A Geometric Effective Nullstellensatz

We present in this paper a geometric theorem which clarifies and extends in several directions work of Brownawell, Kollar and others on the effective Nullstellensatz. To begin with, we work on an arbitrary smooth complex projective variety X, with previous results corresponding to the case when X is projective space. In this setting we prove a local effective Nullstellensatz for ideal sheaves, and a corresponding global division theorem for adjoint-type bundles. We also make explicit the connection with the intersection theory of Fulton and MacPherson. Finally, constructions involving products of prime ideals that appear in earlier work are replaced by geometrically more natural conditions involving order of vanishing along subvarieties. The main technical inputs are vanishing theorems, which are used to give a simple algebro-geometric proof of a theorem of Skoda type, which may be of independent interest.Comment: Introduction expanded, examples added, work of Sombra discusse

The gonality conjecture on syzygies of algebraic curves of large degree

We show that a small variant of the methods used by Voisin in her study of canonical curves leads to a surprisingly quick proof of the gonality conjecture of Green and the second author, asserting that one can read off the gonality of a curve C from its resolution in the embedding defined by any one line bundle of sufficiently large degree. More generally, we establish a necessary and sufficient condition for the asymptotic vanishing of the weight one syzygies of the module associated to an arbitrary line bundle on C