Smoothness, Semistability, and Toroidal Geometry

TL;DR

The paper presents a new proof of a resolution of singularities theorem for algebraic varieties over characteristic zero fields, utilizing semistable reduction and toric geometry techniques.

Contribution

It introduces a novel proof method combining semistable reduction and toric geometry, offering an alternative approach to classical resolution of singularities.

Findings

Provides a new proof of a weak resolution theorem

Utilizes semistable reduction and toric geometry techniques

Offers an alternative to existing proofs by Bogomolov and Pantev

Abstract

We provide a new proof of the following result: Let $X$ be a variety of finite type over an algebraically closed field $k$ of characteristic 0, let $Z\subset X$ be a proper closed subset. There exists a modification $f:X_1 \rar X$, such that $X_1$ is a quasi-projective nonsingular variety and $Z_1 = f^{-1}(Z)_\red$ is a strict divisor of normal crossings. Needless to say, this theorem is a weak version of Hironaka's well known theorem on resolution of singularities. Our proof has the feature that it builds on two standard techniques of algebraic geometry: semistable reduction for curves, and toric geometry. Another proof of the same result was discovered independently by F. Bogomolov and T. Pantev. The two proofs are similar in spirit but quite different in detail.