On the irreducible components of the compactified Jacobian of a ribbon

TL;DR

This paper investigates the structure of the compactified Jacobian of a ribbon, revealing conditions under which certain moduli spaces are not irreducible and identifying the irreducible components involved.

Contribution

It determines the irreducible components of the compactified Jacobian of a ribbon and answers a question posed by Chen and Kass regarding the moduli space of rank 2 semistable vector bundles.

Findings

When g ≥ 4𝑏𝑎𝑟𝑔−2, the moduli space is not an irreducible component.

The irreducible components containing the moduli space are explicitly characterized.

The results complete and extend previous work by Chen and Kass.

Abstract

In this paper we study the irreducible components of the compactified Jacobian of a ribbon $X$ of arithmetic genus $g$ over a smooth curve $X_{\mathrm{red}}$ of genus $\bar{g}$. We prove that when $g\geq 4\bar{g}-2$ the moduli space of rank $2$ semistable vector bundles over $X_{\mathrm{red}}$ is not an irreducible component and we determine the irreducible components in which it is contained. This answers a question of D. Chen and J.L. Kass in [CK] and completes their results.