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. 
Lawrence Ein and Robert Lazarsfeld