Picard group
Encyclopedia
In mathematics
, the Picard group of a ringed space
X, denoted by
, is the group of isomorphism
classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product
. This construction is a global version of the construction of the divisor class group, or ideal class group
, and is much used in algebraic geometry
and the theory of complex manifold
s.
Alternatively, the Picard group can be defined as the sheaf cohomology
group
For integral schemes
the Picard group can be shown to be isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence
gives basic information on the Picard group.
's theories, in particular of divisors on algebraic surface
s.
The Picard group of the spectrum of a Dedekind domain
is its ideal class group.
The invertible sheaves on projective space
for
a field
, are the twisting sheaves
so the Picard group of
is isomorphic to 
. The Picard group of the affine line with two origins over 
is isomorphic to 
.
version of) the Picard group, the Picard scheme, is an important step in algebraic geometry, in particular in the duality theory of abelian varieties. It was constructed by , and also described by and . The Picard variety is dual to the Albanese variety
of classical algebraic geometry.
In the cases of most importance to classical algebraic geometry, for a complete variety V that is non-singular, the connected component
of the identity in the Picard scheme is an abelian variety
written Pic0(V). In the particular case where V is a curve, this neutral component is the Jacobian variety
of V.
The quotient Pic(V)/Pic0(V) is a finitely-generated abelian group denoted NS(V), the Néron–Severi group of V. In other words the Picard group fits into an exact sequence
The fact that the rank is finite is Francesco Severi
's theorem of the base; the rank is the Picard number of V, often denoted ρ(V). Geometrically NS(V) describes the algebraic equivalence classes of divisors
on V; that is, using a stronger, non-linear equivalence relation in place of linear equivalence of divisors, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to numerical equivalence, an essentially topological classification by intersection number
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Picard group of a ringed space
Ringed space
In mathematics, a ringed space is, intuitively speaking, a space together with a collection of commutative rings, the elements of which are "functions" on each open set of the space...
X, denoted by
, is the group of isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product
Tensor 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...
. This construction is a global version of the construction of the divisor class group, or ideal class group
Ideal class group
In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
, and is much used in algebraic geometry
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...
and the theory of complex manifold
Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....
s.
Alternatively, the Picard group can be defined as the sheaf cohomology
Sheaf cohomology
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F...
group
For integral schemes
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
the Picard group can be shown to be isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence
Exponential sheaf sequence
In mathematics, the exponential sheaf sequence is a fundamental short exact sequence of sheaves used in complex geometry.Let M be a complex manifold, and write OM for the sheaf of holomorphic functions on M. Let OM* be the subsheaf consisting of the non-vanishing holomorphic functions. These are...
gives basic information on the Picard group.
Examples
The name is in honour of Charles Émile PicardCharles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
's theories, in particular of divisors on algebraic surface
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
s.
The Picard group of the spectrum of a 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...
is its ideal class group.
The invertible sheaves on projective space
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
for
a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, are the twisting sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
so the Picard group of
is isomorphic to . The Picard group of the affine line with two origins over is isomorphic to .Picard scheme
The construction of a scheme structure on (representable functorRepresentable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
version of) the Picard group, the Picard scheme, is an important step in algebraic geometry, in particular in the duality theory of abelian varieties. It was constructed by , and also described by and . The Picard variety is dual to the Albanese variety
Albanese variety
In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...
of classical algebraic geometry.
In the cases of most importance to classical algebraic geometry, for a complete variety V that is non-singular, the connected component
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
of the identity in the Picard scheme is an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
written Pic0(V). In the particular case where V is a curve, this neutral component is the Jacobian variety
Jacobian variety
In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
of V.
The quotient Pic(V)/Pic0(V) is a finitely-generated abelian group denoted NS(V), the Néron–Severi group of V. In other words the Picard group fits into an exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
The fact that the rank is finite is Francesco Severi
Francesco Severi
Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
's theorem of the base; the rank is the Picard number of V, often denoted ρ(V). Geometrically NS(V) describes the algebraic equivalence classes of divisors
Divisor (algebraic geometry)
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...
on V; that is, using a stronger, non-linear equivalence relation in place of linear equivalence of divisors, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to numerical equivalence, an essentially topological classification by intersection number
Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency...
s.