A vanishing theorem for weight one syzygies

TL;DR

This paper generalizes a vanishing theorem for syzygies from curves to higher-dimensional varieties, linking geometric properties of line bundles to algebraic syzygy groups.

Contribution

It extends the vanishing theorem for asymptotic syzygies from curves to arbitrary smooth projective varieties, relating jet ampleness to Koszul cohomology vanishing.

Findings

Vanishing of K_{p,1}(X, B; L) for p-jet very ample B and large L.

Non-vanishing when B fails to impose independent conditions on certain cycles.

Generalization of syzygy vanishing results to higher dimensions.

Abstract

Inspired by the methods of Voisin, the first two authors recently proved that one could read off the gonality of a curve C from the syzygies of its ideal in any one embedding of sufficiently large degree. This was deduced from from a vanishing theorem for the asymptotic syzygies associated to an arbitrary line bundle B on C. The present paper extends this vanishing theorem to a smooth projective variety X of arbitrary dimension. Specifically, given a line bundle B on X, we prove that if B is p-jet very ample (i.e. the sections of B separate jets of total weight p+1) then the weight one Koszul cohomology group K_{p,1}(X, B; L) vanishes for all sufficiently positive L. In the other direction, we show that if there is a reduced cycle of length p+1 that fails to impose independent conditions on sections of B, then the Koszul group in question is non-zero for very positive L.