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Convexity of multiplicities of filtrations on local rings

Harold Blum, Yuchen Liu, Lu Qi

Abstract

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.

Convexity of multiplicities of filtrations on local rings

Abstract

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions.
Paper Structure (44 sections, 46 theorems, 176 equations)

This paper contains 44 sections, 46 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Multiplicity and geodesics
  3. Applications to volume
  4. Applications to normalized volume
  5. Rees's theorem
  6. Minkowski inequality
  7. Preliminaries
  8. Filtrations
  9. Valuations
  10. Real valuations
  11. Divisorial valuations
  12. Quasi-monomial valuations
  13. Izumi's inequality
  14. Intersection numbers
  15. Definition
  16. ...and 29 more sections

Key Result

Theorem 1.1

Assume $R$ contains a field. If $\mathfrak{a}_{\bullet,0}$ and $\mathfrak{a}_{\bullet,1}$ are $\mathfrak{m}$-filtrations with positive multiplicity, then the function $E(t) \colon [0,1]\to \mathbb{R}$ defined by $E(t) \coloneqq \mathrm{e}(\mathfrak{a}_{\bullet,t})$ satisfies the following propertie

Theorems & Definitions (106)

  • Theorem 1.1
  • Corollary 1.2: Convexity of volume
  • Corollary 1.3: Uniqueness of minimizer
  • Theorem 1.4: Rees's Theorem
  • Corollary 1.5: Minkowski Inequality
  • Remark 1.6: Relation to work of Cutkosky
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 96 more