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On instability of Syzygy Bundles

Snehajit Misra

TL;DR

This paper studies how the (semi)stability of syzygy bundles $M_E$ attached to globally generated vector bundles $E$ on smooth projective varieties changes under polarization. It establishes a criterion that, by twisting and removing a fixed divisor $S$, can produce a destabilizing subbundle $M_{E(d)\otimes \mathcal{O}_X(-S)}$ of $M_{E(d)}$ for large twists when a certain quadratic form in $d$ is positive. On surfaces with Picard number at least $3$ and with an effective cone generated by curves meeting at most once, the authors prove that for any such $E$ with ample determinant $D$, there exists an ample $A$ so that $M_{E(d)}$ is not $A$-stable for $d\gg0$, extending instability phenomena from toric cases to higher rank bundles. This advances understanding of moduli and Brill-Noether-type phenomena for syzygy bundles in higher dimensions and under varying polarizations.

Abstract

In this article, we investigate the instability of syzygy bundles corresponding to globally generated vector bundles on smooth irreducible projective surfaces under change of polarization.

On instability of Syzygy Bundles

TL;DR

This paper studies how the (semi)stability of syzygy bundles attached to globally generated vector bundles on smooth projective varieties changes under polarization. It establishes a criterion that, by twisting and removing a fixed divisor , can produce a destabilizing subbundle of for large twists when a certain quadratic form in is positive. On surfaces with Picard number at least and with an effective cone generated by curves meeting at most once, the authors prove that for any such with ample determinant , there exists an ample so that is not -stable for , extending instability phenomena from toric cases to higher rank bundles. This advances understanding of moduli and Brill-Noether-type phenomena for syzygy bundles in higher dimensions and under varying polarizations.

Abstract

In this article, we investigate the instability of syzygy bundles corresponding to globally generated vector bundles on smooth irreducible projective surfaces under change of polarization.
Paper Structure (3 sections, 4 theorems, 26 equations)

This paper contains 3 sections, 4 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Notations and Conventions
  3. Main results

Key Result

Theorem 1.2

Let $X$ be a smooth surface of Picard number at least 3 having an effective cone generated by curves $C_1,C_2,\cdots, C_m$ with mutual intersection multiplicity of at most 1. Then for every vector bundle $E$ with its determinant bundle $D=\det(E)$ ample, there exists an ample polarisation $A$ such t

Theorems & Definitions (8)

  • Theorem 1.2
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof