divisor

In algebraic geometry and complex analysis, the term divisor plays an important role in the investigation of algebraic varieties or complex manifolds and the functions defined on them. A distinction must be made between the Weil divisor and the Cartier divisor , which in certain cases match.

Originally, the divisor in the one-dimensional case has the meaning of prescribing the zero and pole set of a rational or meromorphic function, and the question arises for which divisors such a realization is possible, which is closely related to the geometry of the variety or manifold is linked.

One-dimensional case

Function theory

definition

Let be a region or a Riemann surface . A mapping is called a divisor in if its carrier in is closed and discrete . The set of all divisors on forms an abelian group with respect to addition, which is denoted by. A partial order is introduced on this group . Be , then you bet , if applies to all . ${\ displaystyle \ Omega \ subseteq \ mathbb {C}}$ ${\ displaystyle D \ colon \ Omega \ rightarrow \ mathbb {Z}}$ ${\ displaystyle \ Omega}$ ${\ displaystyle T: = \ {z \ in \ Omega \,: \, D (z) \ neq 0 \}}$ ${\ displaystyle \ Omega}$ ${\ displaystyle \ Omega}$ ${\ displaystyle \ operatorname {Div} (\ Omega)}$ ${\ displaystyle D, D '\ in \ operatorname {Div} (\ Omega)}$ ${\ displaystyle D \ leq D '}$ ${\ displaystyle D (x) \ leq D '(x)}$ ${\ displaystyle x \ in \ Omega}$

Main divisor

A divisor can be defined for each non-zero meromorphic function by assigning the zero or pole order to each point : ${\ displaystyle f \ colon \ Omega \ to \ mathbb {C}}$ ${\ displaystyle (f)}$ ${\ displaystyle \ Omega}$

${\ displaystyle \ left (f \ right) (x): = {\ begin {cases} 0, & f {\ mbox {holomorphic and not zero in}} x \\ k, & f {\ mbox {has a zero of order }} k {\ mbox {in}} x \\ - k, & f {\ mbox {has a pole of order}} k {\ mbox {in}} x \ end {cases}}}$

A divisor that is equal to the divisor of a meromorphic function is called a main divisor .

The Weierstrass product theorem states that in any divisor a principal divisor is. In a compact Riemann surface, however, this no longer applies and is dependent on the gender of the surface. This is explained in more detail in the article Theorem by Riemann-Roch . ${\ displaystyle \ mathbb {C}}$

Algebraic Curves

Let be a plane algebraic curve . A formal sum is called a divisor in , if except for finitely many . By pointwise addition, the set of all divisors in becomes a free Abelian group . ${\ displaystyle C}$ ${\ displaystyle \ textstyle \ sum _ {P \ in C} a_ {P} \ cdot P, \; a_ {P} \ in \ mathbb {Z}}$ ${\ displaystyle C}$ ${\ displaystyle a_ {P} = 0}$ ${\ displaystyle P}$ ${\ displaystyle C}$

Analogous to the above Definition one defines the divisor of the function for a rational function. A divisor that is equal to the divisor of a rational function is called a main divisor .

In the case of a divisor, the mapping is a divisor in terms of function theory. However, there are divisors in the sense of function theory that do not arise in this way, since there is allowed for an infinite number (which, however, must not have an accumulation point). ${\ displaystyle C = \ mathbb {C}}$ ${\ displaystyle P \ mapsto a_ {P}}$ ${\ displaystyle a_ {P} \ neq 0}$ ${\ displaystyle P}$

general definition

Because divisor

Let be a Noetherian integral separated schema , regular in codimension 1. A prime divisor in is a closed whole sub-schema of codimension one. A Weil divisor (according to André Weil ) is then an element of the freely generated Abelian group of prime divisors and is usually written as a formal sum , with only a finite number different from zero. ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle X}$ ${\ displaystyle \ mathrm {Div} \, X}$ ${\ displaystyle \ textstyle D = \ sum _ {j} a_ {j} \ cdot Y_ {j}, \; a_ {j} \ in \ mathbb {Z},}$ ${\ displaystyle a_ {j}}$

A because divisor is called effective (or positive ) if applies to all . ${\ displaystyle a_ {j} \ geq 0}$ ${\ displaystyle j}$

A because divisor is called a main divisor if it is equal to the divisor of a non-zero rational function: Let be a rational function on , non-zero. For each prime divisor in denote the evaluation of in the discrete evaluation ring that belongs to a generic point of . The evaluation is independent of the choice of the generic point. In the one-dimensional case, the evaluation corresponds to the degree of the zero or pole of at this point. is then called a divisor of and actually defines a Weil divisor, the summands are only different from zero for finitely many prime divisors. ${\ displaystyle f}$ ${\ displaystyle X}$ ${\ displaystyle Y}$ ${\ displaystyle X}$ ${\ displaystyle v (f, Y)}$ ${\ displaystyle f}$ ${\ displaystyle Y}$ ${\ displaystyle f}$ ${\ displaystyle \ textstyle (f): = \ sum _ {Y} v (f, Y) \ cdot Y}$ ${\ displaystyle f}$

Two Weil divisors are called linear equivalent if their difference is a major divisor. The quotient of with respect to this equivalence is the divisor class group and is denoted by. ${\ displaystyle \ mathrm {Div} \, X}$ ${\ displaystyle \ mathrm {Cl} \, X}$

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 a cut in is called the Cartier divisor (after Pierre Cartier ), usually just referred to as a divisor . The set of all cuts forms an Abelian group. ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {O}}}$ ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {M}}}$ ${\ displaystyle X}$ ${\ displaystyle {\ mathcal {D}}: = {\ mathcal {M}} ^ {*} / {\ mathcal {O}} ^ {*}}$ ${\ displaystyle {\ mathcal {D}}}$ ${\ displaystyle \ Gamma (X, {\ mathcal {M}} ^ {*} / {\ mathcal {O}} ^ {*})}$

A Cartier divisor is called the main divisor if it lies in the image of the natural mapping , i.e. is the divisor of a non-vanishing meromorphic function. ${\ displaystyle \ Gamma (X, {\ mathcal {M}} ^ {*}) \ to \ Gamma (X, {\ mathcal {M}} ^ {*} / {\ mathcal {O}} ^ {*} )}$

Two Cartier divisors are called linear equivalent if their quotient is a major divisor. The quotient with respect to this equivalence is denoted by. ${\ displaystyle \ mathrm {CaCl} \, X}$

Relationship between Cartier and Weil divisors

Let be a Noetherian integral separated schema whose local rings are all factorial . Then the group of Weil divisors is isomorphic to the group of Cartier divisors . This isomorphism receives the property of being the main divisor and converts the quotient groups and into one another. ${\ displaystyle X}$ ${\ displaystyle \ mathrm {Div} \, X}$ ${\ displaystyle X}$ ${\ displaystyle \ Gamma (X, {\ mathcal {M}} ^ {*} / {\ mathcal {O}} ^ {*})}$ ${\ displaystyle \ mathrm {Cl} \, X}$ ${\ displaystyle \ mathrm {CaCl} \, X}$

Web links

Wiktionary: Divisor - explanations of meanings, word origins, synonyms, translations

literature

Joseph L. Taylor: Several Complex Variables with Connections to Algebraic Geometry and Lie Groups. American Mathematical Society 2002, ISBN 0-8218-3178-X
William Fulton: Algebraic Curves. An Introduction to Algebraic Geometry. Mathematics lecture note series, 30. Benjamin / Cummings, New York 1969, ISBN 0-201-51010-3
Robin Hartshorne: Algebraic Geometry . Springer-Verlag 1977. ISBN 0-387-90244-9
Reinhold Remmert, Georg Schumacher: Function theory 2. Springer, Berlin 2007, ISBN 978-3-540-57052-3