# 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.

## Key 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.