Generalized determinantal representation of hypersurfaces

TL;DR

This paper extends determinantal representations to sections of determinant line bundles of vector bundles, providing new examples, necessary conditions, and applications to algebraic geometry and linear algebra.

Contribution

It introduces a generalized notion of determinantal representation for hypersurfaces via vector bundles, with conditions for existence and applications to curve degeneracy loci.

Findings

Existence conditions for generalized determinantal representations.

Construction of indecomposable vector bundles on projective planes.

Representation of most degree d curves as degeneracy loci of vector bundle sections.

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.