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

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 $S{U}_C(r)-rightarrow|r\Theta |$ is generically injective for large values of $g$.

19 pages; assumptions in main theorem slightly changed because of a gap in v1; Section 2 rewritten; Section 3 removed

References in corpus (1)

• On cubics and quartics through a canonical curve

Cited by in corpus (2)

• The hyperelliptic theta map and osculating projections
• A Riemann--Kempf singularity theorem for higher rank Brill--Noether loci