Divisor (algebraic geometry)

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In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil). These concepts agree on non-singular varieties.

An algebraic cycle is a higher-dimensional generalization of a divisor.

Divisors in a Riemann surface[edit]

A Riemann surface is a 1-dimensional complex manifold, so its codimension 1 submanifolds are 0-dimensional. The divisors of a Riemann surface are the elements of the free abelian group over the points of the surface.

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

We define the divisor of a meromorphic function f as

(f):=\sum_{z_\nu \in R(f)} s_\nu z_\nu

where R(f) is the set of all zeroes and poles of f, and sν is given by

s_\nu := \left\{ \begin{array}{rl} a  & \ \text{if } z_\nu \text{ is a zero of order }a \\
                           -a & \ \text{if } z_\nu \text{ is a pole of order }a. \end{array} \right.

A divisor that is the divisor of a meromorphic function is called principal. It follows from the fact that a meromorphic function has as many poles as zeroes, that the degree of a principal divisor is 0. Since the divisor of a product is the sum of the divisors, 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.

We define the divisor of a meromorphic 1-form similarly. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the field of meromorphic functions, any two meromorphic 1-forms yield linearly equivalent divisors. The equivalence class of these divisors is called the canonical divisor (usually denoted K).

The Riemann–Roch theorem is an important relation between the divisors of a Riemann surface and its topology.

Weil divisor[edit]

A Weil divisor is a locally finite linear combination (with integral coefficients) of irreducible subvarieties of codimension one. The set of Weil divisors forms an abelian group under addition. In the classical theory, where locally finite is automatic, the group of Weil divisors on a variety of dimension n is therefore the free abelian group on the (irreducible) subvarieties of dimension (n − 1). For example, a divisor on an algebraic curve is a formal sum of its closed points. An effective Weil divisor is then one in which all the coefficients of the formal sum are non-negative.

Cartier divisor[edit]

A Cartier divisor in an algebraic variety X (see below for a scheme case) can be represented by an open cover by affine subsets {U_i} of X, and a collection of rational functions f_i defined on U_i. The functions must be compatible in this sense: on the intersection of two sets in the cover, the quotient of the corresponding rational functions should be regular and invertible. A Cartier divisor is said to be effective if these f_i can be chosen to be regular functions, and in this case the Cartier divisor defines an associated subvariety of codimension 1 by forming the ideal sheaf generated locally by the f_i.

The notion can be described more conceptually with the function field. For each affine open subset U, define M′(U) to be the total quotient ring of OX(U). Because the affine open subsets form a basis for the topology on X, this defines a presheaf on X. (This is not the same as taking the total quotient ring of OX(U) for arbitrary U, since that does not define a presheaf.[1]) The sheaf MX of rational functions on X is the sheaf associated to the presheaf M′, and the quotient sheaf MX* / OX* is the sheaf of local Cartier divisors.

A Cartier divisor is a global section of the quotient sheaf MX*/OX*. We have the exact sequence 1 \to \mathcal O^*_X \to M^*_X \to M^*_X / \mathcal O^*_X \to 1, so, applying the global section functor \Gamma (X, \bullet) gives the exact sequence 1 \to \Gamma (X, O^*_X) \to \Gamma (X, M^*_X) \to \Gamma (X, M^*_X / \mathcal O^*_X) \to H^1(X, \mathcal O^*_X).

A Cartier divisor is said to be principal if it is in the range of the morphism \Gamma (X, M^*_X) \to \Gamma (X, M^*_X / \mathcal O^*_X), that is, if it is the class of a global rational function.

Cartier divisors in nonrigid sheaves[edit]

Of course the notion of Cartier divisors exists in any sheaf. But if the sheaf is not rigid enough, the notion tends to lose some of its interest. For example in a fine sheaf (e.g. the sheaf of real-valued continuous, or smooth, functions on an open subset of a euclidean space, or locally homeomorphic, or diffeomorphic, to such a set, such as a topological manifold), any local section is a divisor of 0, so that the total quotient sheaves are zero, so that the sheaf contains no non-trivial Cartier divisor.

From Cartier divisors to Weil divisor[edit]

There is a natural homomorphism from the group of Cartier divisors to that of Weil divisors, which is an isomorphism for integral separated Noetherian schemes provided that all local rings are unique factorization domains.

In general Cartier behave better than Weil divisors when the variety has singular points.

An example of a surface on which the two concepts differ is a cone, i.e. a singular quadric. At the (unique) singular point, the vertex of the cone, a single line drawn on the cone is a Weil divisor — but is not a Cartier divisor.

The divisor appellation is part of the history of the subject, going back to the DedekindWeber work which in effect showed the relevance of Dedekind domains to the case of algebraic curves.[2] In that case the free abelian group on the points of the curve is closely related to the fractional ideal theory.

From Cartier divisors to line bundles[edit]

The notion of transition map associates naturally to every Cartier divisor D a line bundle (strictly, invertible sheaf) commonly denoted by \mathcal L(D) or sometimes also \mathcal O_X(D).

The line bundle \mathcal L (D) associated to the Cartier divisor D is the sub-bundle of the sheaf MX of rational fractions described above whose stalk at x \in X is given by D_x \in \Gamma (x, M^*_X/\mathcal O^*_X) viewed as a line on the stalk at x of \mathcal O_X in the stalk at x of M_X. The subsheaf thus described is tautologically locally freely monogenous over the structure sheaf \mathcal O_X.

The mapping D \mapsto \mathcal L (D) is a group homomorphism: the sum of divisors corresponds to the tensor product of line bundles, and isomorphism of bundles corresponds precisely to linear equivalence of Cartier divisors. The group of divisors classes modulo linear equivalence therefore injects into the Picard group. The mapping is not surjective for all compact complex manifolds, but surjectivity does hold for all smooth projective varieties. The latter is true because, by the Kodaira embedding theorem, the tensor product of any line bundle with a sufficiently high power of any positive line bundle becomes ample; thus, on any such manifold, any line bundle is the formal difference between two ample line bundles, and any ample line bundle may be viewed as an effective divisor.

Global sections of line bundles and linear systems[edit]

Recall that the local equations of a Cartier divisor D in a variety X give rise to transition maps for a line bundle  \mathcal L (D), and linear equivalences induce isomorphism of line bundles.

Loosely speaking, a Cartier divisor D is said to be effective if it is the zero locus of a global section of its associated line bundle \mathcal L(D). In terms of the definition above, this means that its local equations coincide with the equations of the vanishing locus of a global section.

From the divisor linear equivalence/line bundle isormorphism principle, a Cartier divisor is linearly equivalent to an effective divisor if, and only if, its associate line bundle \mathcal L (D) has non-zero global sections. Two collinear non-zero global sections have the same vanishing locus, and hence the projective space \mathbb P \Gamma (X, \mathcal L (D)) over k identifies with the set of effective divisors linearly equivalent to D.

If X is a projective (or proper) variety over a field k, then \Gamma (X, \mathcal L (D)) is a finite-dimensional k-vector space, and the associated projective space over k is called the complete linear system of D. Its linear subspaces are called linear systems of divisors. The Riemann-Roch theorem for algebraic curves is a fundamental identity involving the dimension of complete linear systems in the setup of projective curves.

\mathbb{Q}-divisors[edit]

Let X be a normal variety. A (Weil) \mathbb{Q}-divisor is a finite formal linear combination of irreducible subvarieties of codimension one of X with rational coefficients. (An \mathbb{R}-divisor is defined similarly.) A \mathbb{Q}-divisor is called effective if the coefficients are nonnegative. A \mathbb{Q}-divisor is called \mathbb{Q}-Cartier if some integral multiple of it is a Cartier divisor. If X is smooth, then any \mathbb{Q}-divisor is \mathbb{Q}-Cartier.

If D = \sum a_j Z_j is a \mathbb{Q}-divisor, then its integer part is the divisor

[D] = \sum [a_j] Z_j

where [a_j] are integer parts of a_j.

See also: multiplier ideal.

Relative Cartier divisors[edit]

An effective Cartier divisor in a scheme X over a ring R is a closed subscheme D of X that (1) is flat over R and (2) the ideal sheaf I(D) of D is locally free of rank one (i.e., invertible sheaf). Equivalently, a closed subscheme D of X is an effective Cartier divisor if there is an open affine cover U_i = \operatorname{Spec} A_i of X and nonzerodivisors f_i \in A_i such that the intersection D \cap U_i is given by the equation f_i = 0 (called local equations) and its ideal sheaf I(D)|_{U_i} = A / f_i A is flat over R and such that they are compatible.

  • If D and D' are effective Cartier divisors, then the sum D + D' is the effective Cartier divisor defined locally as fg = 0 if f, g give local equations for D and D' .
  • If D is an effective Cartier divisor and R \to R' is a ring homomorphism, then D \times_R R' is an effective Cartier divisor in X \times_R R'.
  • If D is an effective Cartier divisor and f: X' \to X a flat morphism over R, then D' = D \times_X X' is an effective Cartier divisor in X' with the ideal sheaf I(D') = f^* (I(D)).

Taking I(D)^{-1} \otimes_{\mathcal{O}_X} - of 0 \to I(D) \to \mathcal{O}_X \to \mathcal{O}_D \to 0 gives the exact sequence

0 \to \mathcal{O}_X \to I(D)^{-1} \to I(D)^{-1} \otimes \mathcal{O}_D \to 0.

This allows one to see global sections of \mathcal{O}_X as global sections of I(D)^{-1}. In particular, the constant 1 on X can be thought of as a section of I(D)^{-1} and D is then the zero locus of this section. Conversely, if L is a line bundle on X and s a global section of it that is a nonzerodivisor on \mathcal{O}_X and if L/\mathcal{O}_X is flat over R, then s = 0 defines an effective Cartier divisor whose ideal sheaf is isomorphic to the inverse of L.

From now on suppose X is a smooth curve (still over R). Let D be an effective Cartier divisor in X and assume it is proper over R (which is immediate if X is proper.) Then \Gamma(D, \mathcal{O}_D) is a locally free R-module of finite rank. This rank is called the degree of D and is denoted by \operatorname{deg} D. It is a locally constant function on \operatorname{Spec} R. If D and D' are proper effective Cartier divisors, then D + D' is proper over R and \operatorname{deg}(D + D') = \operatorname{deg}(D) + \operatorname{deg}(D'). Let f: X' \to X be a finite flat morphism. Then \operatorname{deg}(f^* D) = \operatorname{deg}(f) \operatorname{deg}(D).[3] On the other hand, a base change does not change degree: \operatorname{deg}(D \times_R R') = \operatorname{deg}(D).[4]

A closed subscheme D of X is finite, flat and of finite presentation if and only if it is an effective Cartier divisor that is proper over R.[5]

See also[edit]

Notes[edit]

  1. ^ Kleiman, p. 203
  2. ^ Section VI.6 of Dieudonné (1985).
  3. ^ Katz–Mazur 1985, Lemma 1.2.8.
  4. ^ Katz–Mazur 1985, Lemma 1.2.9.
  5. ^ Katz–Mazur 1985, Lemma 1.2.3.

References[edit]