Combinatorics of compactified universal Jacobians

TL;DR

This paper explores the combinatorial structure of compactified universal Jacobians over the moduli space of stable curves, using orientations on graphs to create compatible stratifications.

Contribution

It introduces a novel combinatorial framework using graph orientations to describe the structure of compactified Jacobians in specific degrees.

Findings

Graded stratifications of Jacobians are constructed using graph orientations.

The stratifications are compatible with the moduli space of stable curves.

Explicit descriptions are provided for the stratifications in terms of graph orientations.

Abstract

We use orientations on stable graphs to express the combinatorial structure of the compactified universal Jacobians in degrees g-1 and g over the moduli space of stable curves, \Mgb, and construct for them graded stratifications compatible with the one of \Mgb. In particular, for a stable curve we exhibit graded stratifications of the compactified Jacobians in terms of totally cyclic, respectively rooted, orientations on subgraphs of its dual graph.