In
mathematics, the Picard group of a
ringed space
In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf of ...
''X'', denoted by Pic(''X''), is the group of
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
classes of
invertible sheaves
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 ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
(or
line bundles) on ''X'', with the
group operation
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. Thes ...
being
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
. This construction is a global version of the construction of the divisor class group, or
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 ...
, and is much used in
algebraic geometry and the theory of
complex manifolds.
Alternatively, the Picard group can be defined as the
sheaf cohomology group
:
For integral
schemes the Picard group is isomorphic to the class group of
Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...
s. For complex manifolds the
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 ''O'M'' for the sheaf of holomorphic functions on ''M''. Let ''O'M''* be ...
gives basic information on the Picard group.
The name is in honour of
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at t ...
's theories, in particular of divisors on
algebraic surfaces.
Examples
* The Picard group of the
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
of a
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 ...
is its ''
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 ...
''.
* The invertible sheaves on
projective space P
''n''(''k'') for ''k'' a
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 ...
, are the
twisting sheaves so the Picard group of P
''n''(''k'') is isomorphic to Z.
*The Picard group of the affine line with two origins over ''k'' is isomorphic to Z.
*The Picard group of the
-dimensional
complex affine space
Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces diff ...
:
, indeed the
exponential 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 ''O'M'' for the sheaf of holomorphic functions on ''M''. Let ''O'M''* be ...
yields the following long exact sequence in cohomology
:
and since
we have
because
is contractible, then
and we can apply the
Dolbeault isomorphism to calculate
by the
Dolbeault-Grothendieck lemma.
Picard scheme
The construction of a scheme structure on (
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets 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 In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field ''K''.
Definition
To an abelian variety ''A'' over a field ''k'', one associates a dual abelian variety ''A''v (over the same field), which i ...
. It was constructed by , and also described by and .
In the cases of most importance to classical algebraic geometry, for a
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In ca ...
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). Thi ...
''V'' over a
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
characteristic zero, the
connected component of the identity in the Picard scheme is an
abelian variety called the Picard variety and denoted Pic
0(''V''). The dual of the Picard variety is the
Albanese variety
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
Precise statement
The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to ...
, and in the particular case where ''V'' is a curve, the Picard variety is naturally isomorphic to 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 ''V''. For fields of positive characteristic however,
Igusa constructed an example of a smooth projective surface ''S'' with Pic
0(''S'') non-reduced, and hence not an
abelian variety.
The quotient Pic(''V'')/Pic
0(''V'') is 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 ...
denoted NS(''V''), the
Néron–Severi group of ''V''. In other words the Picard group fits into an
exact sequence
An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next.
Definition
In the context ...
:
The fact that the rank of NS(''V'') 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 In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of such cycles, and in particular, well-defined inte ...
classes of
divisors
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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
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 (more than 2) curves, and accounting properly for ta ...
s.
Relative Picard scheme
Let ''f'': ''X'' →''S'' be a morphism of schemes. The relative Picard functor (or relative Picard scheme if it is a scheme) is given by:
for any ''S''-scheme ''T'',
:
where
is the base change of ''f'' and ''f''
''T'' * is the pullback.
We say an ''L'' in
has degree ''r'' if for any geometric point ''s'' → ''T'' the pullback
of ''L'' along ''s'' has degree ''r'' as an invertible sheaf over the fiber ''X''
''s'' (when the degree is defined for the Picard group of ''X''
''s''.)
See also
*
Sheaf cohomology
*
Chow variety
In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given projective space. More precisely, the Chow var ...
*
Cartier divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...
*
Holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
*
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 ...
*
Arakelov class group
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of pro ...
*
Group-stack In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a scheme whose sets of points have group structures ...
*
Picard category
Notes
References
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{{Authority control
Geometry of divisors
Scheme theory
Abelian varieties