The wobbly divisors of the moduli space of rank-$2$ vector bundles

TL;DR

This paper investigates the structure of the wobbly locus in the moduli space of semi-stable rank-2 vector bundles with fixed determinant over a complex curve, identifying divisors and their properties.

Contribution

It characterizes the wobbly locus as a union of divisors, computes their classes in the Picard group, and analyzes the degeneracy of maximal subbundles on one of these divisors.

Findings

Wobbly locus forms a union of divisors in the moduli space.

The class of these divisors is explicitly computed in the Picard group.

On one divisor, the set of maximal subbundles is degenerate.

Abstract

Let $X$ be a smooth projective complex curve of genus $g \geq 2$ and let $\M_X(2,\Lambda)$ be the moduli space of semi-stable rank-$2$ vector bundles over $X$ with fixed determinant $\Lambda$. We show that the wobbly locus, i.e., the locus of semi-stable vector bundles admitting a non-zero nilpotent Higgs field is a union of divisors $\Ww_k \subset \M_X(2,\Lambda)$. We show that on one wobbly divisor the set of maximal subbundles is degenerate. We also compute the class of the divisors $\Ww_k$ in the Picard group of $\M_X(2,\Lambda)$.