2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/150082Let $X$ be a non-singular projective curve of genus $g\ge2$ over an algebraically closed field of characteristic zero. Let $\mo$ denote the moduli space of stable bundles of rank $n$ and degree $d$ on $X$ and $\wn $ the Brill-Noether loci in $\mo .$ We prove that, if $0\leq d \leq n $ and $\wn $ is non-empty, then it is irreducible of the expected dimension and smooth outside $\wnn$. We prove further that in this range $\wn$ is non-empty if and only if $d>0$, $n\leq d+(n-k)g$ and $(n,d,k) \not= (n,n,n)$. We also prove irreducibility and non-emptiness for the semistable Brill-Noether loci.34pp and 3 figures. Figures (not included in e-print) may be obtained by sending your mailing address to newstead@liv.ac.uk; complete hard copy also available. LaTeX 2.09Algebraic Geometry14J60 (primary), 14D20 (secondary)text