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The Restricted Picard Functor

Zev Rosengarten

TL;DR

The paper tackles the problem of representability for the Picard functor beyond the proper setting by introducing the restricted Picard functor $\mathrm{Pic}^+_{X/K}$ and proving its almost-representability for regular finite-type $K$-schemes $X$, namely $\mathrm{Pic}^+_{X/K} \simeq G/E$ with $G$ smooth and having finitely generated component group and $E \subset G^t$ étale with $E(K_s)$ free of finite rank. The method combines descent through finite Galois and radicial covers, boundedness of descent data, and de Jong alterations to reduce to tractable cases, culminating in a structural description of $\mathrm{Pic}^+_{X/K}$ and, under extra hypotheses, full representability by a smooth group scheme with a unipotent identity component. The key contributions include a detailed analysis of units and cohomology sheaves under group actions, the introduction of a boundedness framework for descent data, and a robust framework for passing from open subschemes to the whole variety via alterations. These results significantly extend representability phenomena for Picard-type objects to non-proper contexts and provide a concrete description of the restriction of Picard groups to smooth test schemes, with applications to forms of affine spaces and rationally connected varieties.

Abstract

We prove in significant generality the (almost-)representability of the Picard functor when restricted to smooth test schemes. The novelty lies in the fact that we prove such (almost-)representability beyond the proper setting.

The Restricted Picard Functor

TL;DR

The paper tackles the problem of representability for the Picard functor beyond the proper setting by introducing the restricted Picard functor and proving its almost-representability for regular finite-type -schemes , namely with smooth and having finitely generated component group and étale with free of finite rank. The method combines descent through finite Galois and radicial covers, boundedness of descent data, and de Jong alterations to reduce to tractable cases, culminating in a structural description of and, under extra hypotheses, full representability by a smooth group scheme with a unipotent identity component. The key contributions include a detailed analysis of units and cohomology sheaves under group actions, the introduction of a boundedness framework for descent data, and a robust framework for passing from open subschemes to the whole variety via alterations. These results significantly extend representability phenomena for Picard-type objects to non-proper contexts and provide a concrete description of the restriction of Picard groups to smooth test schemes, with applications to forms of affine spaces and rationally connected varieties.

Abstract

We prove in significant generality the (almost-)representability of the Picard functor when restricted to smooth test schemes. The novelty lies in the fact that we prove such (almost-)representability beyond the proper setting.
Paper Structure (8 sections, 43 theorems, 54 equations)

This paper contains 8 sections, 43 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Notation And Conventions
  4. Descent through Galois covers
  5. Boundedness
  6. Descent through radicial covers
  7. Almost-representability of the restricted Picard functor
  8. Structure of the restricted Picard functor

Key Result

Theorem 1.3

If $K$ is a field and $X$ is a regular $K$-scheme of finite type, then there is an isomorphism $\mathop{\mathrm{Pic}}\nolimits^+_{X/K} \simeq G/E$ of sheaves on the smooth-étale site of ${\rm{Spec}}(K)$, where $G$ is a smooth commutative $K$-group scheme with finitely generated component group and $

Theorems & Definitions (94)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Definition 2.2
  • ...and 84 more