Let $X$ be a smooth projective curve of genus $g$ over the field $\mathbb{ C}$. Let $M_{ X}(2,L)$ denote the moduli space of stable rank $2$ vector bundles on $X$ with fixed determinant $L$ of degree $2g-1$. Consider the Brill-Noether subvariety $W^{ 1}_{ X}(2,L)$ of $M_{ X}(2,L)$ which parametrises stable vector bundles having at least two linearly independent global sections. In this article, for generic $X$ and $L$, we show that $W^{ 1}_{ X}(2,L)$ is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when $g\geq 5$. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface.
Categories: math.AG