3,133 research outputs found
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
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