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A complete classification of modular compactifications of the universal Jacobian

Marco Fava, Nicola Pagani, Filippo Viviani

Abstract

We classify all modular compactifications of the universal Jacobian over $\overline{\mathcal{M}}_{g,n}$, both as stacks and as their relative good moduli spaces. Our main result gives a combinatorial parametrization of compactified universal Jacobian stacks by $V$-functions on a stability domain $\mathbb{D}_{g,n}$ of half-vine types (two-components topological types with a chosen side); under this correspondence, fine compactifications are exactly the general $V$-functions. We single out the classical compactified universal Jacobians, namely those induced by numerical polarizations (relative $\mathbb{R}$-line bundles on the universal curve $\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}$), recovering the constructions of Kass-Pagani and Melo in the fine case, and we prove that their good moduli spaces are locally projective over $\overline{\mathcal{M}}_{g,n}$. We determine when two compactified universal Jacobians are isomorphic over $\overline{\mathcal{M}}_{g,n}$ and describe a resolution of the universal family via a compactified Jacobian over $\overline{\mathcal{M}}_{g,n+1}$. Finally, we analyse the poset $Σ_{g,n}$ of compactified universal Jacobians, an extension of the poset of regions of the hyperplane arrangement of classical stability conditions $\mathcal{A}_{g,n}$ studied in Kass-Pagani. We prove that for $n=0$ all compactified universal Jacobians are those constructed by Caporaso. We then give an explicit description of the submaximal elements of $Σ_{g,n}$ for all $n$, generalizing the stability walls in the classical stability space $\mathcal{A}_{g,n}$ from Kass-Pagani's work.

A complete classification of modular compactifications of the universal Jacobian

Abstract

We classify all modular compactifications of the universal Jacobian over , both as stacks and as their relative good moduli spaces. Our main result gives a combinatorial parametrization of compactified universal Jacobian stacks by -functions on a stability domain of half-vine types (two-components topological types with a chosen side); under this correspondence, fine compactifications are exactly the general -functions. We single out the classical compactified universal Jacobians, namely those induced by numerical polarizations (relative -line bundles on the universal curve ), recovering the constructions of Kass-Pagani and Melo in the fine case, and we prove that their good moduli spaces are locally projective over . We determine when two compactified universal Jacobians are isomorphic over and describe a resolution of the universal family via a compactified Jacobian over . Finally, we analyse the poset of compactified universal Jacobians, an extension of the poset of regions of the hyperplane arrangement of classical stability conditions studied in Kass-Pagani. We prove that for all compactified universal Jacobians are those constructed by Caporaso. We then give an explicit description of the submaximal elements of for all , generalizing the stability walls in the classical stability space from Kass-Pagani's work.
Paper Structure (13 sections, 53 theorems, 335 equations, 4 figures)

This paper contains 13 sections, 53 theorems, 335 equations, 4 figures.

Table of Contents

  1. Introduction
  2. V-compactified Jacobians for families of nodal curves
  3. Compactified universal Jacobians
  4. Classical compactified universal Jacobians
  5. Isomorphisms among compactified universal Jacobians
  6. Universal families
  7. The universal family over $\operatorname{TF}_{g,n}$
  8. The case of compactified universal Jacobians
  9. Classification for $n=0$
  10. On the poset of compactified universal Jacobians
  11. The case $n=1$
  12. Maximal and submaximal elements
  13. Example: $g=1$

Key Result

Theorem A

(see Theorem T:cJUniv and Proposition P:Stabgn) There is an anti-isomorphism of posets where $(I_{|X})_Y$ denotes the torsion-free quotient of the restriction of the sheaf $I_{|X}$ on $Y$. Moreover, $\sigma$ is general if and only if $\overline \mathcal{J}_{g,n}(\sigma)$ is fine.

Figures (4)

  • Figure : (1A): Here $r=0$ and $e$ is even: $e/2$ edges between $v_i$ and $v_{i+1}$
  • Figure :
  • Figure : (1A): Here $r=0$ and $e$ is even: $e/2$ edges between $v_i$ and $v_{i+1}$
  • Figure : (1B): Here $r=0$ and $e$ is odd: $(e-1)/2$ edges between $v_i$ and $v_{i+1}$, and $1$ 'diagonal' edge

Theorems & Definitions (137)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Definition 2.1
  • proof
  • Theorem 2.3
  • proof
  • ...and 127 more