Picard varieties

Picard varieties
Name	Picard varieties
Type	Abelian variety
Field	Algebraic geometry
Introduced	1960s

Picard varieties are principally polarized abelian varieties that parametrize algebraically equivalent classes of line bundles on algebraic varieties. Originating in the work of André Weil, Alexander Grothendieck, and Oscar Zariski, Picard varieties connect the geometry of curves, surfaces, and higher-dimensional schemes with the arithmetic of number theory, the theory of abelian varieties, and the language of schemes and étale cohomology. They play central roles in the study of Riemann surfaces, algebraic curves, and moduli problems such as the moduli space of curves.

Introduction

Picard varieties arose from attempts by Bernhard Riemann, Henri Poincaré, and Giuseppe Peano's contemporaries to understand families of line bundles on curves and surfaces, and were formalized through the contributions of André Weil, Jean-Pierre Serre, Alexander Grothendieck, and David Mumford. Early motivations included the Jacobian of a curve studied by Karl Weierstrass and Riemann, the theory of theta functions developed by Friedrich Prym and Igor Shafarevich, and the need to represent Picard functors in the style of Yoneda and representable functors used by Grothendieck in the context of the SGA volumes.

Definitions and basic properties

For a proper connected scheme X over a field k equipped with a rational point, the Picard functor associates to each k-scheme T the group Pic(X ×_k T)/Pic(T). Under hypotheses due to Grothendieck, this functor is representable by a separated group scheme, whose identity component is an abelian variety: the Picard variety. Formal properties were established using tools from fibration theory, the Weil conjectures machinery developed by Pierre Deligne, and duality theories influenced by Serre duality and Grothendieck duality. Key attributes include being projective, connected, and admitting a canonical principal polarization in many cases traced to the theta divisor construction by Mumford and Igusa.

Construction and examples

Classical construction of Picard varieties for a smooth projective curve C over k yields the Jacobian J(C), historically computed by Riemann, Abel, and Jacobi. For surfaces such as K3 surfaces studied by Ernst Kummer and André Weil, the Picard scheme encodes the Néron–Severi group and its continuous part gives the Picard variety after base change. For families over a base S, the relative Picard functor is constructed in SGA6 and treated in works by Mumford, Grothendieck, and Raynaud. Explicit examples include Picard varieties of elliptic curves linked to Mordell–Weil phenomena studied by Gerd Faltings, and higher-dimensional examples arising from products of curves or families considered by Ziv Ran and Claire Voisin. Constructions often invoke the Néron model for degenerations treated by André Néron and the semi-stable reduction of Deligne-Mumford.

Relation to Jacobians and Albanese varieties

For smooth projective curves the Picard variety coincides with the Jacobian J(C) studied by Riemann and Abel; for higher-dimensional varieties, the connected component of the Picard scheme is dual to the Albanese variety Alb(X) constructed from H^0(X, Ω^1) and the Hodge theory of Pierre Deligne and Phillip Griffiths. The duality between Picard and Albanese generalizes classical duality between Poincaré-type constructions and was formalized by Murre and Mumford using the language of Tannakian and Hodge structures studied by Wilfried Schmid and Deligne. The interplay appears in the theory of correspondences examined by André Weil and Alexander Grothendieck.

Functoriality and moduli interpretation

Picard varieties behave functorially under morphisms of varieties, pushforward and pullback operations linked to the derived functors framework of Grothendieck and to the formalism in SGA7. They provide coarse and sometimes fine moduli for line bundles with fixed topological or numerical invariants, interfacing with the moduli of vector bundles work of Narashimhan–Seshadri and the construction of compactified Picard schemes by Caporaso, Esteves, and Alexeev. In arithmetic contexts, functoriality interacts with the absolute Galois group studied by Emil Artin and John Tate via the action on ℓ-adic Tate modules and Tate conjectures.

Applications in algebraic geometry and number theory

Picard varieties are instrumental in proofs of finiteness theorems such as Faltings and in the study of rational points on curves and surfaces investigated by Manin and Skorobogatov. They appear in the study of L-functions and the Birch–Swinnerton-Dyer paradigm when abelian varieties arise from Jacobians. Picard varieties also inform degeneration and compactification problems examined in Deligne–Mumford compactification and in the study of automorphic forms by Langlands and Jacquet–Langlands. They connect to explicit computations in computational algebraic geometry pursued by researchers like Nils Bruin and J. S. Milne.

Cohomological and duality aspects

Cohomological descriptions use H^1(X, O_X) and H^1(X, G_m) via the exponential sequence on complex varieties and the Kummer sequence in étale cohomology developed by Grothendieck and Milne. Duality theories for Picard varieties draw on Serre duality, Grothendieck duality, and the theory of pure Hodge structures by Pierre Deligne, with comparisons provided by the Hodge conjecture context considered by Griffiths and Green. ℓ-adic realizations link the Tate module of a Picard variety to étale cohomology groups used by Grothendieck in the proof strategy of the Weil conjectures; motivic interpretations are pursued in the frameworks proposed by Grothendieck and later by Vladimir Voevodsky.

Category:Algebraic geometry