Nonemptiness of Brill-Noether loci

Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the Brill-Noether locus $\text{bns}$ is non-empty if $d=n{d}^{\mathrm{\prime }}+d^{\mathrm{\prime }\mathrm{\prime }}$ with $0<d^{\mathrm{\prime }\mathrm{\prime }}<2n$, $1\le s\le g$, $d^{\mathrm{\prime }}\ge (s-1)(s+g)\mathrm{/}s$, $n\le d^{\mathrm{\prime }\mathrm{\prime }}+(n-k)g$, $(d^{\mathrm{\prime }\mathrm{\prime }},k)\mathrm{\ne }(n,n)$. These results hold for an arbitrary curve of genus $\ge 2$, and allow us to construct a region in the associated Brill-Noether $\pa$-map'' of points for which the Brill-Noether loci are non-empty. Even for the generic case, the region so constructed extends beyond that defined by the so-called Teixidor parallelograms.‘’ For hyperelliptic curves, the same methods give more extensive and precise results.