A simpler proof of toroidalization of morphisms from 3-folds to surfaces

TL;DR

This paper presents a simpler, more conceptual proof that morphisms from 3-folds to surfaces can be transformed into toroidal morphisms through blow-ups, improving upon a previous longer and more complex proof.

Contribution

It offers a clearer, more elegant proof of toroidalization of morphisms from 3-folds to surfaces over characteristic zero fields, simplifying the original complex argument.

Findings

Proof of toroidalization is simpler and more conceptual

Sequences of blow-ups can achieve toroidal morphisms

Enhanced clarity in the global reduction process

Abstract

We give a simpler and more conceptual proof that a morphism from a 3-fold to a surface, over an algebraically closed field of characteristic 0, can be made into a toroidal morphism by sequences of blow ups of nonsingular subvarieties above the domain and range. Our original proof, which is much longer and more complicated, appeared in Springer Lecture Notes in 2002. The final version of this paper is much clearer, thanks to the encouragement of the referee. In particular, Section 5 on global reduction has been completely rewritten. The paper will appear in the Annals of the Fourier Institute.