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Chai's conjecture for semiabelian Jacobians

Otto Overkamp

TL;DR

The paper proves Chai's conjecture for Jacobians, establishing that the base change conductor c(G) of G = Pic^0_{C/K} splits additively as c(G) = c(T) + c(E) for the toric and abelian parts. It develops a Raynaud-style description of the Néron lft-model for Jacobians via Picard functors on seminormal models, extends these ideas to not-necessarily geometrically reduced curves, and provides explicit structure results for Jacobians in imperfect residue characteristic. It also analyzes the exactness of the Néron model sequence and constructs a perfect obstruction-theory-like complex L(C) to capture when exactness holds, presenting a concrete counterexample to exactness. Finally, it gives existence criteria for Néron lft-models in terms of seminormality, reducedness, and a graph-theoretic forest condition on the dual graph, thereby strengthening the understanding of Jacobians in the non-smooth and imperfect setting.

Abstract

We prove Chai's conjecture on the additivity of the base change conductor of semiabelian varieties in the case of Jacobians of proper curves. This includes the first infinite family of non-trivial wildly ramified examples. Along the way, we extend Raynaud's construction of the Néron lft-model of a Jacobian in terms of the Picard functor to arbitrary seminormal curves (beyond which Jacobians admit no Néron lft-models, as shown by our more general structural results). Finally, we investigate the structure of Jacobians of (not necessarily geometrically reduced) proper curves over fields of degree of imperfection at most one and prove two conjectures about the existence of Néron models and Néron lft-models due to Bosch-Lütkebohmert-Raynaud for Jacobians of general proper curves in the case of perfect residue fields, thus strengthening the author's previous results in this situation.

Chai's conjecture for semiabelian Jacobians

TL;DR

The paper proves Chai's conjecture for Jacobians, establishing that the base change conductor c(G) of G = Pic^0_{C/K} splits additively as c(G) = c(T) + c(E) for the toric and abelian parts. It develops a Raynaud-style description of the Néron lft-model for Jacobians via Picard functors on seminormal models, extends these ideas to not-necessarily geometrically reduced curves, and provides explicit structure results for Jacobians in imperfect residue characteristic. It also analyzes the exactness of the Néron model sequence and constructs a perfect obstruction-theory-like complex L(C) to capture when exactness holds, presenting a concrete counterexample to exactness. Finally, it gives existence criteria for Néron lft-models in terms of seminormality, reducedness, and a graph-theoretic forest condition on the dual graph, thereby strengthening the understanding of Jacobians in the non-smooth and imperfect setting.

Abstract

We prove Chai's conjecture on the additivity of the base change conductor of semiabelian varieties in the case of Jacobians of proper curves. This includes the first infinite family of non-trivial wildly ramified examples. Along the way, we extend Raynaud's construction of the Néron lft-model of a Jacobian in terms of the Picard functor to arbitrary seminormal curves (beyond which Jacobians admit no Néron lft-models, as shown by our more general structural results). Finally, we investigate the structure of Jacobians of (not necessarily geometrically reduced) proper curves over fields of degree of imperfection at most one and prove two conjectures about the existence of Néron models and Néron lft-models due to Bosch-Lütkebohmert-Raynaud for Jacobians of general proper curves in the case of perfect residue fields, thus strengthening the author's previous results in this situation.
Paper Structure (15 sections, 36 theorems, 81 equations)

This paper contains 15 sections, 36 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Excellence and degrees of imperfection
  4. Stein reduced morphisms
  5. General background
  6. Chai's conjecture for Jacobians
  7. Degrees and compatible rigidification
  8. Semi-factorial models and Picard functors
  9. Relationship with the Néron model
  10. Proof of Chai's conjecture for Jacobians
  11. Exactness properties
  12. An example
  13. Existence of Néron (lft-)models
  14. The structure of Jacobians in degree of imperfection 1
  15. Existence results

Key Result

Theorem 1.1

(Theorem Chaithm) Let $C$ be a proper curvei. e. purely one-dimensional scheme over $K$ and assume that $G:=\mathop{\mathrm{\mathrm{Pic}}}\nolimits^0_{C/K}$ is semiabelian. Then Chai's conjecture holds for $G.$

Theorems & Definitions (40)

  • Theorem 1.1
  • Lemma 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 3.1
  • Proposition 3.2
  • Definition 3.3
  • ...and 30 more