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 arbitrary codimension-1 subvarieties and those which can locally be defined by one equation: this is the distinction between Weil divisors and Cartier divisors. In topological terms, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. Only Cartier divisors correspond to line bundles. On a smooth variety (or more generally a regular scheme), 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 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.

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.

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, ord_p(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

$(f):=\sum_{p \in X} \operatorname{ord}_p(f) p,$

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, K_X. The genus g of X can be read from the canonical divisor: namely, K_X 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.

Weil divisors

Definition

Let X be an integral Noetherian scheme. For example, X could be an algebraic variety over a field. An irreducible divisor on X is an irreducible closed subset Z of codimension 1 in X. A Weil divisor on X is a finite linear combination with integer coefficients of irreducible divisors on X:

$\sum_Z n_Z Z,$

where only finitely many n_Z are nonzero. For example, a divisor on an algebraic curve is a formal sum of its closed points. A Weil divisor D is effective if all the coefficients are nonnegative. One writes D ≥ D' if the difference D − D' is effective.

Write k(X) for the field of rational functions on X. If Z ⊂ X is an irreducible closed subset of codimension one, the local ring O_X,Z of X at the generic point of Z means the ring of rational functions on X which are regular at the generic point of Z (or equivalently, on a nonempty open subset of Z). The local ring O_X,Z has dimension 1. For a nonzero function f in O_X,Z, the order of vanishing of f along Z is defined by

$\operatorname{ord}_Z(f) = \operatorname{length}_{\mathcal{O}_{X,Z}}({\mathcal{O}_{X,Z}}/(f)),$

using that O_X,Z/(f) has finite length as a module over O_X,Z.^[2] More generally, the order of vanishing along Z of any nonzero rational function f on X is defined by:

$\operatorname{ord}_Z(f/g) = \operatorname{ord}_Z(f) - \operatorname{ord}_Z(g),$

where f and g are regular at the generic point of Z. In particular, if X is normal, then the local ring O_X,Z is a discrete valuation ring, and this function ord_Z: k(X)* → Z is the corresponding valuation. For a nonzero rational function f on X, the divisor of zeros of f is defined as:

$(f) = \sum_Z \operatorname{ord}_Z (f) Z$

This is a finite sum, and hence a divisor on X. It is also called the principal divisor defined by f. (By construction, the coefficients of the divisor (f) can be negative if f has poles.)

If D is a Weil divisor on a normal scheme X, then the sheaf O(D) on X is defined by: for any open subset U of X, a section of O(D) over U is a rational function whose poles on U are at most given by D.^[3] That is:

$H^0(U, O(D)) := \{ f \in k(X) : f = 0 \text{ or }(f) + D \ge 0 \text{ on } U \}.$

If D is principal, so D is the divisor of a rational function g, then O(D) is isomorphic to the sheaf O_X of regular functions via $f \mapsto fg$. Conversely, if O(D) is isomorphic to O_X as an O_X-module, then D is principal. It follows that D is locally principal (that is, D is Cartier, as defined below) 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.^[3] (Some authors say "locally factorial".) In particular, every regular scheme is factorial.^[4] On a factorial scheme X, every Weil divisor D is locally principal, and so O(D) is a line bundle.^[5] In general, however, a Weil divisor on a normal scheme need not be locally principal; see the examples of quadric cones, below.

For any Weil divisor D on a normal scheme X, the sheaf O(D) is a reflexive sheaf. Every reflexive sheaf of rank 1 (called a divisorial sheaf) arises in this way from some Weil divisor.

Divisor class group

Let Div(X) be the abelian group of Weil divisors on an integral scheme X. Since principal divisors form a subgroup, one can form the quotient abelian group:

$\operatorname{Cl}(X) = \operatorname{Div}(X)/\{ (f) : f \in k(X)^* \},$

called the divisor class group of X. 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 special case of the Chow groups; namely, Cl(X) is the Chow group CH_n−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.^[6] (These facts are special cases of the localization sequence for Chow groups.)

Examples

• Let k be a field, and let n be a positive integer. Since the polynomial ring k[x₁,...,x_n] is a unique factorization domain, the divisor class group of affine space Aⁿ over k is equal to zero.^[5] 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.^[7] 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 Pic⁰_X/k.^[8] For k of characteristic zero, Pic⁰_X/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.
• 

  The affine quadric cone xy = z².
  Let X be the 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.^[9]
• 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.^[10]

The canonical divisor

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. Writing j for the inclusion U → X, the restriction homomorphism:

$j^*: \operatorname{Cl}(X) \to \operatorname{Cl}(U) = \operatorname{Pic}(U)$

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 K_X 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(K_X) on X is the direct image sheaf j_*Ωⁿ_U, where n is the dimension of X.

Cartier divisors

A Cartier divisor on an integral Noetherian scheme X can be described by a collection of open subschemes U_i of X whose union is X, together with a nonzero rational function f_i on U_i for each i. The functions must be compatible in the sense that on the intersection of two sets in the cover, the quotient of the corresponding rational functions is a unit (an invertible regular function). Two Cartier divisors are considered the same if they are related by multiplying the rational functions f_i by units, or by passing from one open covering to a refinement.

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 f_i on the open sets U_i. 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.

Cartier divisors can be defined more formally in terms of sheaves. On an integral Noetherian scheme X, the sheaf of rational functions M_X means the constant sheaf on X associated to the field k(X). The sheaf O_X of regular functions is a subsheaf of M_X, and hence there is a short exact sequence

$0\to O_X^* \to M_X^* \to M_X^*/O_X^*\to 0$

of sheaves on X, writing R* for the group of units in a sheaf of rings R. Then a Cartier divisor on X is simply a global section of the sheaf M_X*/O_X* over X.

More generally, a Cartier divisor on any scheme (not necessarily integral and Noetherian) is defined to be a section of M_X*/O_X*, where the sheaf of rational functions M_X is defined as follows.^[11] For each open subset U of X, define M′(U) = S_U⁻¹O(U), where S_U is the multiplicative subset of regular functions f on U such that the germ of f in the local ring O_U,p is a non-zerodivisor for each point p in U. The sheaf M_X of rational functions on X is the sheaf of rings associated to the presheaf M′.

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

$H^0(X, M^*_X) \to H^0(X, M^*_X / \mathcal O^*_X) \to H^1(X, \mathcal O^*_X) = \operatorname{Pic}(X).$

A Cartier divisor is said to be principal if it is in the image of the homomorphism H⁰(X, M_X*) → H⁰(X, M_X*/O_X*), 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,^[12] but it can fail in general (even for proper schemes over C), which lessens the interest of Cartier divisors in full generality.^[13]

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

$c_1\colon \operatorname{Pic}(X) \to \operatorname{Cl}(X),$

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) → H_2n-2^BM(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:

$\begin{array}{ccc} \operatorname{Pic}(X) & \to & H^2(X,\mathbf{Z})\\ \downarrow & & \downarrow \\ \operatorname{Cl}(X) & \to & H_{2n-2}^{\operatorname{BM}}(X,\mathbf{Z}). \end{array}$

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 f_i 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 f_i = 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.^[14] These observations lead to several notions of positivity for Cartier divisors (or line bundles), such as ample divisors and nef divisors.^[15]

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 K_X and its positive multiples. The Kodaira dimension of X is a key birational invariant, measuring the growth of the vector spaces H⁰(X, mK_X) (meaning H⁰(X, O(mK_X))) 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.

Q-divisors

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 = ∑ a_j Z_j is a Q-divisor, then its round-down is the divisor

$\lfloor D\rfloor = \sum \lfloor a_j \rfloor Z_j,$

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).^[16] The dimension bound here is optimal, as shown by the example of the 3-dimensional quadric cone, above.

Notes

1. Dieudonné (1985), section VI.6.
2. Stacks Project, Tag 0BE0.
3. Kollár (2013), Notation 1.2.
4. Stacks Project, Tag 0AFW.
5. Hartshorne (1977), Proposition II.6.2.
6. Hartshorne (1977), Proposition II.6.5.
7. Stacks Project, Tag 02RS.
8. Kleiman (2005), Theorems 2.5 and 5.4, Remark 6.19.
9. Hartshorne (1977), Example II.6.5.2.
10. Hartshorne(1977), Exercise II.6.5.
11. Stacks Project, Tag 02AQ
12. Grothendieck, EGA IV, Part 4, Proposition 21.3.4, Corollaire 21.3.5.
13. Lazarsfeld (2004), Example 1.1.6.
14. Hartshorne (1977), Theorem II.7.1.
15. Lazarsfeld (2004), Chapter 1.
16. Grothendieck, SGA 2, Corollaire XI.3.14.

References

• Dieudonné, Jean (1985), History of Algebraic Geometry, Wadsworth Mathematics Series, Translated by Judith D. Sally, Belmont, CA: Wadsworth International Group, ISBN 0-534-03723-2, MR 0780183
• Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
• Grothendieck, Alexander; Raynaud, Michèle (2005) [1968], Laszlo, Yves, ed., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques 4, Paris: Société Mathématique de France, arXiv:math/0511279, ISBN 978-2-85629-169-6, MR 2171939
• Section II.6 of Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics 52, New York, Heidelberg: Springer-Verlag, ISBN 0-387-90244-9, MR 0463157
• Kleiman, Steven (2005), "The Picard scheme", Fundamental Algebraic Geometry, Math. Surveys Monogr. 123, Providence, R.I.: American Mathematical Society, pp. 235–321, arXiv:math/0504020, MR 2223410
• Kollár, János (2013), Singularities of the Minimal Model Program, Cambridge University Press, ISBN 978-1-107-03534-8, MR 3057950
• Lazarsfeld, Robert (2004), Positivity in Algebraic Geometry 1, Berlin: Springer-Verlag, ISBN 3-540-22533-1, MR 2095471

External links

• The Stacks Project Authors, The Stacks Project