Koszul modules with vanishing resonance in algebraic geometry

Abstract

We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace \(K\subseteq \bigwedge ^2 V\) , where V is 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 instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion.