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A Riemann--Kempf singularity theorem for higher rank Brill--Noether loci (1711.02370v1)
Published 7 Nov 2017 in math.AG
Abstract: Given a vector bundle $V$ over a curve $X$, we define and study a surjective rational map $\mathrm{Hilb}d (\mathbb{P} V ) - \mathrm{Quot}{0, d} ( V* )$ generalising the natural map $\mathrm{Sym}d X \to \mathrm{Quot}{0, d} ({\mathcal O}X)$. We then give a generalisation of the geometric Riemann--Roch theorem to vector bundles of higher rank over $X$. We use this to give a geometric description of the tangent cone to the Brill--Noether locus $Br{r, d}$ at a suitable bundle $E$ with $h0 (E) = r+n$. This gives a generalisation of the Riemann--Kempf singularity theorem. As a corollary, we show that the $n$th secant variety of the rank one locus of $\mathbb{P} \mathrm{End} E$ is contained in the tangent cone.