Plucker forms and the theta map

TL;DR

This paper introduces Pl"ucker forms for pairs of vector bundles and subspaces, and uses this concept to analyze the theta map on moduli spaces of vector bundles, proving its generic injectivity under certain conditions.

Contribution

It develops the notion of Pl"ucker forms for pairs of vector bundles and applies it to establish the generic injectivity of the theta map on moduli spaces.

Findings

Theta map $ heta_r$ is generically injective for general curves when genus is much larger than rank.

Introduces the concept of Pl"ucker form for pairs $(E,S)$ in vector bundle theory.

Provides new tools for studying the geometry of moduli spaces of vector bundles.

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 $g >> r$.