Uniform Bounds and Symbolic Powers on Smooth Varieties
Algebraic Geometry
2009-10-31 v3 Commutative Algebra
Complex Variables
Abstract
We show how multiplier ideals can be used to obtain uniform multiplicative bounds for certain families of ideals on a smooth complex algebraic variety. In particular we prove a quick but rather surprising result about symbolic powers of radical ideals on such a variety. Specifically, let I be a radical ideal sheaf on a smooth variety X defining a reduced subscheme Z of X and suppose that every irreducible component of Z has codimension at most e in X. Given an integer m > 0 suppose that f is a function germ that vanishes to order at least e.m at a general point of each irreducible component of Z . Then in fact f lies in the m-th power I^m of I.
Cite
@article{arxiv.math/0005098,
title = {Uniform Bounds and Symbolic Powers on Smooth Varieties},
author = {Lawrence Ein and Robert Lazarsfeld and Karen E. Smith},
journal= {arXiv preprint arXiv:math/0005098},
year = {2009}
}
Comments
Section added containing generalizations and questions suggested by work of Hochster and Huneke