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Generalized determinantal representation of hypersurfaces

A. El Mazouni, D. S. Nagaraj, Supravat Sarkar

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.

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 there is an indecomposable vector bundle of rank on such that almost all curves of degree of arise as the degeneracy loci of a pair of holomorphic sections of , upto an automorphism of . We use this result to obtain a linear algebraic application.
Paper Structure (7 sections, 11 theorems, 60 equations)

This paper contains 7 sections, 11 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Proof of The Theorem \ref{['thm1']}
  3. Proof of The Theorem \ref{['thm2']}
  4. An algebraic application
  5. Proof of Theorem \ref{['thm3']}
  6. Proof of Theorem \ref{['thm4']}
  7. Proof of Theorem \ref{['thm5']}

Key Result

Theorem 1.1

Let $N$ be the rank two vector bundle on $\mathbb{P}^2$ obtained by taking the quotient of ${\mathcal{O}}_{\mathbb{P}^2}^2 \oplus {\mathcal{O}}_{\mathbb{P}^2}(1)$ by the subbundle $(x,y,z^2){\mathcal{O}}_{\mathbb{P}^2}(-1).$ Then $N$ is abundant.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • ...and 12 more