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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 Divisor s. For complex manifolds the Exponential Sheaf Sequence gives basic information on the Picard group.

One can also think of the Picard group as a group of holomorphic line bundles over X.
The group operation is the tensor multiplication of line bundles.


EXAMPLES

The name is in honour of Charles-Emile Picard '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

:\mathbb{P}^n_k,

for k a Field , are the twisting Sheaves

:\mathcal{O}(m),

so the Picard group of \mathbb{P}^n_k is isomorphic to \mathbb{Z}. The Picard group of the affine line with two origins over k is also isomorphic to \mathbb{Z}.


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 carried out by Alexander Grothendieck at the beginning of the 1960s. See also David Mumford 's book ''Lectures on Curves on an Algebraic Surface''. 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 Picard scheme is an Abelian Variety written

Pic


and the quotient

Pic


is a Finitely-generated Abelian Group ''NS''(''V''), the Néron-Severi group of ''V''.

In other words the Picard group fits into an Exact Sequence

:1 o \mathrm{Pic}^0(V) o\mathrm{Pic}(V) o \mathrm{NS}(V) o 0

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.


REFERENCES