Picard group

Picard group
Name	Picard group
Field	Algebraic geometry
Introduced by	Émile Picard
Notation	Pic(X)

Picard group The Picard group is a fundamental invariant in Algebraic geometry and Complex geometry that classifies isomorphism classes of invertible sheaves on a scheme or line bundles on a variety. It connects to classical objects such as the Jacobian variety, the Néron–Severi group, and the class group of a Dedekind domain, and appears in the contexts of Riemann–Roch theorem, Hodge theory, and the study of elliptic curves. The Picard group plays a central role in moduli problems involving vector bundles, in the formulation of dualities like Serre duality, and in arithmetic questions related to the Brauer group.

Definition and basic properties

For a scheme X, the Picard group is the group of isomorphism classes of line bundles (invertible sheaves) on X with tensor product as the group law; typical notation is Pic(X). Over a Complex manifold or a smooth projective curve C, line bundles correspond to holomorphic line bundles and divisor classes, relating Pic(C) to the Jacobian variety J(C) and to degree maps to the integers. Basic properties include functoriality under pullback by a morphism f: X → Y, compatibility with restriction to open subschemes such as affine covers like Spec Z-patches, and relations with cohomology groups such as H^1(X, O_X^×) and H^2(X, G_m). For projective varieties like Projective space P^n, Pic(P^n) is generated by O(1), while for abelian varieties like Complex toruses the Picard group links to polarizations and the Rosati involution.

Examples and computations

On a smooth projective curve C over a field k, Pic^0(C) is isomorphic to the Jacobian variety J(C), and Pic(C) decomposes as Z × Pic^0(C) via the degree map. For P^n over a field such as C or F_p, Pic(P^n) ≅ Z generated by O(1), a fact used in the study of Grassmannians and Veronese embedding. For a Dedekind scheme like Spec of the ring of integers O_K of a number field K, the Picard group coincides with the ideal class group Cl(O_K). For singular varieties such as nodal curves or normal surface singularities studied by Kodaira and Artin, the Picard group can have nontrivial components detected by the Weil divisor class group and by resolution maps to smooth models like blow-ups studied by Zariski. Computational techniques use Čech cohomology, Leray spectral sequence for maps to Spec k, and the exponential sequence on complex varieties involving O_X and O_X^×; these tools relate Pic groups to H^1(X, O_X) and H^2(X, Z) as seen in Hodge decomposition.

Relation to line bundles and divisor class groups

The Picard group classifies isomorphism classes of line bundles; under mild hypotheses on a normal integral scheme X, there is a map from the group of Cartier divisors Cartier(X) to Pic(X) which becomes an isomorphism modulo principal divisors, linking Pic(X) to the Cartier divisor class group. For proper varieties over algebraically closed fields such as k = C, the Néron–Severi group NS(X) = Pic(X)/Pic^0(X) is finitely generated by the Néron–Severi theorem and is related to intersection theory on Chow groups and to the Mori cone in birational geometry studied by Mori and Kollár. In arithmetic geometry contexts involving Shimura varieties or modular curves, the distinction between Weil divisors and Cartier divisors affects Picard computations and the study of line bundles with automorphic interpretations like the Hodge bundle.

Functoriality and exact sequences

Picard groups are contravariantly functorial for pullback along morphisms of schemes; pushforward operations occur in the context of derived functors and higher direct image sheaves used by Grothendieck in the formulation of the Picard functor. Exact sequences connecting Pic(X), Pic(U) for open U ⊂ X, and Pic(Z) for closed subschemes Z arise from localization sequences and from long exact sequences in cohomology derived from the Kummer sequence and from the exponential sequence on complex manifolds involving H^i(X, Z) and H^i(X, O_X). For a short exact sequence of group schemes or of sheaves like 1 → μ_n → G_m → G_m → 1, one obtains connecting homomorphisms linking Pic to the Brauer group Br(X) and to cohomology groups H^2(X, μ_n). The Hochschild–Serre spectral sequence relates Pic of a variety to Pic of its base change under coverings such as Galois extensions studied by Artin and Grothendieck.

Picard scheme and Picard functor

To parameterize families of line bundles, one considers the Picard functor and its representability by the Picard scheme or Picard variety when conditions like properness and flatness hold; this construction was developed by Grothendieck and refined by Mumford and Matsusaka. For a proper smooth scheme X over a base S, the Picard functor Pic_{X/S} is often representable by a group scheme whose connected component of the identity is an abelian scheme when X is projective with geometrically integral fibers; examples include the Jacobian J(C) representing Pic^0 for families of curves in the work of Jacobi and Weil. The Picard scheme carries additional structure such as the Poincaré line bundle on X × Pic^0(X) and interacts with duality theories like the Fourier–Mukai transform on Derived categorys considered by Mukai.

Applications in algebraic geometry and number theory

Picard groups appear in the classification of algebraic surfaces via the Enriques–Kodaira classification and in birational geometry through the study of ample and nef line bundles, ampleness criteria like Kodaira vanishing theorem and the Lefschetz hyperplane theorem. In arithmetic geometry, Picard groups of models of varieties over Spec Z and of arithmetic surfaces connect to the arithmetic of number fields, to height pairings on Néron–Tate heights for abelian varietys, and to the study of rational points via descent and obstruction theories involving the Brauer–Manin obstruction. Moduli problems for vector bundles on curves use determinant line bundles and theta divisors related to Pic, featuring prominently in the work on Geometric Invariant Theory by Mumford and in the theory of moduli stacks like the stack of principal bundles studied by Behrend and Laumon.

Category:Algebraic geometry