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

Abstract

Given a vector bundle V over a curve X, we define and study a surjective rational map Hilbd(PV ) 99K Quot0,d(V ∗) generalising the natural map SymdX → Quot0,d(OX). 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 Brr,d at a suitable bundle E with h0(E) = r + k. This gives a generalisation of the Riemann–Kempf singularity theorem. As a corollary, we show that the kth secant variety of the rank one locus of PEndE is contained in the tangent cone.