Codimension of Brill-Noether Loci on Abelian Surfaces and the First Segre Invariant

Let $X$ be an abelian surface of Picard rank 1 with ample generator $H$. We show that, for a Mukai vector $\mathbf{v}=(r,mH,χ)$ with $r,m\geq 1$ and $χ\leq 0$, the first Brill-Noether loci $\{E \;|\; H^0(E)\neq 0\}$ in $M_X(\mathbf{v})$ is nonempty of the expected codimension $1-χ$. As an applicatio