Geography of Brill-Noether loci for small slopes

Abstract

Let $X$ be a non-singular projective curve of genus $g\ge 2$ 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 $0\le d\le n$ 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 $d>0$, $n\le d+(n-k)g$ and $(n,d,k)\ne (n,n,n)$. 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 [email protected]; complete hard copy also available. LaTeX 2.09

References in corpus (3)

• Stable pairs on curves and surfaces
• Higher cohomology triples and holomorphic extensions
• On the Brill-Noether Problem for Vector Bundles

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