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

Abstract

Let $X$ be a smooth projective complex curve of genus $g\ge 2$ and let $\M_X(2,Λ)$ 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 $\Ww_k \subset \M_X(2,Λ)$. 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 $\M_X(2,Λ)$.