Toroidalization of Locally Toroidal Morphisms from N-folds to Surfaces

TL;DR

This paper proves that any locally toroidal morphism from an N-fold to a surface can be modified into a toroidal morphism, advancing the understanding of the toroidalization conjecture in algebraic geometry.

Contribution

It establishes the positive resolution of the toroidalization problem for locally toroidal morphisms from N-folds to surfaces.

Findings

Any locally toroidal morphism from an N-fold to a surface can be toroidalized.

Supports the toroidalization conjecture in a specific case.

Provides a method for modifying morphisms into toroidal form.

Abstract

The toroidalization conjecture of D. Abramovich, K. Karu, K. Matsuki, and J. Wlodarczyk asks whether any given morphism of nonsingular varieties over an algebraically closed field of characteristic zero can be modified into a toroidal morphism. Following a suggestion by Dale Cutkosky, we define the notion of \emph{locally toroidal} morphisms and ask whether any locally toroidal morphism can be modified into a toroidal morphism. In this paper, we answer the question in the affirmative when the morphism is between any arbitrary variety and a surface.