Picard scheme

Picard scheme
Name	Picard scheme
Field	Algebraic Geometry
Introduced	1960s
Introduced by	Alexander Grothendieck

Picard scheme The Picard scheme is a fundamental object in algebraic geometry linking line bundles on a scheme with families of group schemes; it organizes isomorphism classes of invertible sheaves into a representable group-valued moduli object. Its study intersects the work of Alexander Grothendieck, Jean-Pierre Serre, David Mumford, Igor Shafarevich, and John Tate, and connects to classical constructions like the Jacobian variety, the Néron model, and the theory of Abelian varieties. The Picard scheme plays a central role in problems arising in the Riemann–Roch theorem, the theory of moduli of curves, and arithmetic questions such as the Birch and Swinnerton-Dyer conjecture.

Introduction

The historical emergence of the Picard scheme stems from attempts to globalize the Picard group in families: Grothendieck framed the problem in the language of scheme theory developed in works like Éléments de géométrie algébrique and pursued representability of the Picard functor. Subsequent contributions by Oscar Zariski, Federico Enriques, André Weil, Riemann, and Max Noether influenced its classical antecedents. The modern perspective situates the Picard scheme alongside the Hilbert scheme, the Chow scheme, the Moduli space of curves, and construction methods echoed in the theory of stacks and fppf topology.

Definition and basic properties

One begins with a proper morphism f: X → S of schemes and considers the relative Picard functor Pic_{X/S} sending an S-scheme T to the group Pic(X_T)/Pic(T). The Picard scheme, when it exists, is a group scheme representing the sheafification of Pic_{X/S} in the fppf topology or the étale topology, thereby relating to representability results by Grothendieck, Mumford, and others. Key properties include its behavior under base change, its decomposition into connected components often denoted by a neutral component and a component group, and the existence of the identity component Pic^0 which is an Abelian variety when X is a smooth projective geometrically integral curve over a field. The Picard scheme interacts with the Néron–Severi group, the Tate module, and phenomena like reduction of abelian varieties.

Construction and representability

Grothendieck proved representability results using techniques from cohomology, derived functors, and relative duality, building on the Existence theorem frameworks in SGA seminars. Mumford developed geometric invariant theory tools and the notion of cohomologically flatness in families to establish representability in many cases; Serre's duality and results of Raynaud refined the treatment in nonreduced situations. The proof strategy often involves forming the Picard functor as a sheaf, constructing a candidate scheme via the Quot scheme or the Hilbert scheme, then checking the group law and smoothness using criteria from Grothendieck's descent theory and Artin's algebraization theorem. Representability depends on hypotheses like properness, flatness, and cohomological conditions (e.g., vanishing of higher direct images) that echo in the work of Matsusaka and Deligne.

Examples and computations

For a smooth projective curve C over a field, the Picard scheme decomposes into components indexed by degree and Pic^0(C) is the Jacobian J(C), an Abelian variety constructed classicaly by Bernhard Riemann and modernly via functorial methods of Weil. For a projective line P^1 the Picard scheme is isomorphic to the constant group scheme Z and Pic^0 is trivial. For a smooth projective surface S the Picard scheme reflects the Néron–Severi group and can have nonreduced structure as shown in examples by Michael Artin and Raynaud; singular curves and degenerations give rise to generalized Jacobians studied by Rosenlicht and Serre. Explicit computations employ techniques from the Riemann–Roch theorem, Abel–Jacobi map, and intersection theory developed by Federer and William Fulton.

Relation to Picard functor and Jacobians

The Picard functor is the presheaf whose sheafification is the target for representability; it generalizes the classical Picard group of invertible sheaves. The Jacobian of a curve is a manifestation of the identity component of the Picard scheme and connects intimately to the Abel map, the Torelli theorem, and the construction of Prym varieties. The relationship further interacts with the Néron model for Abelian varieties over Dedekind schemes, the Weil pairing, and the study of rational points in Diophantine geometry influenced by Gerd Faltings and Shou-Wu Zhang.

Cohomological and deformation aspects

Cohomology controls the tangent space of the Picard scheme: for X over a field, the tangent space at the identity identifies with H^1(X, O_X), while obstruction classes live in H^2(X, O_X). Deformation theory articulated by Michael Artin, Grothendieck, and Deligne provides criteria for smoothness and dimension counts, with Hodge theory and results of Pierre Deligne constraining behavior in characteristic zero and crystalline cohomology by Jean-Marc Fontaine influencing p-adic phenomena. Nonreducedness and infinitesimal automorphisms appear in surfaces and higher-dimensional schemes, studied using the Cotangent complex and derived methods advanced by Bernhard Toen and Jacob Lurie.

Applications in algebraic geometry and number theory

The Picard scheme underlies construction and study of moduli spaces such as the Picard variety families, the Moduli of vector bundles, and feeds into the geometric Langlands program influenced by work at Institut des Hautes Études Scientifiques and Princeton University. It provides tools for arithmetic questions involving Galois representations, the Tate conjecture, and the arithmetic of Abelian varieties central to investigations by Andrew Wiles and Barry Mazur. In enumerative geometry and intersection theory, Picard schemes govern line bundle variation used in calculations by Kontsevich and Maxim Kontsevich’s collaborators, and they appear in contemporary research on derived categories and mirror symmetry pursued at institutions like Harvard University and Courant Institute.

Category:Algebraic geometry