Embedded desingularization of toric varieties

TL;DR

This paper introduces a novel combinatorial approach to achieve embedded desingularization of toric varieties, resulting in regular strict transforms with normal crossings.

Contribution

It develops a new method using combinatorial blowing-ups to desingularize toric varieties in an embedded setting.

Findings

Successfully produces regular strict transforms with normal crossings.

Provides a finite sequence of combinatorial blow-ups for desingularization.

Ensures the final variety is smooth and has simple normal crossing divisors.

Abstract

We present a new method to achieve an embedded desingularization of a toric variety. Let $W$ be a regular toric variety defined by a fan $\Sigma$ and $X\subset W$ be a toric embedding. We construct a finite sequence of combinatorial blowing-ups such that the final strict transforms $X'\subset W'$ are regular and $X'$ has normal crossing with the exceptional divisor.