Picard Variety
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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or
line 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 organisin ...
s) 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. 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 Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and the theory of
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
s. Alternatively, the Picard group can be defined as the
sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
group :H^1 (X, \mathcal_X^).\, 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 Mum ...
s. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on
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 (as a complex manifold, when it is non-singular) and so of di ...
s.


Examples

* The Picard group of the spectrum of a Dedekind domain is its '' ideal class group''. * The invertible sheaves on
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
P''n''(''k'') for ''k'' a field, are the
twisting Twist may refer to: In arts and entertainment Film, television, and stage * ''Twist'' (2003 film), a 2003 independent film loosely based on Charles Dickens's novel ''Oliver Twist'' * ''Twist'' (2021 film), a 2021 modern rendition of ''Olive ...
sheaves \mathcal(m),\, 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 n-dimensional complex affine space: \operatorname(\mathbb^n)=0, indeed the exponential sequence yields the following long exact sequence in cohomology : \dots\to H^1(\mathbb^n,\underline)\to H^1(\mathbb^n,\mathcal_) \to H^1(\mathbb^n,\mathcal^\star_)\to H^2(\mathbb^n,\underline)\to\cdots and since H^k(\mathbb^n,\underline)\simeq H_^k(\mathbb^n;\mathbb) we have H^1(\mathbb^n,\underline)\simeq H^2(\mathbb^n,\underline)\simeq 0 because \mathbb^n is contractible, then H^1(\mathbb^n,\mathcal_) \simeq H^1(\mathbb^n,\mathcal^\star_) and we can apply the
Dolbeault isomorphism In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault ...
to calculate H^1(\mathbb^n,\mathcal_)\simeq H^1(\mathbb^n,\Omega^0_)\simeq H^_(\mathbb^n)=0 by the
Dolbeault-Grothendieck lemma In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let ''M'' be a complex manifold. Then the Dolbeault cohomo ...
.


Picard scheme

The construction of a scheme structure on ( representable functor 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 . In the cases of most importance to classical algebraic geometry, for a non-singular complete variety ''V'' over a field of characteristic zero, the connected component of the identity in the Picard scheme is an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 func ...
called the Picard variety and denoted Pic0(''V''). The dual of the Picard variety is the Albanese variety, and in the particular case where ''V'' is a curve, the Picard variety is naturally isomorphic to the Jacobian variety of ''V''. For fields of positive characteristic however, Igusa constructed an example of a smooth projective surface ''S'' with Pic0(''S'') non-reduced, and hence not an
abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic 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 func ...
. The quotient Pic(''V'')/Pic0(''V'') is a finitely-generated abelian group denoted NS(''V''), the NĂ©ron–Severi group of ''V''. In other words the Picard group fits into an exact sequence :1\to \mathrm^0(V)\to\mathrm(V)\to \mathrm(V)\to 1.\, The fact that the rank of NS(''V'') is finite is
Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...
'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 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 Mumfo ...
, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to numerical equivalence, an essentially topological classification by intersection numbers.


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'', :\operatorname_(T) = \operatorname(X_T)/f_T^*(\operatorname(T)) where f_T: X_T \to T is the base change of ''f'' and ''f''''T'' * is the pullback. We say an ''L'' in \operatorname_(T) has degree ''r'' if for any geometric point ''s'' → ''T'' the pullback s^*L 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 In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when i ...
* Chow variety *
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 Mum ...
* Holomorphic line bundle * Ideal class group * Arakelov class group * Group-stack *
Picard category Picard may refer to: *Picardy, a region of France *Picard language, a language of France *Jean-Luc Picard, a fictional character in the ''Star Trek'' franchise Places * Picard, California, USA * Picard, Quebec, Canada * Picard (crater), a lunar ...


Notes


References

* * * * * * * {{Authority control Geometry of divisors Scheme theory Abelian varieties