In
algebraic geometry, divisors are a generalization of
codimension-1 subvarieties of
algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for
Pierre Cartier and
André Weil by
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
). Both are derived from the notion of divisibility in the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s and
algebraic number fields.
Globally, every codimension-1 subvariety of
projective space is defined by the vanishing of one
homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables; ...
; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a
complete intersection.) Locally, every codimension-1 subvariety of a
smooth variety In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smo ...
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 play the role of
homology classes, while Cartier divisors represent
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
classes. On a smooth variety (or more generally a
regular scheme
In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth..
For an example of a regul ...
), a result analogous to
Poincaré duality
In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
says that Weil and Cartier divisors are the same.
The name "divisor" goes back to the work of
Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and
the axiomatic foundations of arithmetic. His ...
and
Weber, who showed the relevance of
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 ...
s to the study of
algebraic curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s. The group of divisors on a curve (the
free abelian group generated by all divisors) is closely related to the group of
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 ...
s for a Dedekind domain.
An
algebraic cycle In mathematics, an algebraic cycle on an algebraic variety ''V'' is a formal linear combination of subvarieties of ''V''. These are the part of the algebraic topology of ''V'' that is directly accessible by algebraic methods. Understanding the a ...
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
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
is a 1-dimensional
complex manifold, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
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
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
coefficients. The degree of a divisor on ''X'' is the sum of its coefficients.
For any nonzero
meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The ...
''f'' on ''X'', one can define the order of vanishing of ''f'' at a point ''p'' in ''X'', ord
''p''(''f''). 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 (''f'') are also called principal divisors. Since (''fg'') = (''f'') + (''g''), 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
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of meromorphic functions on ''X'' with poles at most given by ''D'', called ''H''
0(''X'', ''O''(''D'')) 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 (because a meromorphic function cannot have more zeros than poles). If ''D'' has positive degree, then the dimension of ''H''
0(''X'', ''O''(''mD'')) grows linearly in ''m'' for ''m'' sufficiently large. The
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It rel ...
is a more precise statement along these lines. On the other hand, the precise dimension of ''H''
0(''X'', ''O''(''D'')) 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 In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers ...
. To define it, one first defines the divisor of a nonzero meromorphic
1-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ...
along the lines above. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''g'' of ''X'' can be read from the canonical divisor: namely, ''K''
''X'' has degree 2''g'' − 2. The key trichotomy among compact Riemann surfaces ''X'' is whether the canonical divisor has negative degree (so ''X'' has genus zero), zero degree (genus one), or positive degree (genus at least 2). For example, this determines whether ''X'' has a
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
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
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
CP
1.
Weil divisors
Let ''X'' be an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
locally Noetherian scheme. A prime divisor or irreducible divisor on ''X'' is an
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
closed subscheme ''Z'' of
codimension 1 in ''X''. A Weil divisor on ''X'' is a
formal sum In mathematics, a formal sum, formal series, or formal linear combination may be:
*In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients.
*In linear algebra, an ...
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
In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally th ...
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''(''X'') is the
field of rational functions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
, then the local ring
is a
discrete valuation ring
In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.
This means a DVR is an integral domain ''R'' which satisfies any one of the following equivalent conditions:
# ''R'' i ...
, 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
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with ref ...
on ''X''. Concretely it may be defined as subsheaf of the sheaf of rational functions
[Kollár (2013), Notation 1.2.]
:
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'' (for example ''D'' can be a reduced divisor or a prime divisor), 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 In algebraic geometry, a reflexive sheaf is a coherent sheaf that is isomorphic to its second dual (as a sheaf of modules) via the canonical map. The second dual of a coherent sheaf is called the reflexive hull of the sheaf. A basic example of a re ...
, and since
is defined as a subsheaf of
it is a fractional ideal sheaf (see below). 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 (again, see below), 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(''X'') is the quotient of Div(''X'') 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
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...
; namely, Cl(''X'') is the Chow group CH
''n''−1(''X'') of (''n''−1)-dimensional cycles.
Let ''Z'' be a closed subset of ''X''. If ''Z'' is irreducible of codimension one, then Cl(''X'' − ''Z'') is isomorphic to the quotient group of Cl(''X'') by the class of ''Z''. If ''Z'' has codimension at least 2 in ''X'', then the restriction Cl(''X'') → Cl(''X'' − ''Z'') is an isomorphism. (These facts are special cases of the
localization sequence for Chow groups.)
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''.
Examples
* Let ''k'' be a field, and let ''n'' be a positive integer. Since the polynomial ring ''k''
1, ..., ''xn''">'x''1, ..., ''xn''is a unique factorization domain, the divisor class group of affine space A
''n'' over ''k'' is equal to zero.
[ Since projective space P''n'' over ''k'' minus a hyperplane ''H'' is isomorphic to A''n'', it follows that the divisor class group of P''n'' is generated by the class of ''H''. From there, it is straightforward to check that Cl(P''n'') is in fact isomorphic to the integers Z, generated by ''H''. Concretely, this means that every codimension-1 subvariety of P''n'' is defined by the vanishing of a single homogeneous polynomial.
* Let ''X'' be an algebraic curve over a field ''k''. Every closed point ''p'' in ''X'' has the form Spec ''E'' for some finite extension field ''E'' of ''k'', and the degree of ''p'' is defined to be the degree of ''E'' over ''k''. Extending this by linearity gives the notion of degree for a divisor on ''X''. If ''X'' is a projective curve over ''k'', then the divisor of a nonzero rational function ''f'' on ''X'' has degree zero. As a result, for a projective curve ''X'', the degree gives a homomorphism deg: Cl(''X'') → Z.
* For the projective line P1 over a field ''k'', the degree gives an isomorphism Cl(P1) ≅ Z. For any smooth projective curve ''X'' with a ''k''-]rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
, the degree homomorphism is surjective, and the kernel is isomorphic to the group of ''k''-points on the Jacobian variety
In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
of ''X'', which is an abelian variety of dimension equal to the genus of ''X''. It follows, for example, that the divisor class group of a complex elliptic curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
is an uncountable
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
abelian group.
* Generalizing the previous example: for any smooth projective variety ''X'' over a field ''k'' such that ''X'' has a ''k''-rational point, the divisor class group Cl(''X'') is an extension of a finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
, the Néron–Severi group, by the group of ''k''-points of a connected group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...
For ''k'' of characteristic zero, is an abelian variety, the Picard variety of ''X''.
*For ''R'' the ring of integers of a number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
, the divisor class group Cl(''R'') := Cl(Spec ''R'') is also called the ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of , and is its subgroup of principal ideals. The class group is a mea ...
of ''R''. It is a finite abelian group. Understanding ideal class groups is a central goal of algebraic number theory.
* Let ''X'' be the quadric
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension ''D'') in a -dimensional space, and it is de ...
cone of dimension 2, defined by the equation ''xy'' = ''z''2 in affine 3-space over a field. Then the line ''D'' in ''X'' defined by ''x'' = ''z'' = 0 is not principal on ''X'' near the origin. Note that ''D'' ''can'' be defined as a set by one equation on ''X'', namely ''x'' = 0; but the function ''x'' on ''X'' vanishes to order 2 along ''D'', and so we only find that 2''D'' is Cartier (as defined below) on ''X''. In fact, the divisor class group Cl(''X'') is isomorphic to the cyclic group Z/2, generated by the class of ''D''.
* Let ''X'' be the quadric cone of dimension 3, defined by the equation ''xy'' = ''zw'' in affine 4-space over a field. Then the plane ''D'' in ''X'' defined by ''x'' = ''z'' = 0 cannot be defined in ''X'' by one equation near the origin, even as a set. It follows that ''D'' is not Q-Cartier on ''X''; that is, no positive multiple of ''D'' is Cartier. In fact, the divisor class group Cl(''X'') is isomorphic to the integers Z, generated by the class of ''D''.
The canonical divisor
Let ''X'' be a normal variety over a perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds:
* Every irreducible polynomial over ''k'' has distinct roots.
* Every irreducible polynomial over ''k'' is separable.
* Every finite extension of ''k' ...
. The smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
locus ''U'' of ''X'' is an open subset whose complement has codimension at least 2. Let ''j'': ''U'' → ''X'' be the inclusion map, then the restriction homomorphism:
:
is an isomorphism, since ''X'' − ''U'' has codimension at least 2 in ''X''. For example, one can use this isomorphism to define the canonical divisor In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over the complex numbers ...
''K''''X'' of ''X'': it is the Weil divisor (up to linear equivalence) corresponding to the line bundle of differential forms of top degree on ''U''. Equivalently, the sheaf on ''X'' is the direct image sheaf In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topo ...
where ''n'' is the dimension of ''X''.
Example: Let ''X'' = P''n'' be the projective ''n''-space with the homogeneous coordinates ''x''0, ..., ''xn''. Let ''U'' = . Then ''U'' is isomorphic to the affine ''n''-space with the coordinates ''yi'' = ''xi''/''x''0. Let
:
Then ω is a rational differential form on ''U''; thus, it is a rational section of which has simple poles along ''Zi'' = , ''i'' = 1, ..., ''n''. Switching to a different affine chart changes only the sign of ω and so we see ω has a simple pole along ''Z''0 as well. Thus, the divisor of ω is
:
and its divisor class is
: