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A complete theory of smoothable compactified Jacobians of nodal curves

Marco Fava, Nicola Pagani, Filippo Viviani

TL;DR

The paper develops a complete theory for smoothable compactified Jacobians of nodal curves by encoding degenerations through V-stability conditions on the dual graph and Picard-Type (PT) assignments. It proves a bijection between V-stability and PT-assignments, and shows that V-compactified Jacobians are precisely the smoothable ones, with a minimal-positet of orbits controlled by a degeneracy subset. This yields two characterizations of V-compactified Jacobians: they are the smoothable objects arising from some V-stability, and among those with a fixed degeneracy, they have the smallest possible number of irreducible components. The results extend and complete prior work on fine and canonical compactified Jacobians, providing a broad, graph-theoretic framework (via $\Gamma_X$ and its subgraphs) to classify all modular degenerations of Jacobians and to understand their geometry and moduli via Kirchhoff-type theorems and BD-sets.

Abstract

We introduce and study a new class of compactified Jacobians for nodal curves, that we call compactified Jacobians of vine type, or simply V-compactified Jacobians. This class is strictly larger than the class of classical compactified Jacobians, as constructed by Oda-Seshadri, Simpson, Caporaso and Esteves. We characterize V-compactified Jacobians as the compactified Jacobians that can arise as limits of Jacobians of smooth curves under a one-parameter smoothing of the nodal curve, extending previous works on fine compactified Jacobians by Pagani-Tommasi and Viviani to the case of all compactified Jacobians.

A complete theory of smoothable compactified Jacobians of nodal curves

TL;DR

The paper develops a complete theory for smoothable compactified Jacobians of nodal curves by encoding degenerations through V-stability conditions on the dual graph and Picard-Type (PT) assignments. It proves a bijection between V-stability and PT-assignments, and shows that V-compactified Jacobians are precisely the smoothable ones, with a minimal-positet of orbits controlled by a degeneracy subset. This yields two characterizations of V-compactified Jacobians: they are the smoothable objects arising from some V-stability, and among those with a fixed degeneracy, they have the smallest possible number of irreducible components. The results extend and complete prior work on fine and canonical compactified Jacobians, providing a broad, graph-theoretic framework (via and its subgraphs) to classify all modular degenerations of Jacobians and to understand their geometry and moduli via Kirchhoff-type theorems and BD-sets.

Abstract

We introduce and study a new class of compactified Jacobians for nodal curves, that we call compactified Jacobians of vine type, or simply V-compactified Jacobians. This class is strictly larger than the class of classical compactified Jacobians, as constructed by Oda-Seshadri, Simpson, Caporaso and Esteves. We characterize V-compactified Jacobians as the compactified Jacobians that can arise as limits of Jacobians of smooth curves under a one-parameter smoothing of the nodal curve, extending previous works on fine compactified Jacobians by Pagani-Tommasi and Viviani to the case of all compactified Jacobians.

Paper Structure

This paper contains 25 sections, 50 theorems, 314 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. V-stability conditions on graphs
  4. Notation on graphs
  5. Definition of V-stability conditions
  6. The semistable set of a V-stability condition
  7. Operations on V-stability conditions
  8. PT-assignments
  9. A generalisation of Kirchhoff's theorem
  10. Equivalence of V-stability conditions and PT-assignments
  11. Compactified Jacobians
  12. Notation on nodal curves
  13. Torsion-free, rank-1 sheaves on nodal curves
  14. Specializations
  15. Isotrivial specializations
  16. ...and 10 more sections

Key Result

Theorem A

(see Theorems T:VcJ-smoo and T:cla-smoo) Let $X$ be a connected nodal curve and let $\overline \mathcal{J}_X^d$ be a compactified Jacobian stack of $X$ of degree $d$. Then the following conditions are equivalent:

Figures (2)

  • Figure 1: The four points of $\Theta_R$ and their specializations: the horizontal arrows are ordinary specializations while the vertical arrows are isotrivial specializations.
  • Figure 2:

Theorems & Definitions (141)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Remark 2.2
  • Example 2.3: Classical V-stability conditions
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Definition 2.7
  • ...and 131 more