A remark on the generic vanishing of Koszul cohomology

TL;DR

This paper provides a new sufficient condition for the vanishing of Koszul cohomology groups in algebraic geometry, simplifying the approach to the Maximal Rank Conjecture for quadrics by reducing it to cases with zero Brill-Noether number.

Contribution

It introduces an inductive criterion for Koszul cohomology vanishing and demonstrates its application to the Maximal Rank Conjecture, streamlining the proof process.

Findings

Established a sufficient condition for Koszul cohomology vanishing

Reduced the proof of the Maximal Rank Conjecture to cases with Brill-Noether number zero

Provided a new inductive approach for studying Koszul cohomology in algebraic geometry

Abstract

We give a sufficient condition to study the vanishing of certain Koszul cohomology groups for general pairs $(X,L)\in W^r_{g,d}$ by induction. As an application, we show that to prove the Maximal Rank Conjecture (for quadrics), it suffices to check all cases with the Brill-Noether number $\rho=0$.