Divisor (algebraic geometry)

The term of the divisor plays an important role in studying algebraic varieties or Complex manifolds and the functions defined thereon in algebraic geometry, and complex analysis. Must be distinguished here the Weil divisor and the Cartier divisor, which coincide in certain cases.

Originally the divisor is the one-dimensional case, the meaning to prescribe the zero and pole set of a rational or meromorphic function, and it begs the question, for what divisors such an implementation is possible, which closely related to the geometry of the variety and diversity is associated.

• 2.1 Weil divisor
• 2.2 Cartier divisor
• 2.3 Relationship between Cartier and Weil divisors

One-dimensional case

Function theory

Definition

Be an area or a Riemann surface. A mapping is called a divisor in if you completed in support and is discrete. The set of all divisors on forms with respect to addition is an abelian group, which is denoted by. In this group one introduces a partial order. Be, then placed on, if for all.

Principal divisor

For each nonzero meromorphic function can be defined a divisor by the divisor assigns to each point from the zero or the Polstellenordnung:

A divisor is equal to the divisor of a meromorphic function, ie, principal divisor.

The Weierstrass product theorem states that every divisor is a divisor. In a compact Riemann surface, however, this is no longer valid and is the genus of the surface dependent. This is explained in the article theorem of Riemann -Roch closer.

Algebraic curves

Be a plane algebraic curve. A formal sum is called the divisor in case except for a finite number. By pointwise addition is the set of all divisors in to a free Abelian group.

Analogous to the above-mentioned Definition is defined for a rational function of the divisor, and a divisor of the function which is equal to the divisor of a rational function, called principal divisor.

In the case of a divisor is the picture one divisor in the sense of the theory of functions.

General definition

Weil divisor

Be a noetherian integrity -separated scheme, regular in codimension 1 in A divisor is a completed whole sub - scheme of codimension one. Because a divisor ( by André Weil) is then an element of the generated free Abelian group of divisors and is usually written as a formal sum, whereby only a finite number different from zero.

• A Weil divisor is called effective ( or positive) if and only if for all.

• A Weil divisor is called principal divisor if it is equal to the divisor of a nonzero rational function: Let a rational function, different from zero. For each prime divisor in denote the evaluation of the discrete valuation ring, which belongs to a generic point of. The rating is independent of the choice of the generic point. In the one-dimensional case corresponds to the evaluation of the degree of the zero or pole of at this point. then is called a divisor of and actually defines a Weil divisor, the summands are different only for a finite number of prime divisors of zero.

• Two Weil divisors are called linearly equivalent if their difference is a principal divisor. The quotient of the equivalence with respect to the divisor and is designated by.

Cartier divisor

Let be a complex manifold or an algebraic variety and denote the sheaf of holomorphic or algebraic functions and denote the sheaf of meromorphic or rational functions. The quotient sheaf is called the sheaf of divisors, and one cut in is called Cartier divisor ( by Pierre Cartier ), usually referred to only as the divisor. The set of all sections forms an Abelian group.

• A Cartier divisor is called principal divisor, if it lies in the image of the natural picture, so the divisor is a non-zero meromorphic function.

• Two Cartier divisors are called linearly equivalent if their quotient is a principal divisor. The ratio with respect to this equivalence is designated.

Relationship between Cartier and Weil divisors

Be a noetherian integrity -separated scheme whose local rings are all factorial. Then the group of Weil divisors on is isomorphic to the group of Cartier divisors. This isomorphism obtains the property of being divisor and the quotient resulting groups and merge into one another.