Divisor (algebraic geometry)
Encyclopedia
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.
of points on 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
where R(f) is the set of all zeroes and poles of f, and sν is given by
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 class of equivalence 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.
(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.
, and a collection of rational function
s
defined on 
. 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 
can be chosen to be regular function
s, and in this case the Cartier divisor defines an associated subvariety of codimension 1 by forming the ideal sheaf generated locally by the
.
The notion can be described more conceptually with the function field
. For each affine open subset U, define K′(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.) The sheaf KX of rational functions on X is the sheaf associated to the presheaf K′, and the quotient sheaf is the sheaf of local Cartier divisors.
A Cartier divisor is a global section of the quotient sheaf KX*/OX*. We have the exact sequence
, so, applying the global section functor 
gives the exact sequence 
.
A Cartier divisor is said to be principal if it is in the range of the morphism
, that is, if it is the class of a global rational function.
, 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.
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 Dedekind
–Weber work which in effect showed the relevance of Dedekind domain
s to the case of algebraic curve
s. In that case the free abelian group on the points of the curve is closely related to the fractional ideal
theory.
(strictly, invertible sheaf
) commonly denoted by OX(D).
The line bundle
associated to the Cartier divisor D is the sub-bundle of the sheaf KX of rational fractions described above whose stalk at 
is given by 
viewed as a line on the stalk at x of 
in the stalk at x of 
. The subsheaf thus described is tautollogically locally freely monogenous over the structure sheaf 
.
The application
is a group morphism: 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 in the Picard group The mapping is not always surjective.
in a variety 
can be thought of as a the transition maps of a line bundle 
, and linear equivalence as 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
. 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
has non-zero global sections. Two colinear non-zero global sections have the same vanishing locus, and hence the projective space 
over k identifies with the set of effective divisors linearly equivalent to 
.
If the space
is noetherian, then 
is finite dimensional and is called the complete linear system of 
. Its subspaces are called linear systems of divisors, and constitute a fundamental tool in algebraic geometry. The Riemann-Roch theorem for algebraic curves is a fundamental identity involving the dimension of complete linear systems in the simple setup of projective curves.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, divisors are a generalization of codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier
Pierre Cartier (mathematician)
Pierre Cartier is a mathematician. An associate of the Bourbaki group and at one time a colleague of Alexander Grothendieck, his interests have ranged over algebraic geometry, representation theory, mathematical physics, and category theory....
and André Weil
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
). These concepts agree on non-singular varieties.
Divisors in a Riemann surface
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 groupFree abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
of points on 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
Meromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
f as
where R(f) is the set of all zeroes and poles of f, and sν is given by
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 class of equivalence of these divisors is called the canonical divisor (usually denoted K).
The Riemann–Roch theorem
Riemann–Roch theorem
The Riemann–Roch theorem is an important tool in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles...
is an important relation between the divisors of a Riemann surface and its topology.
Weil divisor
A Weil divisor is a locally finite linear combinationLinear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
(with integral
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
coefficients) of irreducible subvarieties of codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
one. The set of Weil divisors forms an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
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
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
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
A Cartier divisor can be represented by an open cover by affine sets, and a collection of rational functionRational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s
defined on . 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 can be chosen to be regular functionRegular function
In mathematics, a regular function is a function that is analytic and single-valued in a given region. In complex analysis, any complex regular function is known as a holomorphic function...
s, and in this case the Cartier divisor defines an associated subvariety of codimension 1 by forming the ideal sheaf generated locally by the
.The notion can be described more conceptually with the function field
Function field (scheme theory)
In algebraic geometry, the function field KX of a scheme Xis a generalization of the notion of a sheaf of rational functions on a variety. In the case of varieties, such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, KX is the set of...
. For each affine open subset U, define K′(U) to be the total quotient ring
Total quotient ring
In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of a domain to commutative rings that may have zero divisors. The construction embeds the ring in a larger ring, giving every non-zerodivisor of the...
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.) The sheaf KX of rational functions on X is the sheaf associated to the presheaf K′, and the quotient sheaf is the sheaf of local Cartier divisors.
A Cartier divisor is a global section of the quotient sheaf KX*/OX*. We have the exact sequence
, so, applying the global section functor gives the exact sequence .A Cartier divisor is said to be principal if it is in the range of the morphism
, that is, if it is the class of a global rational function.Cartier divisors in non rigid sheaves
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 spaceEuclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, or locally homeomorphic, or diffeomorphic, to such a set, such as a topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
), 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
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
Mathematical singularity
In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
.
An example of a surface on which the two concepts differ is a cone, i.e. a singular quadric
Quadric
In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
. 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 Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
–Weber work which in effect showed the relevance of Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
s to the case of algebraic curve
Algebraic curve
In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
s. In that case the free abelian group on the points of the curve is closely related to the fractional ideal
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
theory.
From Cartier divisors to line bundles
The notion of transition map associates naturally to every Cartier divisor D a line bundleLine bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...
(strictly, invertible sheaf
Invertible sheaf
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
) commonly denoted by OX(D).
The line bundle
associated to the Cartier divisor D is the sub-bundle of the sheaf KX of rational fractions described above whose stalk at is given by viewed as a line on the stalk at x of in the stalk at x of . The subsheaf thus described is tautollogically locally freely monogenous over the structure sheaf .The application
is a group morphism: the sum of divisors corresponds to the tensor productTensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
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 in the Picard group The mapping is not always surjective.
Global sections of line bundles and linear systems
Recall that the local equations of a Cartier divisor in a variety can be thought of as a the transition maps of a line bundle , and linear equivalence as 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
. 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
has non-zero global sections. Two colinear non-zero global sections have the same vanishing locus, and hence the projective space over k identifies with the set of effective divisors linearly equivalent to .If the space
is noetherian, then is finite dimensional and is called the complete linear system of . Its subspaces are called linear systems of divisors, and constitute a fundamental tool in algebraic geometry. The Riemann-Roch theorem for algebraic curves is a fundamental identity involving the dimension of complete linear systems in the simple setup of projective curves.See also
- nef divisor
- ample divisor
- Theta-divisorTheta-divisorIn mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers by the zero locus of the associated Riemann theta-function...
- Linear system of divisorsLinear system of divisorsIn algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family....