Jumping coefficients of multiplier ideals

By L. Ein, R. Lazarsfeld, Karen E. Smith, Dror Varolin. 2003

146 citations recorded by the source corpus.

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 \xi is a jumping number of f = 0 lying in the interval (0, 1], then -\xi is 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9ea0c3fb3bee4964f5f52850047fcf87e0e134a3

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Citations in the corpus

Discreteness and rationality of F-jumping numbers on singular varieties (2009, background): “More generally, the sequence of jumping numbers contains a great deal of information about the singularities of X and elements of a ,s ee [8, 10 ,34,49].”
Jumping coefficients and spectrum of a hyperplane arrangement (2009, background): “It is known that the jumping coefficients are closely related to the roots of the bfunction, see [ 12 ] and (1.13) below.”
Bernstein–Sato polynomials of arbitrary varieties (2004, background): “These are generalizations of the hypersurface case, see for example [5], [8], [11], [14], [26], [28].”
Bernstein–Sato polynomials of hyperplane arrangements (2006, background): “Let I0 be the ideal sheaf of {0} ⊂ C , and J (X,αD) be the multiplier ideal sheaf, see [15], [20].”
Multiplier ideals, $b$-function, and spectrum of a hypersurface singularity (2004, background): “It is shown by Ein et al. (see [ELSV04]) that any jumping coefficients which are less than one are roots of the Bernstein–Sato polynomial bf (s) (i.e. the b-function) up to a sign.”