ Hartshorne 1977, Ch III. Theorem 5.2
  18. ^ Hartshorne 1977, Ch III. Exercise 5.2
  19. ^ Hartshorne 1977, Ch IV. Theorem 1.3
  20. ^ Hartshorne, Ch III. Remark 7.5.
  21. ^ Kollár 1996, Ch I 1.4
  22. ^ Eisenbud–Harris 2000, VI 2.2
  23. ^ Griffiths-Adams, IV. 1. 10. Corollary H
  24. ^ Griffiths-Adams, IV. 1. 10. Corollary I
  25. ^ Hartshorne, Appendix B. Theorem 2.1

References[edit]

External links[edit]