Generalized determinantal representation of hypersurfaces

Abstract

In this article we extend the notion of determinantal representation of hypersurfaces to the determinantal representation of sections of the determinant line bundle of a vector bundle. We give several examples, and prove some necessary conditions for existence of determinantal representation. As an application, we show that for any integer $d \geq 1,$ there is an indecomposable vector bundle $E_d$ of rank $2$ on $\mathbb{P}^2$ such that almost all curves of degree $d$ of $\mathbb{P}^2$ arise as the degeneracy loci of a pair of holomorphic sections of $E_d$, upto an automorphism of $\mathbb{P}^2$. We use this result to obtain a linear algebraic application.