A Riemann--Kempf singularity theorem for higher rank Brill--Noether loci

TL;DR

This paper extends classical theorems to higher rank vector bundles over curves, providing a geometric description of Brill--Noether loci and their tangent cones, generalizing the Riemann--Kempf singularity theorem.

Contribution

It introduces a new geometric framework for higher rank Brill--Noether loci and generalizes the Riemann--Kempf singularity theorem for these cases.

Findings

Describes tangent cones to Brill--Noether loci at specific bundles.

Shows the secant variety of rank one locus is contained in the tangent cone.

Provides a higher rank generalization of the Riemann--Kempf theorem.

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 $B^r_{r, d}$ at a suitable bundle $E$ with $h^0 (E) = r+n$. 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.