On the irreducible components of the compactified Jacobian of a ribbon

Abstract

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

Keywords:

• Compactified Jacobians
• ribbons
• generalized line bundles
• multiple curves

2010 Mathematics Subject Classification:

• 14D20
• 14H40
• 14H60

Acknowledgments

This article is born as an aside to my doctoral thesis, which is in progress and is about the compactified Jacobian of a primitive multiple curve of multiplicity ≥3. I am grateful to my supervisor, Filippo Viviani, who introduced me to the articles [2] and [4] and more generally to the subject and, moreover, gave me various suggestions about the exposition. I am also grateful to Edoardo Sernesi: my knowledge of one of the key ingredients of the proof, namely Segre-Nagata Theorem (i.e., Fact 2(i)) is due to his unpublished notes about algebraic curves which he distributed confidentially in a preliminary version during a doctoral course about Brill–Noether theory. I would thank also the anonymous referee for his useful comments.

A.m.D.g.

Disclosure statement

No potential conflict of interest was reported by the author.