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

Let X$X$ be a smooth projective curve of genus g$g$ over the field C$\mathbb {C}$. Let MX(2,L)$M_{X}(2,L)$ denote the moduli space of stable rank 2 vector bundles on X$X$ with fixed determinant L$L$ of degree 2g−1$2g-1$. Consider the Brill–Noether subvariety WX1(2,L)$W^{1}_{X}(2,L)$ of MX(2,L)$M_{X}(2,L)$ which parameterizes stable vector bundles having at least two linearly independent global sections. In this paper, for generic X$X$ and L$L$, we show that WX1(2,L)$W^{1}_{X}(2,L)$ is stably‐rational when g=3$g=3$, unirational when g=4$g=4$, and rationally chain connected by Hecke curves, when g≥5$g\ge 5$. We also show triviality of low‐dimensional rational Chow groups of an associated Brill–Noether hypersurface.