Koszul modules and Green's conjecture
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.
Keywords
Cite
@article{arxiv.1810.11635,
title = {Koszul modules and Green's conjecture},
author = {Marian Aprodu and Gavril Farkas and Stefan Papadima and Claudiu Raicu and Jerzy Weyman},
journal= {arXiv preprint arXiv:1810.11635},
year = {2020}
}
Comments
minor edits, 42 pages, to appear in Invent. Math