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Compactified Jacobians as Mumford models

Karl Christ, Sam Payne, Tif Shen

TL;DR

The paper connects nonarchimedean geometry and tropical methods to the theory of compactified Jacobians for one-parameter smoothings of nodal curves. It proves that relative compactified Jacobians $\overline J_{\mathcal{X}}(\phi)$ are Mumford models of the generic fiber ${\operatorname{Pic}}^d(\mathcal{X}_K)$, obtained from polytopal decompositions of the skeleton of the Jacobian's Berkovich analytification. These decompositions, identified with Namikawa-type stratifications by stability data $(S,\underline{d})$, organize the moduli into toric charts and illuminate the special fiber via a wall-and-chamber structure on polarizations; in degree $d=g$ the break divisor decomposition yields a canonical, unique compactified Jacobian. The construction unifies tropical geometry, Néron-type uniformization, and toric degeneration theory to describe both global and local structures of compactified Jacobians in families. It also clarifies how specialization maps between compactified Jacobians arise from the stability wall structure, and provides explicit tropical correspondences for theta- and break-divisor phenomena.

Abstract

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slop stability, and it is induced by the tropical break divisor decomposition.

Compactified Jacobians as Mumford models

TL;DR

The paper connects nonarchimedean geometry and tropical methods to the theory of compactified Jacobians for one-parameter smoothings of nodal curves. It proves that relative compactified Jacobians are Mumford models of the generic fiber , obtained from polytopal decompositions of the skeleton of the Jacobian's Berkovich analytification. These decompositions, identified with Namikawa-type stratifications by stability data , organize the moduli into toric charts and illuminate the special fiber via a wall-and-chamber structure on polarizations; in degree the break divisor decomposition yields a canonical, unique compactified Jacobian. The construction unifies tropical geometry, Néron-type uniformization, and toric degeneration theory to describe both global and local structures of compactified Jacobians in families. It also clarifies how specialization maps between compactified Jacobians arise from the stability wall structure, and provides explicit tropical correspondences for theta- and break-divisor phenomena.

Abstract

We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that in degree g there is a unique compactified Jacobian encoding slop stability, and it is induced by the tropical break divisor decomposition.

Paper Structure

This paper contains 29 sections, 23 theorems, 64 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Line bundles and torsion free sheaves
  3. Multidegrees
  4. Dual graphs
  5. Partial normalizations and blowups
  6. Compactified Jacobians
  7. Polarizations and semistability
  8. Polystability
  9. Compactified Jacobians
  10. The special fiber
  11. The relative setting
  12. The wall and chamber decompostion on the space of polarizations
  13. Extensions of line bundles on semistable models
  14. Skeletons of $\mathcal{X}_K^\mathrm{an}$ and semistable models
  15. Extension of line bundles
  16. ...and 14 more sections

Key Result

Theorem 1.1

The compactified Jacobian $\overline J_{\mathcal{X}}(\phi)$ is a Mumford model of ${\operatorname{{Pic}}}^d(\mathcal{X}_K)$.

Figures (1)

  • Figure 1: The one-dimensional family of equivalent divisors in Example \ref{['ex:polystable family']}. All divisors are $\phi$-semistable, and the ones in the middle two figures are in addition $\phi$-polystable.

Theorems & Definitions (63)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • ...and 53 more