Koszul modules with vanishing resonance in algebraic geometry

TL;DR

This paper explores the applications of a vanishing result for Koszul modules in algebraic geometry, impacting various areas such as cohomology stabilization, moduli spaces, and degeneracy loci, and linking stability of vector bundles to resonance.

Contribution

It introduces new applications of Koszul module vanishing results to diverse problems in algebraic geometry, including cohomology, moduli spaces, and vector bundle stability.

Findings

Vanishing of Koszul modules aids in cohomology stabilization.

Applications to divisors on moduli spaces of curves.

Resonance governs stability of positive rank 2 vector bundles.

Abstract

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of a vector space. Previously Koszul modules of finite length have been used to give a proof of Green's Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.