Nonemptiness of Brill-Noether loci

Abstract

Let $X$ be a non-singular algebraic curve of genus $g$. We prove that the Brill-Noether locus $\bns $ is non-empty if $d= nd&#39; +d&#39;&#39; $ with $0< d&#39;&#39; <2n$, $1\le s\le g$, $d&#39;\geq (s-1)(s+g)/s $, $n\leq d&#39;&#39;+(n-k)g$, $(d&#39;&#39;,k)\ne(n,n)$. 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&#39;&#39; 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.&#39;&#39; For hyperelliptic curves, the same methods give more extensive and precise results.