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. Both are derived from the notion of divisibility in the integers and algebraic number fields.
Globally, every codimension-1 subvariety of a 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. 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 property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties and the corresponding line bundles.
On singular varieties, this property can also 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 correspond to homology cycles, while Cartier divisors correspond to cohomology classes defined by line bundles. On a smooth variety, 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. The group of divisors on a curve 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.

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. 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 are also called principal divisors. Since = +, 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 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. If D has positive degree, then the dimension of H⁰ 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 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, zero degree, or positive degree. 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

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,
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. A Weil divisor D is effective if all the coefficients are non-negative. One writes if the difference 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 is a formal sum of prime numbers with integer coefficients and therefore corresponds to a non-zero fractional ideal in Q. 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 has Krull dimension one. If is non-zero, then the order of vanishing of f along Z, written, is the length of This length is finite, and it is additive with respect to multiplication, that is,. If k is the field of rational functions on X, then any non-zero may be written as a quotient, where g and h are in and the order of vanishing of f is defined to be. With this definition, the order of vanishing is a function. If X is normal, then the local ring is a discrete valuation ring, and the function 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. 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 is a homomorphism, and in particular its image is a subgroup of the group of all Weil divisors.
Let X be a normal integral Noetherian scheme. Every Weil divisor D determines a coherent sheaf on X. Concretely it may be defined as subsheaf of the sheaf of rational functions
That is, a nonzero rational function f is a section of over U if and only if for any prime divisor Z intersecting U,
where n_Z 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
since is an effective divisor and so is regular thanks to the normality of X. Conversely, if is isomorphic to as an -module, then D is principal. It follows that D is locally principal if and only if is invertible; that is, a line bundle.
If D is an effective divisor that corresponds to a subscheme of X, then the ideal sheaf of the subscheme D is equal to This leads to an often used short exact sequence,
The sheaf cohomology of this sequence shows that contains information on whether regular functions on D are the restrictions of regular functions on X.
There is also an inclusion of sheaves
This furnishes a canonical element of namely, the image of the global section 1. This is called the canonical section and may be denoted s_D. While the canonical section is the image of a nowhere vanishing rational function, its image in vanishes along D because the transition functions vanish along D. When D is a smooth Cartier divisor, the cokernel of the above inclusion may be identified; see #Cartier divisors below.
Assume that X is a normal integral separated scheme of finite type over a field. Let D be a Weil divisor. Then is a rank one reflexive sheaf, and since is defined as a subsheaf of it is a fractional ideal sheaf. 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, and because the singular locus has codimension at least two, the closure of the Cartier divisor is a Weil divisor.

Divisor class group

The Weil divisor class group Cl is the quotient of Div 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 is the Chow group CH_n−1 of -dimensional cycles.
Let Z be a closed subset of X. If Z is irreducible of codimension one, then Cl is isomorphic to the quotient group of Cl by the class of Z. If Z has codimension at least 2 in X, then the restriction Cl → Cl is an isomorphism.
On a normal integral Noetherian scheme X, two Weil divisors D, E are linearly equivalent if and only if and are isomorphic as -modules. Isomorphism classes of reflexive sheaves on X form a monoid with product given as the reflexive hull of a tensor product. Then defines a monoid isomorphism from the Weil divisor class group of X to the monoid of isomorphism classes of rank-one reflexive sheaves on X.