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A new proof of monomialisation from 3-folds to surfaces

Yueting Jiang

TL;DR

The paper delivers a new proof of Cutkosky's monomialisation theorem for dominant morphisms from a smooth 3-fold to a surface by exploiting log-Fitting ideals and a Rank Theorem–style framework in algebraic geometry. It introduces and develops the notions of log-rank adapted and strongly prepared morphisms, providing intrinsic characterisations through log-Fitting ideals and enabling a staged reduction process. The approach reduces an arbitrary dominant morphism to a log-rank adapted form, then to a strongly prepared form, and finally invokes a known monomialisation result to achieve the desired monomial morphism. This clarifies the key structural mechanisms behind monomialisation and broadens the toolkit for tackling resolution-type problems, though a global higher-dimensional generalisation remains open.

Abstract

In this paper, we give a new proof of the foundational result, due to S. Cutkosky, on the existence of a monomialisation of a morphism from a 3-fold to a surface. Our proof brings to the fore the notion of log-Fitting ideals, and requires us to develop new methods related to Rank Theorems and log-Fitting ideals.

A new proof of monomialisation from 3-folds to surfaces

TL;DR

The paper delivers a new proof of Cutkosky's monomialisation theorem for dominant morphisms from a smooth 3-fold to a surface by exploiting log-Fitting ideals and a Rank Theorem–style framework in algebraic geometry. It introduces and develops the notions of log-rank adapted and strongly prepared morphisms, providing intrinsic characterisations through log-Fitting ideals and enabling a staged reduction process. The approach reduces an arbitrary dominant morphism to a log-rank adapted form, then to a strongly prepared form, and finally invokes a known monomialisation result to achieve the desired monomial morphism. This clarifies the key structural mechanisms behind monomialisation and broadens the toolkit for tackling resolution-type problems, though a global higher-dimensional generalisation remains open.

Abstract

In this paper, we give a new proof of the foundational result, due to S. Cutkosky, on the existence of a monomialisation of a morphism from a 3-fold to a surface. Our proof brings to the fore the notion of log-Fitting ideals, and requires us to develop new methods related to Rank Theorems and log-Fitting ideals.

Paper Structure

This paper contains 13 sections, 14 theorems, 9 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of main definitions
  4. Locally principal ideal sheaves
  5. The geometric rank and the Rank Theorem
  6. Quasi-prepared morphisms
  7. Strongly prepared morphisms
  8. Log-Fitting ideals and log-rank
  9. Definition of log-Fitting ideals
  10. Definition of log-rank
  11. Intrinsic characterisation of strongly prepared morphisms
  12. Log-rank adapted morphisms
  13. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1.1

Let $\Phi:(X,E)\rightarrow (S,F)$ be a dominant morphism of pairs (Definition def:MorphismPairs) between $\mathbb{K}$-varieties (Definition kv) with dim$X=3$ and dim$S=2$. Then $\Phi$ admits a monomialisation as recalled in Definition MOM, i.e. there exists a commutative diagram: with $\sigma,\tau$ finite sequences of blowings-up and $\Phi^{\prime}$ a monomial morphism (Definition AMM).

Theorems & Definitions (47)

  • Theorem 1.1: cutkosky2002monomialization
  • Definition 2.1: $\mathbb{K}$-variety
  • Remark 2.2
  • Definition 2.3: Morphism of pairs
  • Definition 2.4: Dominant algebraic monomial morphism, see cutkosky2004monomialization
  • Remark 2.5: On the definition of monomial morphisms, cf. bbmono
  • Definition 2.6: Monomialization of morphisms
  • Lemma 2.7
  • proof
  • Definition 2.8
  • ...and 37 more