Rank two bundles with canonical determinant and four sections

Abstract

Let $C$ be a smooth irreducible complex projective curve of genus $g$ and let $B^k(2,K_C)$ be the Brill–Noether locus parametrizing classes of (semi)-stable vector bundles $E$ of rank two with canonical determinant over $C$ with $h^0(C,E)\ge k$. We show that $B^4(2,K_C)$ has an irreducible component $\mathcal B$ of dimension $3g-13$ on a general curve $C$ of genus $g\ge 8$. Moreover, we show that for the general element $[E]$ of $\mathcal B$, $E$ fits into an exact sequence $0\rightarrow {\mathcal {O}}_C(D)\rightarrow E\rightarrow K_C(-D)\rightarrow 0$ with $D$ a general effective divisor of degree three, and the corresponding coboundary map $\partial : H^0(C,K_C(-D))\rightarrow H^1(C,{\mathcal {O}}_C(D))$ has cokernel of dimension three.