Introduction

The technique of torification is a powerful tool to reduce certain questions in algebraic geometry (at least locally) to toric varieties and thereby to combinatorial questions about convex cones and lattice polytopes. Landmark results proved using this technique include the Weak Factorization Theorem for birational morphisms by Wodarczyk et al. or the higher-dimensional Strong McKay Correspondence by Batyrev.

The aim of this seminar is to understand the proof of one of the first applications of this principle to the problem of semistable reduction for families of higher-dimensional varieties. Here, the combinatorial part alone is of independent interest.

We start with a crash course on toric varieties. We proceed by reducing the general theorem to a question about certain triangulations, and eventually solve this combinatorial problem.