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Uniform Bounds and Symbolic Powers on Smooth Varieties

Lawrence Ein·Robert Lazarsfeld·Karen E. Smith
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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.

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Lawrence Ein, Robert Lazarsfeld, Karen E. Smith. 2000-10-04. Uniform Bounds and Symbolic Powers on Smooth Varieties. https://doi.org/10.1007/s002220100121

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