Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0303002

Jumping Coefficients of Multiplier Ideals

Lawrence Ein·Robert Lazarsfeld·Karen E. Smith·Dror Varolin
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.AGmath.ACmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lawrence Ein, Robert Lazarsfeld, Karen E. Smith, Dror Varolin. 2003-02-28. Jumping Coefficients of Multiplier Ideals. https://arxiv.org/abs/math/0303002

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lawson--Deligne Classes and Applications

We construct the integral Lawson--Deligne map of weight $q=n-p-k-1$ on smooth complex projective $n$-folds using filtered currents. It lifts the Friedlander--Mazur cycle class, recovers the reduced generalized Abel--Jacobi invariant on homologically trivial classes, and is compatible with algebraic correspondences. A Picard--Fuchs separation argument applied to the conic and van Geemen normal functions on the mirror quintic determines explicit regulator subspaces modulo the full rational period group. For prescribed elliptic moduli and a suitable mirror-quintic fiber, the subspace generated by their $a$- and $b$-loop products has dimension twice the $\Q$-dimension of the period-monomial space. Moduli $i\sqrt{\ell_j}$ for distinct primes $\ell_j$ give $2^{k+1}$ independent images on varieties of dimension $p+k+2$; one repeated imaginary quadratic modulus gives dimension four for every $k\geq1$. Compatibility with known projective-bundle and blow-up decompositions yields independent exceptional subspaces on smooth rational varieties. We also compare the higher Chow composite with the Bloch--KLM regulator after lowering the Hodge filtration. The KLM representative reduces to a cut-current class, and equality with the Lawson composite is proved in degree zero and for constant-unit decomposable classes. The general positive-degree comparison is reduced to an explicit filtered-realization condition.

math.AG↗
arXiv2026

Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.

math.AG↗