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Nonemptiness of Brill-Noether loci

L. Brambila-Paz·V. Mercat·P. E. Newstead·F. Ongay
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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' +d'' $ with $0< d'' <2n$, $1\le s\le g$, $d'\geq (s-1)(s+g)/s $, $n\leq d''+(n-k)g$, $(d'',k)\ne(n,n)$. 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.

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L. Brambila-Paz, V. Mercat, P. E. Newstead, F. Ongay. 1999-04-26. Nonemptiness of Brill-Noether loci. https://arxiv.org/abs/math/9904147

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