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 Poincare 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. The group of divisors on a curve (the free abelian group on its set of points) is closely related to the group of fractional ideals for a Dedekind domain.
An algebraic cycle is a higher-dimensional generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.
- 1 Divisors on a Riemann surface
- 2 Weil divisors
- 3 Divisor class group
- 4 Cartier divisors
- 5 Functoriality
- 6 The first Chern class
- 7 Global sections of line bundles and linear systems
- 8 Q-divisors
- 9 The Grothendieck–Lefschetz hyperplane theorem
- 10 Notes
- 11 References
- 12 External links
Divisors on a Riemann surface
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.
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.
where the collection is locally finite. If X is quasi-compact, local finiteness is equivalent to being finite. The group of all Weil divisors is denoted Div(X). A Weil divisor D is effective if all the coefficients are non-negative. One writes D ≥ D′ if the difference D − D′ is effective.
For example, a divisor on an algebraic curve over a field is a formal sum of finitely many closed points. A divisor on Spec Z is a formal sum of prime numbers and therefore corresponds to a proper non-zero ideal in Z. A similar characterization is true for divisors on , where K is a number field.
If Z ⊂ X is a prime divisor, then the local ring OX,Z has Krull dimension one. If f ∈ OX,Z is non-zero, then the order of vanishing of f along Z, written ordZ(f), is the length of OX,Z / (f). This length is finite, and it is additive with respect to multiplication, that is, ordZ(fg) = ordZ(f) + ordZ(g). If k(X) is the field of rational functions on X, then any non-zero f ∈ k(X) may be written as a quotient g / h, where g and h are in OX,Z, and the order of vanishing of f is defined to be ordZ(g) - ordZ(h). With this definition, the order of vanishing is a function ordZ : k(X)* → Z. If X is normal, then the local ring OX,Z is a discrete valuation ring, and the function ordZ is the corresponding valuation. For a non-zero rational function f on X, the principal Weil divisor associated to f is defined to be the Weil divisor
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.
Every Weil divisor determines a subsheaf of the sheaf of rational functions. Specifically, if D is a Weil divisor, then the sheaf of modules O(D) (also denoted L(D)) is defined on an open subset U to be those rational functions whose poles are at most those specified by D:
That is, when a prime divisor Z appears to positive order in D, then the functions in O(D) may have poles along Z of the same or less order than the Z-coefficient of D. When the coefficient of Z is negative, then the functions in O(D) are required to have zeroes of order equal to or greater than the Z-coefficient of D. When X is a normal Noetherian scheme and D is an effective divisor, then O(D) is a subsheaf of the sheaf of regular functions, and in particular each of its sections is a regular function. If D is principal, so D is the divisor of a rational function g, then O(D) is isomorphic to the sheaf OX of regular functions via . Conversely, if O(D) is isomorphic to OX as an OX-module, then D is principal and corresponds to the rational function which is the image of the global section 1 under the isomorphism OX → O(D).
For normal integral separated schemes of finite type over a field, the sheaves O(D) are rank one reflexive sheaves. Conversely, every rank one reflexive sheaf corresponds to a Weil divisor: The sheaf can be restricted to the regular locus, where it becomes free and so corresponds to a Cartier divisor (see below), and because the singular locus has codimension at least two, the closure of the Cartier divisor is a Weil divisor.
A Weil divisor D is Cartier if and only if the sheaf O(D) is locally free of rank one, that is, a line bundle. A Noetherian scheme X is called factorial if all local rings of X are unique factorization domains. (Some authors say "locally factorial".) In particular, every regular scheme is factorial. On a factorial scheme X, every Weil divisor D is locally principal, and so O(D) is a line bundle. In general, however, a Weil divisor on a normal scheme need not be locally principal; see the examples of quadric cones, below.
Divisor class group
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(X − Z) 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(X − Z) is an isomorphism. (These facts are special cases of the localization sequence for Chow groups.)
- Let k be a field, and let n be a positive integer. Since the polynomial ring k[x1,...,xn] is a unique factorization domain, the divisor class group of affine space An over k is equal to zero. Since projective space Pn over k minus a hyperplane H is isomorphic to An, it follows that the divisor class group of Pn is generated by the class of H. From there, it is straightforward to check that Cl(Pn) is in fact isomorphic to the integers Z, generated by H. Concretely, this means that every codimension-1 subvariety of Pn is defined by the vanishing of a single homogeneous polynomial.
- Let X be an algebraic curve over a field k. Every closed point p in X has the form Spec E for some finite extension field E of k, and the degree of p is defined to be the degree of E over k. Extending this by linearity gives the notion of degree for a divisor on X. If X is a projective curve over k, then the divisor of a nonzero rational function f on X has degree zero. As a result, for a projective curve X, the degree gives a homomorphism deg: Cl(X) → Z.
- For the projective line P1 over a field k, the degree gives an isomorphism Cl(P1) ≅ Z. For any smooth projective curve X with a k-rational point, the degree homomorphism is surjective, and the kernel is isomorphic to the group of k-points on the Jacobian variety of X, which is an abelian variety of dimension equal to the genus of X. It follows, for example, that the divisor class group of a complex elliptic curve is an uncountable abelian group.
- Generalizing the previous example: for any smooth projective variety X over a field k such that X has a k-rational point, the divisor class group Cl(X) is an extension of a finitely generated abelian group, the Néron–Severi group, by the group of k-points of a connected group scheme Pic0X/k. For k of characteristic zero, Pic0X/k is an abelian variety, the Picard variety of X.
- For R the ring of integers of a number field, the divisor class group Cl(R) := Cl(Spec R) is also called the ideal class group of R. It is a finite abelian group. Understanding ideal class groups is a central goal of algebraic number theory.
- quadric cone of dimension 2, defined by the equation xy = z2 in affine 3-space over a field. Then the line D in X defined by x = z = 0 is not principal on X near the origin. Note that D can be defined as a set by one equation on X, namely x = 0; but the function x on X vanishes to order 2 along D, and so we only find that 2D is Cartier (as defined below) on X. In fact, the divisor class group Cl(X) is isomorphic to the cyclic group Z/2, generated by the class of D.
- Let X be the quadric cone of dimension 3, defined by the equation xy = zw in affine 4-space over a field. Then the plane D in X defined by x = z = 0 cannot be defined in X by one equation near the origin, even as a set. It follows that D is not Q-Cartier on X; that is, no positive multiple of D is Cartier. In fact, the divisor class group Cl(X) is isomorphic to the integers Z, generated by the class of D.
The canonical divisor
is an isomorphism, since X − U has codimension at least 2 in X. For example, one can use this isomorphism to define the canonical divisor KX of X: it is the Weil divisor (up to linear equivalence) corresponding to the line bundle of differential forms of top degree on U. Equivalently, the sheaf O(KX) on X is the direct image sheaf j*ΩnU, where n is the dimension of X.
Let X be an integral Noetherian scheme. Then X has a sheaf of rational functions MX. All regular functions are rational functions, which leads to a short exact sequence
A Cartier divisor on X is a global section of MX* / OX*. An equivalent description is that a Cartier divisor is a collection , where fi is a section of MX* on Ui, the elements fi agree with each other on the overlaps Ui ∩ Uj, and two such collections are equivalent if, after passing to a common refinement of their open covers, the rational functions are equal up to multiplication by elements of OX*.
Cartier divisors also have a sheaf-theoretic description. A fractional ideal sheaf is a sub-OX-module of MX. A fractional ideal sheaf J is invertible if, for each x in X, there exists an open neighborhood U of x on which the restriction of J to U is equal to OU⋅f, where f is in MX*(U) and the product is taken in MX. Each Cartier divisor defines an invertible fractional ideal sheaf using the description of the Cartier divisor as a collection , and conversely, invertible fractional ideal sheaves define Cartier divisors.
A Cartier divisor on an integral Noetherian scheme X determines a Weil divisor on X in a natural way, by taking the divisor of zeros of the functions fi on the open sets Ui. 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.
By the exact sequence above, there is an exact sequence of sheaf cohomology groups:
A Cartier divisor is said to be principal if it is in the image of the homomorphism H0(X, MX*) → H0(X, MX*/OX*), that is, if it is the divisor of a rational function on X. Two Cartier divisors are linearly equivalent if their difference is principal. Every line bundle L on X on an integral Noetherian scheme is the class of some Cartier divisor. As a result, the exact sequence above identifies the Picard group of line bundles on an integral Noetherian scheme X with the group of Cartier divisors modulo linear equivalence. This holds more generally for reduced Noetherian schemes, or for quasi-projective schemes over a Noetherian ring, but it can fail in general (even for proper schemes over C), which lessens the interest of Cartier divisors in full generality.
Effective Cartier divisors
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 U ∩ D = 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.
There is a good theory of families of effective Cartier divisors. Let φ : X → S be a morphism. A relative effective Cartier divisor for X over S is an effective Cartier divisor D on X which is flat over S. Because of the flatness assumption, for every S′ → S, there is a pullback of D to X ×S S′, and this pullback is an effective Cartier divisor. In particular, this is true for the fibers of φ.
Let φ : X → Y 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 Z is a prime Weil divisor on X, then is a closed irreducible subscheme of Y. Depending on φ, it may or may not be a prime Weil divisor. For example, if φ is the blow up of a point in the plane and Z is the exceptional divisor, then its image is not a Weil divisor. Therefore φ*Z is defined to be if that subscheme is a prime divisor and is defined to be the zero divisor otherwise. Extending this by linearity will, assuming X is quasi-compact, define a homomorphism Div(X) → Div(Y) called the pushforward. (If X is not quasi-compact, then the pushforward may fail to be a locally finite sum.) This is a special case of the pushforward on Chow groups.
If Z is a Cartier divisor, then under mild hypotheses on φ, there is a pullback φ*Z. Sheaf-theoretically, when there is a pullback map φ−1MY → MX, then this pullback can be used to define pullback of Cartier divisors. In terms of local sections, the pullback of is defined to be . Pullback is always defined if φ is dominant, but it cannot be defined in general. For example, if X = Z and φ is the inclusion of Z into Y, then φ*Z is undefined because the corresponding local sections would be everywhere zero. (The pullback of the corresponding line bundle, however, is defined.)
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.
The first Chern class
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. (For a variety X over a field, the Chern classes of any vector bundle on X act by cap product on the Chow groups of X, and the homomorphism here can be described as L ↦ c1(L) ∩ [X].) 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 c1: Pic(X) → Cl(X) 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: Cl(X) → H2n-2BM(X,Z). (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: Pic(X) → H2(X,Z). 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:
For X smooth over C, both vertical maps are isomorphisms.
Global sections of line bundles and linear systems
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 X → Pn. These observations lead to several notions of positivity for Cartier divisors (or line bundles), such as ample divisors and nef divisors.
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.
If D = ∑ aj Zj is a Q-divisor, then its round-down is the divisor
where ⌊a⌋ is the greatest integer less than or equal to a. The sheaf O(D) is then defined to be O(⌊D⌋).
The Grothendieck–Lefschetz hyperplane theorem
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). The dimension bound here is optimal, as shown by the example of the 3-dimensional quadric cone, above.
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