Tangent cones to generalised theta divisors and generic injectivity of the theta map

TL;DR

This paper demonstrates how to recover a stable vector bundle from the tangent cone to its associated theta divisor on a Petri general curve, and proves the generic injectivity of the theta map for large genus values.

Contribution

It provides a constructive method to recover vector bundles from tangent cones and sharpens the known results on the injectivity of the theta map for large genus.

Findings

Bundle E can be recovered from the tangent cone to its theta divisor.

The theta map is generically injective for genus g > (2r+2)(2r+1).

The proof offers a constructive approach and improves previous theorems.

Abstract

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $\Theta_E$ at ${\mathcal O}_C$. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r \Theta|$ is generically injective for large values of $g$.