Picard Functor
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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 ...
''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 ...
(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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), 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 e ...
. 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 ...
, and is much used in algebraic geometry 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 ...
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 whe ...
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 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 th ...
'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 color ...
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 necessari ...
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 ...
''. * 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 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 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 dif ...
: \operatorname(\mathbb^n)=0, 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 th ...
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 coh ...
to calculate H^1(\mathbb^n,\mathcal_)\simeq H^1(\mathbb^n,\Omega^0_)\simeq H^_(\mathbb^n)=0 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 an ...
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 ...
. 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 ...
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). This ...
''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 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 functi ...
called the Picard variety and denoted Pic0(''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 t ...
, 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 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 functi ...
. The quotient Pic(''V'')/Pic0(''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 conte ...
: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 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 ...
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'', :\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 whe ...
*
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 va ...
*
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 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 ...
*
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 ...
* Arakelov class group *
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 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