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 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.

Tangent developable surfaces and the equations defining algebraic curves

“…The actual write-up in [2] is a bit long and complicated, in part because the authors work to extend their results as far as possible to positive characteristics, and in part because they are fastidious in checking that the maps that come up are the expected ones. Here we focus on the essential geometric ideas that seem to underlie their computations.…”

“…Computing the syzygies of T . The first step in the argument of [2] is to understand the tangent developable T = Tan(C) and its syzygies in terms of more familiar and computable objects. This culminates in Theorem 3.3 below, which describes the relevant syzygies linear algebraically.…”

“…A substantial body of experimental evidence supported this proposal, but in spite of considerable effort nobody was able to push through the required computations. In a recent preprint [2], however, Aprodu, Farkas, Papadima, Raicu and Weyman have succeeded in doing so. Their work is the subject of the present report.…”

“…We work throughout over the complex numbers. In particular, we completely ignore contributions of [2] to understanding what parts of Green's conjecture work in positive characteristics.…”

“…We thank the authors of [2] for sharing an early draft of their paper. We have profited from correspondence and conversations with David Eisenbud, Gabi Farkas, Claudiu Raicu, Frank Schreyer and Claire Voisin.…”