  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.

Algebraic Geometry;Commutative Algebra;Complex Variables