Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicities of Jumping Numbers

Suchitra Pande

TL;DR

The paper advances the understanding of jumping numbers of multiplier ideals on smooth varieties by proving that the multiplicity function $m(c+n)$ is a polynomial in $n$ of degree $h-1$ (where $h$ is the codimension of the relevant component), which implies that $m(c)$ is a quasi-polynomial in $c$. It introduces the Rees coefficient $\rho_c$ as the leading coefficient and links its positivity to contributions from Rees valuations, establishing a precise correspondence between growth of multiplicities and divisorial/valuation data. The authors provide explicit formulas in the monomial case via Newton polyhedra, relate the sum of Rees coefficients to the Hilbert-Samuel multiplicity, and show that the associated Poincaré series is a rational function. Together, these results generalize known two-dimensional phenomena to higher dimensions and yield practical computational tools, including corollaries that compute jumping numbers from blow-ups without full resolutions. The work thus deepens the structural understanding of singularities through multiplier ideals, valuations, and intersection theory, with potential algorithmic applications.

Abstract

We study multiplicities of jumping numbers of multiplier ideals in a smooth variety of arbitrary dimension. We prove that the multiplicity function is a quasi-polynomial, hence proving that the Poincaré series is a rational function. We further study when the various components of the quasi-polynomial have the highest possible degree and relate it to jumping numbers contributed by Rees valuations. Finally, we study the special case of monomial ideals.

Multiplicities of Jumping Numbers

TL;DR

The paper advances the understanding of jumping numbers of multiplier ideals on smooth varieties by proving that the multiplicity function is a polynomial in of degree (where is the codimension of the relevant component), which implies that is a quasi-polynomial in . It introduces the Rees coefficient as the leading coefficient and links its positivity to contributions from Rees valuations, establishing a precise correspondence between growth of multiplicities and divisorial/valuation data. The authors provide explicit formulas in the monomial case via Newton polyhedra, relate the sum of Rees coefficients to the Hilbert-Samuel multiplicity, and show that the associated Poincaré series is a rational function. Together, these results generalize known two-dimensional phenomena to higher dimensions and yield practical computational tools, including corollaries that compute jumping numbers from blow-ups without full resolutions. The work thus deepens the structural understanding of singularities through multiplier ideals, valuations, and intersection theory, with potential algorithmic applications.

Abstract

We study multiplicities of jumping numbers of multiplier ideals in a smooth variety of arbitrary dimension. We prove that the multiplicity function is a quasi-polynomial, hence proving that the Poincaré series is a rational function. We further study when the various components of the quasi-polynomial have the highest possible degree and relate it to jumping numbers contributed by Rees valuations. Finally, we study the special case of monomial ideals.

Paper Structure

This paper contains 17 sections, 27 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Review of Multiplier Ideals and Intersection Theory
  3. Multiplier Ideals and Jumping Numbers
  4. Candidate Jumping Numbers:
  5. Numerical Intersection Theory
  6. The Polynomial Nature of Multiplicities
  7. Properties of the Rees coefficient
  8. Criteria for Positivity
  9. Jumping numbers contributed by divisors
  10. Rees valuations:
  11. Proof of Theorem \ref{['mainthm41']}:
  12. Step 1:
  13. Step 2:
  14. On the valuations contributing jumping numbers.
  15. Relation to Hilbert-Samuel Multiplicity
  16. ...and 2 more sections

Key Result

Theorem 2.4

(Local Vanishing Theorem) Let $\mathfrak{a} \subset \mathcal{O}_{X}$ be an ideal, and $\mu: Y \to X$ be a log resolution of $\mathfrak{a}$ and let $\mathfrak{a} \cdot \mathcal{O}_{Y} = \mathcal{O}_{Y}(-F)$ for an effective divisor $F$. Then for any $c >0$, the following higher direct images vanish:

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 2.7
  • Proposition 2.8
  • proof
  • Definition 3.2
  • ...and 54 more