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Functorial monomialization and uniqueness of centers for relative principalization

Dan Abramovich, Michael Temkin, Jarosław Włodarczyk

TL;DR

The paper addresses functorial monomialization and uniqueness of centers for relative principalization in characteristic 0. It combines a functorial Kummer cover, Rees-algebra flattening, and resolution to produce a monomialization that is compatible with regular base changes and group actions. The main contributions are the identification of uniquely determined Q-ideals J and J^w after monomialization and the establishment of functorial relative principalization and logarithmic reduction, with compatibility for localization on the base. These results provide a robust, functorial framework for principalization in logarithmic geometry and for reducing families of varieties in a way that respects base change and symmetry.

Abstract

Theorem 1.2.6 of [ATW20] provides a relatively functorial logarithmic principalization of ideals on relative logarithmic orbifolds $X\to B$ in characteristic 0, relying on a delicate monomialization theorem for Kummer ideals. The paper [AdSTW25] provides a parallel avenue through weighted blowings up. In this paper we show that, if $X\to B$ is proper, monomialization of both Kummer and weighted logarithmic centers can be carried out in a manner which is functorial for base change by regular morphisms. This implies in particular logarithmic relative principalization of ideals and logarithmically smooth reduction of proper families of varieties in characteristic 0 in a manner equivariant for group actions and compatible with localization on the base.

Functorial monomialization and uniqueness of centers for relative principalization

TL;DR

The paper addresses functorial monomialization and uniqueness of centers for relative principalization in characteristic 0. It combines a functorial Kummer cover, Rees-algebra flattening, and resolution to produce a monomialization that is compatible with regular base changes and group actions. The main contributions are the identification of uniquely determined Q-ideals J and J^w after monomialization and the establishment of functorial relative principalization and logarithmic reduction, with compatibility for localization on the base. These results provide a robust, functorial framework for principalization in logarithmic geometry and for reducing families of varieties in a way that respects base change and symmetry.

Abstract

Theorem 1.2.6 of [ATW20] provides a relatively functorial logarithmic principalization of ideals on relative logarithmic orbifolds in characteristic 0, relying on a delicate monomialization theorem for Kummer ideals. The paper [AdSTW25] provides a parallel avenue through weighted blowings up. In this paper we show that, if is proper, monomialization of both Kummer and weighted logarithmic centers can be carried out in a manner which is functorial for base change by regular morphisms. This implies in particular logarithmic relative principalization of ideals and logarithmically smooth reduction of proper families of varieties in characteristic 0 in a manner equivariant for group actions and compatible with localization on the base.

Paper Structure

This paper contains 10 sections, 5 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Results
  4. Remarks
  5. Uniqueness of non-weighted centers
  6. Monomialization
  7. A Kummer cover
  8. Flattening
  9. Resolution
  10. Proof of monomialization

Key Result

Proposition 1.2.1

Let $X \to B$ be an integral and proper relative logarithmic orbifold, with $B$ regular. Consider a ${\mathbb{Q}}$-ideal $J$, respectively $J^w$, locally of the form with ${\mathcal{F}}^{\leq 1}({\mathcal{I}}_\infty) ={\mathcal{I}}_\infty$. There is a functorial birational morphism $B' \to B$, with saturated pullback $X' = (X \times_BB')^{\rm sat}$, so that $J':= J{\mathcal{O}}_{X'}$, respectivel

Theorems & Definitions (8)

  • Proposition 1.2.1
  • Proposition 1.2.2
  • Lemma 2.0.1
  • proof
  • proof : Proof of Proposition \ref{['Prop:uniqueness']}
  • Lemma 3.1.1
  • Lemma 3.4.1
  • proof