Jumping Coefficients of Multiplier Ideals

Abstract

We study in this paper some local invariants attached via multiplier ideals to an effective divisor or ideal sheaf on a smooth complex variety. First considered (at least implicitly) by Libgober and by Loeser and Vaquie, these jumping coefficients consist of an increasing sequence of rational numbers beginning with the log canonical threshold of the divisor or ideal in question. They encode interesting geometric and algebraic information, and we show that they arise naturally in several different contexts. Given a polynomial f having only isolated singularities, results of Varchenko, Loeser and Vaquie imply that if ξis a jumping number of f = 0 lying in the interval (0, 1], then -ξis a root of the Bernstein-Sato polynomial of f. We adapt an argument of Kollar to show prove that this holds also when the singular locus of f has positive dimension. In a more algebraic direction, we show that the number of such jumping coefficients bounds the uniform Artin-Rees number of the principal ideal (f) in the sense of Huneke: in the case of isolated singularities, this in turn leads to bounds involving the Milnor and Tyurina numbers of f . Along the way, we establish a general result relating multiplier to Jacobian ideals. We also explore the extension of these ideas to the setting of graded families of ideals. The paper contains many concrete examples.