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 ${\mathrm{Hilb}}^d(\mathbb{P}V)-{\mathrm{Quot}}^{0,d}(V^{\ast })$ 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 $B_{r,d}^r$ 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.

19 pages

References in corpus (3)

• Tangent cones to generalised theta divisors and generic injectivity of the theta map
• On cubics and quartics through a canonical curve
• The Osculating cone to special Brill-Noether Loci