Projective variety

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. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of Pn.

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

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

is called the homogeneous coordinate ring of X. Basic invariants of X such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring.

Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there are no points "missing". The converse is not true in general, but Chow's lemma describes the close relation of these two notions. Showing that a variety is projective is done by studying line bundles or divisors on X.

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 of Poincaré duality. It also leads to the Riemann-Roch theorem for projective curves, i.e., projective varieties of dimension 1. The theory of projective curves is particularly rich, including a classification by the genus 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 Pn with prescribed Hilbert polynomial. Hilbert schemes, of which Grassmannians are special cases, are also projective schemes in their own right. Geometric invariant theory offers another approach. The classical approaches include the Teichmüller space and Chow varieties.

A particularly rich theory, reaching back to the classics, is available for complex projective varieties, i.e., when the polynomials defining X have complex coefficients. Broadly, the GAGA 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 of holomorphic vector bundles (more generally coherent analytic sheaves) on X 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 as Hodge theory.

Variety and scheme structure

Variety structure

Let k be an algebraically closed field. The basis of the definition of projective varieties is projective space Pn, which can be defined in different, but equivalent ways:

• as the set of all lines through the origin in kn+1 (i.e., one-dimensional sub-vector spaces of kn+1)
• as the set of tuples , modulo the equivalence relation
for any . The equivalence class of such a tuple is denoted by
and referred to as a homogeneous coordinate.

A projective variety is, by definition, a closed subvariety of Pn, where closed refers to the Zariski topology.[2] In general, closed subsets of the Zariski topology are defined to be the zero-locus of polynomial functions. Given a polynomial , the condition

does not make sense for arbitrary polynomials, but only if f is homogeneous, i.e., the total degree of all the monomials (whose sum is f) is the same. In this case, the vanishing of

is independent of the choice of .

Therefore, projective varieties arise from homogeneous prime ideals P of , and setting

.

X can be shown to be an algebraic variety by showing that the intersection of X with the affine subspaces

is an affine variety.

Projective schemes

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., Pn(k) is a scheme which it is a union of (n + 1) copies of the affine n-space kn. More generally,[3] projective space over a ring A is the union of the affine schemes

in such a way the variables match up as expected. The set of closed points of , for algebraically closed fields 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.[4] For example, if A is a ring, then

If R is a quotient of by a homogeneous ideal I, then the canonical surjection induces the closed immersion

Compared to projective varieties, the condition that the ideal I be a prime ideal was dropped. This leads to a much more flexible notion: on the one hand the topological space may have multiple irreducible components. Moreover, there may be nilpotent functions on X.

Closed subschemes of correspond bijectively to the homogeneous ideals I of that is saturated; i.e., .[5] 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

where is the symmetric algebra of .[6] It is the projectivization of V; i.e., it parametrizes lines in V. There is a canonical surjective map , which is defined using the chart described above.[7] 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

;

it is called the complete linear system of D.

Projective space over a noetherian scheme S is defined as a fiber product

If is the twisting sheaf of Serre on , we let denote the pullback of to ; that is, for the canonical map

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

followed by the projection to S.

Relation to complete varieties

By definition, a variety is complete, if it is proper over k. The valuative 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:

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

for any projective variety X over k.[9] This fact is an algebraic analogue of Liouville'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 includes affine 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 globally regular functions on a projective variety.

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 , is discussed further below.

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

As a consequence, the fiber product of projective varieties is again projective. The Plücker embedding exhibits a Grassmannian as a projective variety. Flag varieties such as the quotient of the general linear group modulo the subgroup of upper triangular matrices, are also projective, which is an important fact in the theory of algebraic groups.[10]

Hilbert polynomial, degree and dimension

As the prime ideal P defining a projective variety X is homogeneous, the so-called homogeneous coordinate ring

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

The Hilbert polynomial measures the rate how grows as a function of n. The degree of P is the 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 and its Hilbert polynomial is ; its arithmetic genus is zero.

Projective curves

Further information: 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 singularities of curves can be resolved by 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 fields are isomorphic. The study of finite extensions of

or equivalently smooth projective curves over is an important branch in algebraic number theory.[11]

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 P2. In general, any (smooth) projective 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.

A smooth complete curve of genus greater than or equal to two is called a hyperelliptic curve if there is a finite morphism of degree two.[12]

Projective hypersurfaces

Every irreducible closed subset of Pn of codimension one is a hypersurface; i.e., the zero set of some homogeneous irreducible polynomial.[13]

Abelian varieties

Another important invariant of a projective variety X is the Picard group of X, the set of isomorphism classes of line bundles on X. It is isomorphic to and therefore an intrinsic notion (independent of embedding). For example, the Picard group of Pn is isomorphic to Z via the degree map. The kernel of 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 varieties, in honor of Niels Abel. In marked contrast to affine algebraic groups such as , 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 surfaces.

Line bundle and divisors

Main article: Ample line bundle

The number of particular properties of projective varieties makes it desirable to have efficient criteria to show that a given variety is projective. Such criteria can be formulated using the notion of very ample line bundles.

Let X be a scheme over a ring A. Suppose there is a morphism

.

Then, along this map, the Serre twisting sheaf pulls-back to a line bundle L on X, which is generated by the global sections .[14] Conversely, any line bundle L which is generated by global sections defines a morphism

which in homogeneous coordinates is given by This map is such that and . Furthermore, is a closed immersion if and only if are affine and are surjective.[15]

A line bundle (or invertible sheaf) on a scheme X over S is said to be very ample relative to S if there is an immersion (i.e., an open immersion followed by a closed immersion)

for some n so that pullbacks to 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 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.

Cohomology of coherent sheaves

Main article: coherent sheaf

Let X be a projective scheme over a field (or, more generally over a Noetherian ring A). Cohomology of coherent sheaves on X satisfies the following important theorems due to Serre:

1. is a finite-dimensional k-vector space for any p.
2. There exists an integer (depending on ; see also Castelnuovo–Mumford regularity) such that
for all and p > 0, where is the twisting with a power of a very ample line bundle

These results are proven reducing to the case using the isomorphism

where in the right-hand side is viewed as a sheaf on the projective space by extension by zero.[16] The result then follows by a direct computation for n any integer, and for arbitrary reduces to this case without much difficulty.[17]

As a corollay to 1. above, if f is a projective morphism from a noetherian scheme to a noetherian ring, then the higher direct image is coherent. The same result holds for proper morphisms f, as can be shown with the aid of Chow's lemma.

Sheaf cohomology groups Hi on a noetherian topological space vanish for i strictly greater than the dimension of the space. Thus the quantity, called the Euler characteristic of ,

is a well-defined integer (for X projective). One can then show for some polynomial P over rational numbers.[18] Applying this procedure to the structure sheaf , 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

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

The arithmetic genus of a hypersurface of degree d is in . In particular, a smooth curve of degree d in P2 has arithmetic genus . This is the genus formula.

Smooth projective varieties

Let X be a smooth projective variety where all of its irreducible components have dimension n. In this situation, the canonical sheaf ωX, defined as the sheaf of Kähler differentials of top degree (i.e., algebraic n-forms), is a line bundle.

Serre duality

Serre duality states that for any locally free sheaf on X,

where the superscript prime refers to the dual space and is the dual sheaf of . A generalization to projective, but not necessarily smooth schemes is known as Verdier duality.

Riemann-Roch theorem

For a (smooth projective) curve X, 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 . By definition, the geometric genus of X is the dimension of H0(X, ωX). Serre duality thus implies that the arithmetic genus and the geometric genus coincide. They will simply be called the genus of X.

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

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 . Then the Riemann–Roch theorem states: if g is a genus of X,

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

,

which can be readily proved.[19] A generalization of the Riemann-Roch theorem to higher dimension is the Hirzebruch-Riemann-Roch theorem, as well as the far-reaching Grothendieck-Riemann-Roch theorem.

Hilbert schemes

Hilbert schemes parametrize all closed subvarieties of a projective scheme X in the sense that the points (in the functorial sense) of H correspond to the closed subschemes of X. As such, the Hilbert scheme is an example of a moduli space, i.e., a geometric object whose points parametrize other geometric objects. More precisely, the Hilbert scheme parametrizes closed subvarieties whose Hilbert polynomial equals a prescribed polynomial P.[20] It is a deep theorem of Grothendieck that there is a scheme[21] over k such that, for any k-scheme T, there is a bijection

The closed subscheme of that corresponds to the identity map is called the universal family.

For , the Hilbert scheme is called the Grassmannian of r-planes in and, if X is a projective scheme, is called the Fano scheme of r-planes on X.[22]

Complex projective varieties

In this section, all algebraic varieties are complex 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 variety X yields a complex analytic space, denoted . Moreover, geometric properties of X are reflected by the ones of . For example, the latter is a complex manifold iff X is smooth; it is compact iff X is proper over C.

Relation to complex Kähler manifolds

Complex projective space is a Kähler manifold. This implies that, for any projective algebraic variety X, X(C) is a compact Kähler manifold. The converse is not in general true, but the Kodaira embedding theorem gives a criterion for a Kähler manifold to be projective.

In low dimensions, there are the following results:

GAGA and Chow's theorem

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 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.[24]
• Every holomorphic line bundle on a projective variety is a line bundle of a divisor.[25]

Chow's theorem can be shown via Serre's GAGA principle. Its main theorem 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

are isomorphisms for all i and all coherent sheaves on X.[26]

Complex tori vs. complex abelian varieties

The complex manifold associated to an abelian variety A over C is a compact complex Lie group. These can be shown to be of the form

and are also referred to as complex tori. Here, g is the dimension of the torus and L is a lattice (also referred to as period lattice).

According to the uniformization theorem already mentioned above, any torus of dimension 1 arises from an abelian variety of dimension 1, i.e., from an elliptic curve. In fact, the Weierstrass's elliptic function attached to L satisfies a certain differential equation and as a consequence it defines a closed immersion:[27]

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

Kodaira vanishing

The fundamental Kodaira vanishing theorem states that for an ample line bundle on a smooth projective variety X over a field of characteristic zero,

for i > 0, or, equivalently by Serre duality for i < n.[28] 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.[29]

Related notions

Closed subvarieties of weighted projective spaces are known as weighted projective varieties.[30]

Notes

1. Kollár & Moduli, Ch I.
2. Shafarevich, Igor R. (1994), Basic Algebraic Geometry 1: Varieties in Projective Space, Springer
3. Mumford 1999, pg. 82
4. Hartshorne 1977, Section II.5
5. Mumford 1999, pg. 111
6. This definition differs from Eisenbud–Harris 2000, III.2.3 but is consistent with the other parts of Wikipedia.
7. cf. the proof of Hartshorne 1977, Ch II, Theorem 7.1
8. Hartshorne 1977, Ch II. Exercise 4.5
9. Humphreys, James (1981), Linear algebraic groups, Springer, Theorem 21.3
10. Rosen, Michael (2002), Number theory in Function Fields, Springer
11. Hartshorne & 1977 Ch IV, Exercise 1.7.
12. 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.
13. Hartshorne 1977, Ch II, Theorem 7.1
14. Hartshorne 1977, Ch II, Proposition 7.2
15. This is not difficult:(Hartshorne 1977, Ch III. Lemma 2.10) consider a flasque resolution of and its zero-extension to the whole projective space.
16. Hartshorne 1977, Ch III. Theorem 5.2
17. Hartshorne 1977, Ch III. Exercise 5.2
18. Hartshorne 1977, Ch IV. Theorem 1.3
19. Kollár 1996, Ch I 1.4
20. To make the construction work, one needs to allow for a non-variety.
21. Eisenbud & Harris 2000, VI 2.2
22. Hartshorne 1977, Appendix B. Theorem 3.4.
23. Griffiths-Adams, IV. 1. 10. Corollary H
24. Griffiths-Adams, IV. 1. 10. Corollary I
25. Hartshorne 1977, Appendix B. Theorem 2.1
26. Mumford 1970, pg. 36
27. Hartshorne 1977, Ch III. Remark 7.15.
28. Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems, Birkhäuser
29. Dolgachev, Igor (1982), "Weighted projective varieties", Group actions and vector fields (Vancouver, B.C., 1981), Lecture Notes in Math., 956, Berlin: Springer, pp. 34–71, doi:10.1007/BFb0101508, MR 0704986