Some results on the generic vanishing of Koszul cohomology via deformation theory

TL;DR

This paper investigates how Koszul cohomology groups associated with linear series on singular nodal curves deform to smooth curves, providing partial results on their generic vanishing through deformation-obstruction theory.

Contribution

It introduces a deformation-obstruction framework for Koszul cohomology on singular curves and computes obstruction classes for their deformation to smooth curves.

Findings

Obstruction classes for Koszul cohomology on singular curves are computed.

Partial results on the generic vanishing of Koszul cohomology groups are obtained.

The study advances understanding of Koszul cohomology deformation behavior on singular curves.

Abstract

We study the deformation-obstruction theory of Koszul cohomology groups of $g^r_d$'s on singular nodal curves. We compute the obstruction classes for Koszul cohomology classes on singular curves to deform to a smooth one. In the case the obstructions are nontrivial, we obtain some partial results for generic vanishing of Koszul cohomology groups.