Picard group

A group of classes of invertible sheaves (or line bundles). More precisely, let $ (X,\mathcal{O}_{X}) $ be a ringed space. A sheaf $ \mathcal{L} $ of $ \mathcal{O}_{X} $-modules is called invertible if and only if it is locally isomorphic to the structure sheaf $ \mathcal{O}_{X} $. The set of classes of isomorphic invertible sheaves on $ X $ is denoted by $ \operatorname{Pic}(X) $. The tensor product $ \mathcal{L} \otimes_{\mathcal{O}_{X}} \mathcal{L}' $ defines an operation on the set $ \operatorname{Pic}(X) $, making it an Abelian group called the Picard group of $ X $. The group $ \operatorname{Pic}(X) $ is naturally isomorphic to the cohomology group $ {H^{1}}(X,\mathcal{O}_{X}^{*}) $, where $ \mathcal{O}_{X}^{*} $ is the sheaf of invertible elements in $ \mathcal{O}_{X} $.

For a commutative ring $ A $, the Picard group $ \operatorname{Pic}(A) $ is the group of classes of invertible $ A $-modules; $ \operatorname{Pic}(A) \cong \operatorname{Pic}(\operatorname{Spec}(A)) $. For a Krull ring, the group $ \operatorname{Pic}(A) $ is closely related to the divisor class group for this ring.

The Picard group of a complete normal algebraic variety $ X $ has a natural algebraic structure (see Picard scheme). The reduced connected component of the zero of $ \operatorname{Pic}(X) $ is denoted by $ {\operatorname{Pic}^{0}}(X) $ and is called the Picard variety for $ X $; it is an algebraic group (an Abelian variety if $ X $ is a complete non-singular variety). The quotient group $ \operatorname{Pic}(X) / {\operatorname{Pic}^{0}}(X) $ is called the Néron–Severi group, and it has a finite number of generators; its rank is called the Picard number. In the complex case, where $ X $ is a smooth projective variety over $ \mathbb{C} $, the group $ {\operatorname{Pic}^{0}}(X) $ is isomorphic to the quotient group of the space $ {H^{0}}(X,\Omega_{X}) $ of holomorphic $ 1 $-forms on $ X $ by the lattice $ {H^{1}}(X,\mathbb{Z}) $.

References

[1]	D. Mumford, “Lectures on curves on an algebraic surface”, Princeton Univ. Press (1966). MR0209285 Zbl 0187.42701

References

[a1]	R. Hartshorne, “Algebraic geometry”, Springer (1977), pp. 91. MR0463157 Zbl 0367.14001