Koszul modules and Green's conjecture

TL;DR

This paper proves a vanishing theorem for Koszul modules and confirms Green's conjecture for certain rational curves, providing new results especially in positive characteristic and offering a different proof in characteristic zero.

Contribution

It introduces a characteristic-independent Hermite reciprocity and characterizes syzygy behavior of tangential varieties, advancing understanding of Green's conjecture.

Findings

Green's conjecture holds for g-cuspidal rational curves in specified characteristic ranges.

New proof of Green's conjecture for general canonical curves in characteristic zero.

Explicit, characteristic-independent Hermite reciprocity established.

Abstract

We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describe an explicit, characteristic-independent version of Hermite reciprocity for sl_2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.