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 Mₗ (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^1ₗ (2, L) of Mₗ (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^1ₗ (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.