Plucker forms and the theta map

Abstract

In this paper we introduce the elementary notion of Pl\"ucker form of a pair $(E,S)$, where $E$ is a vector bundle of rank $r$ on a smooth, irreducible, complex projective variety $X$ and $S \subset H^0(E)$ is a subspace of dimension $rm$. We apply this notion to the study of theta map $\theta_r$ on the moduli space $SU_X(r,0)$ of semistable vector bundles of rank $r$ and trivial determinant on a curve $X$ of genus $g$. We prove that $\theta_r$ is generically injective if $X$ is general and $g >> r$.