Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are ultimately derived from the notion of divisibility in the integers and algebraic number fields.

The background is that codimension-1 subvarieties are understood much better than higher-codimension subvarieties. This happens in both global and local ways. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this good property, much of algebraic geometry studies an arbitrary variety by analyzing its codimension-1 subvarieties and the corresponding line bundles.

On singular varieties, this good property can fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors. Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same.

The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves.[1] The group of divisors on a curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain.

An algebraic cycle is a higher-codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.

A Riemann surface is a 1-dimensional complex manifold, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a compact Riemann surface X is the free abelian group on the points of X.

Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients. The degree of a divisor on X is the sum of its coefficients.

For any nonzero meromorphic function f on X, one can define the order of vanishing of f at a point p in X, ordp(f). It is an integer, negative if f has a pole at p. The divisor of a nonzero meromorphic function f on the compact Riemann surface X is defined as

which is a finite sum. Divisors of the form (f) are also called principal divisors. Since (fg) = (f) + (g), the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called linearly equivalent.

On a compact Riemann surface, the degree of a principal divisor is zero; that is, the number of zeros of a meromorphic function is equal to the number of poles, counted with multiplicity. As a result, the degree is well-defined on linear equivalence classes of divisors.

Given a divisor D on a compact Riemann surface X, it is important to study the complex vector space of meromorphic functions on X with poles at most given by D, called H0(X, O(D)) or the space of sections of the line bundle associated to D. The degree of D says a lot about the dimension of this vector space. For example, if D has negative degree, then this vector space is zero (because a meromorphic function cannot have more zeros than poles). If D has positive degree, then the dimension of H0(X, O(mD)) grows linearly in m for m sufficiently large. The Riemann–Roch theorem is a more precise statement along these lines. On the other hand, the precise dimension of H0(X, O(D)) for divisors D of low degree is subtle, and not completely determined by the degree of D. The distinctive features of a compact Riemann surface are reflected in these dimensions.

One key divisor on a compact Riemann surface is the canonical divisor. To define it, one first defines the divisor of a nonzero meromorphic 1-form along the lines above. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two nonzero meromorphic 1-forms yield linearly equivalent divisors. Any divisor in this linear equivalence class is called the canonical divisor of X, KX. The genus g of X can be read from the canonical divisor: namely, KX has degree 2g − 2. The key trichotomy among compact Riemann surfaces X is whether the canonical divisor has negative degree (so X has genus zero), zero degree (genus one), or positive degree (genus at least 2). For example, this determines whether X has a Kähler metric with positive curvature, zero curvature, or negative curvature. The canonical divisor has negative degree if and only if X is isomorphic to the Riemann sphere CP1.

Let X be an integral locally Noetherian scheme. A prime divisor or irreducible divisor on X is an integral closed subscheme Z of codimension 1 in X. A Weil divisor on X is a formal sum over the prime divisors Z of X,

It can be shown that this sum is locally finite and hence that it indeed defines a Weil divisor. The principal Weil divisor associated to f is also notated (f). If f is a regular function, then its principal Weil divisor is effective, but in general this is not true. The additivity of the order of vanishing function implies that

Consequently div is a homomorphism, and in particular its image is a subgroup of the group of all Weil divisors.

where nZ is the coefficient of Z in D. If D is principal, so D is the divisor of a rational function g, then there is an isomorphism

The Weil divisor class group Cl(X) is the quotient of Div(X) by the subgroup of all principal Weil divisors. Two divisors are said to be linearly equivalent if their difference is principal, so the divisor class group is the group of divisors modulo linear equivalence. For a variety X of dimension n over a field, the divisor class group is a Chow group; namely, Cl(X) is the Chow group CHn−1(X) of (n−1)-dimensional cycles.

Let Z be a closed subset of X. If Z is irreducible of codimension one, then Cl(XZ) is isomorphic to the quotient group of Cl(X) by the class of Z. If Z has codimension at least 2 in X, then the restriction Cl(X) → Cl(XZ) is an isomorphism.[6] (These facts are special cases of the localization sequence for Chow groups.)

Let X be a normal variety over a perfect field. The smooth locus U of X is an open subset whose complement has codimension at least 2. Let j: UX be the inclusion map, then the restriction homomorphism:

By the exact sequence above, there is an exact sequence of sheaf cohomology groups:

Assume D is an effective Cartier divisor. Then there is a short exact sequence

This sequence is derived from the short exact sequence relating the structure sheaves of X and D and the ideal sheaf of D. Because D is a Cartier divisor, O(D) is locally free, and hence tensoring that sequence by O(D) yields another short exact sequence, the one above. When D is smooth, OD(D) is the normal bundle of D in X.

If X is normal, a Cartier divisor is determined by the associated Weil divisor, and a Weil divisor is Cartier if and only if it is locally principal.

A Noetherian scheme X is called factorial if all local rings of X are unique factorization domains.[5] (Some authors say "locally factorial".) In particular, every regular scheme is factorial.[14] On a factorial scheme X, every Weil divisor D is locally principal, and so O(D) is always a line bundle.[7] In general, however, a Weil divisor on a normal scheme need not be locally principal; see the examples of quadric cones above.

Effective Cartier divisors are those which correspond to ideal sheaves. In fact, the theory of effective Cartier divisors can be developed without any reference to sheaves of rational functions or fractional ideal sheaves.

Let X be a scheme. An effective Cartier divisor on X is an ideal sheaf I which is invertible and such that for every point x in X, the stalk Ix is principal. It is equivalent to require that around each x, there exists an open affine subset U = Spec A such that UD = Spec A / (f), where f is a non-zero divisor in A. The sum of two effective Cartier divisors corresponds to multiplication of ideal sheaves.

Let φ : XY be a morphism of integral locally Noetherian schemes. It is often—but not always—possible to use φ to transfer a divisor D from one scheme to the other. Whether this is possible depends on whether the divisor is a Weil or Cartier divisor, whether the divisor is to be moved from X to Y or vice versa, and what additional properties φ might have.

If φ is flat, then pullback of Weil divisors is defined. In this case, the pullback of Z is φ*Z = φ−1(Z). The flatness of φ ensures that the inverse image of Z continues to have codimension one. This can fail for morphisms which are not flat, for example, for a small contraction.

For an integral Noetherian scheme X, the natural homomorphism from the group of Cartier divisors to that of Weil divisors gives a homomorphism

known as the first Chern class.[15] The first Chern class is injective if X is normal, and it is an isomorphism if X is factorial (as defined above). In particular, Cartier divisors can be identified with Weil divisors on any regular scheme, and so the first Chern class is an isomorphism for X regular.

Explicitly, the first Chern class can be defined as follows. For a line bundle L on an integral Noetherian scheme X, let s be a nonzero rational section of L (that is, a section on some nonempty open subset of L), which exists by local triviality of L. Define the Weil divisor (s) on X by analogy with the divisor of a rational function. Then the first Chern class of L can be defined to be the divisor (s). Changing the rational section s changes this divisor by linear equivalence, since (fs) = (f) + (s) for a nonzero rational function f and a nonzero rational section s of L. So the element c1(L) in Cl(X) is well-defined.

For a complex variety X of dimension n, not necessarily smooth or proper over C, there is a natural homomorphism, the cycle map, from the divisor class group to Borel–Moore homology:

The latter group is defined using the space X(C) of complex points of X, with its classical (Euclidean) topology. Likewise, the Picard group maps to integral cohomology, by the first Chern class in the topological sense:

The two homomorphisms are related by a commutative diagram, where the right vertical map is cap product with the fundamental class of X in Borel–Moore homology:

A Cartier divisor is effective if its local defining functions fi are regular (not just rational functions). In that case, the Cartier divisor can be identified with a closed subscheme of codimension 1 in X, the subscheme defined locally by fi = 0. A Cartier divisor D is linearly equivalent to an effective divisor if and only if its associated line bundle O(D) has a nonzero global section s; then D is linearly equivalent to the zero locus of s.

Let X be a projective variety over a field k. Then multiplying a global section of O(D) by a nonzero scalar in k does not change its zero locus. As a result, the projective space of lines in the k-vector space of global sections H0(X, O(D)) can be identified with the set of effective divisors linearly equivalent to D, called the complete linear system of D. A projective linear subspace of this projective space is called a linear system of divisors.

One reason to study the space of global sections of a line bundle is to understand the possible maps from a given variety to projective space. This is essential for the classification of algebraic varieties. Explicitly, a morphism from a variety X to projective space Pn over a field k determines a line bundle L on X, the pullback of the standard line bundle O(1) on Pn. Moreover, L comes with n+1 sections whose base locus (the intersection of their zero sets) is empty. Conversely, any line bundle L with n+1 global sections whose common base locus is empty determines a morphism XPn.[16] These observations lead to several notions of positivity for Cartier divisors (or line bundles), such as ample divisors and nef divisors.[17]

For a divisor D on a projective variety X over a field k, the k-vector space H0(X, O(D)) has finite dimension. The Riemann–Roch theorem is a fundamental tool for computing the dimension of this vector space when X is a projective curve. Successive generalizations, the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Riemann–Roch theorem, give some information about the dimension of H0(X, O(D)) for a projective variety X of any dimension over a field.

Because the canonical divisor is intrinsically associated to a variety, a key role in the classification of varieties is played by the maps to projective space given by KX and its positive multiples. The Kodaira dimension of X is a key birational invariant, measuring the growth of the vector spaces H0(X, mKX) (meaning H0(X, O(mKX))) as m increases. The Kodaira dimension divides all n-dimensional varieties into n+2 classes, which (very roughly) go from positive curvature to negative curvature.

Let X be a normal variety. A (Weil) Q-divisor is a finite formal linear combination of irreducible codimension-1 subvarieties of X with rational coefficients. (An R-divisor is defined similarly.) A Q-divisor is effective if the coefficients are nonnegative. A Q-divisor D is Q-Cartier if mD is a Cartier divisor for some positive integer m. If X is smooth, then every Q-divisor is Q-Cartier.

The Lefschetz hyperplane theorem implies that for a smooth complex projective variety X of dimension at least 4 and a smooth ample divisor Y in X, the restriction Pic(X) → Pic(Y) is an isomorphism. For example, if Y is a smooth complete intersection variety of dimension at least 3 in complex projective space, then the Picard group of Y is isomorphic to Z, generated by the restriction of the line bundle O(1) on projective space.

Grothendieck generalized Lefschetz's theorem in several directions, involving arbitrary base fields, singular varieties, and results on local rings rather than projective varieties. In particular, if R is a complete intersection local ring which is factorial in codimension at most 3 (for example, if the non-regular locus of R has codimension at least 4), then R is a unique factorization domain (and hence every Weil divisor on Spec(R) is Cartier).[18] The dimension bound here is optimal, as shown by the example of the 3-dimensional quadric cone, above.