A note on the Brill-Noether loci of small codimension in moduli space of stable bundles

Let X be a smooth projective curve of genus g over the field C. Let MX(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 W¹X(2,L) of MX(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 W¹X(2,L) is stably-rational when $g=3$, unirational when $g=4$, and rationally chain connected by Hecke curves, when g≥ 5. We also show triviality of low dimensional rational Chow groups of an associated Brill-Noether hypersurface.