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arXiv · 1810.11635

Koszul modules and Green's conjecture

Marian Aprodu·Gavril Farkas·Stefan Papadima·Claudiu Raicu·Jerzy Weyman
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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.

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Marian Aprodu, Gavril Farkas, Stefan Papadima, Claudiu Raicu, Jerzy Weyman. 2018-10-27. Koszul modules and Green's conjecture. https://doi.org/10.1007/s00222-019-00894-1

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