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 $d=n{d}^{\prime }+d^{\prime \prime }$ with $0<d^{\prime \prime }<2n$, $1\le s\le g$, $d^{\prime }\ge (s-1)(s+g)/s$, $n\le d^{\prime \prime }+(n-k)g$, $(d^{\prime \prime },k)\ne (n,n)$. 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'' 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.

AMSLatex file, 22 pages, 12 figures