Length, multiplicity, and multiplier ideals

Details Length, multiplicity, and multiplier ideals Length, multiplicity, and multiplier ideals

2004-09-26

arxiv.org/abs/math/0409500v1

Mathematics

Algebraic Geometry

15 pages; to appear in Trans. Amer. Math. Soc

Scientific paper

Let (R,m) be an n-dimensional regular local ring, essentially of finite type over a field of characteristic zero. In this paper we study the relationship between the singularities of the scheme defined by an m-primary ideal I of R and the multiplier ideals J(I^c), with c varying among the positive rational numbers. In particular, we prove that, for every nonnegative integer k, the Samuel multiplicity of I is greater or equal to (n+k)^n/c^n whenever J(I^c) is contained in the (k+1)-th power of the maximal ideal m. This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Musta\c{t}\v{a} and the author of this paper (arXiv:math.AG/0205171). A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

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