Koszul modules and Green’s conjecture

By M. Aprodu, G. Farkas, Stefan Papadima, Claudiu Raicu, Jerzy Weyman. Inventiones Mathematicae, 2018

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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 $${{\mathbf {k}}}$$k, with $${\text {char}}({{\mathbf {k}}})=0$$char(k)=0 or $${\text {char}}({{\mathbf {k}}})\ge \frac{g+2}{2}$$char(k)≥g+22. 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 $${{\mathfrak {s}}}{{\mathfrak {l}}}_2$$sl2-representations; (2) we completely characterize, in arbitrary characteristics, the (non-)vanishing behavior of the syzygies of the tangential variety to a rational normal curve.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

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