A vanishing theorem for weight one syzygies

Lawrence Ein; Robert Lazarsfeld; David Yang

doi:10.2140/ant.2016.10.1965

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arXiv · 1512.06384

A vanishing theorem for weight one syzygies

Lawrence Ein·Robert Lazarsfeld·David Yang

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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.

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Lawrence Ein, Robert Lazarsfeld, David Yang. 2016-09-21. A vanishing theorem for weight one syzygies. https://doi.org/10.2140/ant.2016.10.1965

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