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A new class of compactified Jacobians for families of reduced curves

Marco Fava, Nicola Pagani, Filippo Viviani

TL;DR

The paper develops a unified framework to construct and study compactified Jacobians for families of reduced curves via V-stability (vine-type stability) on torsion-free rank-1 sheaves. It defines relative V-compactified Jacobians as open substacks of the stack of torsion-free rank-1 sheaves, with good moduli spaces, and proves their fundamental properties, including openness, properness-type criteria, and the relation to classical compactifications through numerical polarizations. It analyzes the combinatorics of stability via degeneracy sets, the poset of V-stability conditions, and their behavior in families, and then proves that for families with planar singularities these spaces have especially nice geometric features (flatness, connected fibers, LCI singularities, trivial dualizing sheaves) and are robust under base change. In doing so, it both recovers classical constructions (Oda–Seshadri, Caporaso, Esteves) as special cases and provides a broad, flexible framework for constructing new compactified Jacobians in families, with applications to moduli, deformation theory, and related invariants. The work lays the groundwork for a three-paper program (FPV, FPV3) aimed at completeness results for nodal curves and a universal classification over moduli stacks, while highlighting the role of planarity and stability combinatorics in controlling geometric properties.

Abstract

This is the first paper of a series of three. Here we give an abstract definition of the relative compactified Jacobian of a family of reduced curves. We prove that, under some mild assumptions on the family of curves, the fibres of the relative Jacobian are schemes (and not just algebraic spaces). We define V-stability conditions, and use them to construct relative compactified Jacobians. This extends the classical methods to produce modular compactifications of the Jacobian. To conclude, we show that, in the case when the curves have at worst planar singularities, the compactified Jacobians constructed from V-stability conditions have the same good properties of the classical ones.

A new class of compactified Jacobians for families of reduced curves

TL;DR

The paper develops a unified framework to construct and study compactified Jacobians for families of reduced curves via V-stability (vine-type stability) on torsion-free rank-1 sheaves. It defines relative V-compactified Jacobians as open substacks of the stack of torsion-free rank-1 sheaves, with good moduli spaces, and proves their fundamental properties, including openness, properness-type criteria, and the relation to classical compactifications through numerical polarizations. It analyzes the combinatorics of stability via degeneracy sets, the poset of V-stability conditions, and their behavior in families, and then proves that for families with planar singularities these spaces have especially nice geometric features (flatness, connected fibers, LCI singularities, trivial dualizing sheaves) and are robust under base change. In doing so, it both recovers classical constructions (Oda–Seshadri, Caporaso, Esteves) as special cases and provides a broad, flexible framework for constructing new compactified Jacobians in families, with applications to moduli, deformation theory, and related invariants. The work lays the groundwork for a three-paper program (FPV, FPV3) aimed at completeness results for nodal curves and a universal classification over moduli stacks, while highlighting the role of planarity and stability combinatorics in controlling geometric properties.

Abstract

This is the first paper of a series of three. Here we give an abstract definition of the relative compactified Jacobian of a family of reduced curves. We prove that, under some mild assumptions on the family of curves, the fibres of the relative Jacobian are schemes (and not just algebraic spaces). We define V-stability conditions, and use them to construct relative compactified Jacobians. This extends the classical methods to produce modular compactifications of the Jacobian. To conclude, we show that, in the case when the curves have at worst planar singularities, the compactified Jacobians constructed from V-stability conditions have the same good properties of the classical ones.
Paper Structure (14 sections, 47 theorems, 233 equations, 5 figures)

This paper contains 14 sections, 47 theorems, 233 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Reduced curves
  4. Torsion-free, rank-1 sheaves on reduced curves
  5. Specializations
  6. Isotrivial specializations
  7. V-stability conditions
  8. Numerical polarizations
  9. The poset of V-stability conditions
  10. V-stability conditions for families
  11. The category of semistable sheaves
  12. V-compactified Jacobians
  13. Projectivity and Schematicity of compactified Jacobian spaces
  14. V-compactified Jacobians for reduced curves with planar singularities

Key Result

Theorem A

(Theorem T:VcJ) Let $\pi:X\to S$ be a family of connected reduced curves over a quasi-separated and locally Noetherian algebraic space $S$. For any V-stability condition $\mathfrak{s}= \{\mathfrak{s}^s\}$ on $X/S$ of characteristic $\chi$, the substack $\overline \mathcal{J}_{X/S}(\mathfrak{s})$ of (see Lemma-Definition LD:VStab) is a compactified Jacobian stack of $X/S$ of characteristic $\chi$,

Figures (5)

  • 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: $\Theta$-completeness of $\overline \mathcal{J}_{X/S}(\mathfrak{s})\to S$.
  • Figure 3: The four points of $\operatorname{ST}_R$ and their specializations: the solid arrows are ordinary specializations while the squiggly arrows are isotrivial specializations.
  • Figure 4: S-completeness of $\overline \mathcal{J}_{X/S}(\mathfrak{s})\to S$.
  • Figure 5: Existence part of the valuative criterion of properness for $\overline \mathcal{J}_{X/S}(\mathfrak{s})\to S$.

Theorems & Definitions (125)

  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Example 3.4
  • Lemma 3.5
  • proof
  • ...and 115 more