# Gaku Liu: Semistable reduction in characteristic 0

**Time: **
Tue 2020-02-11 11.00 - 11.50

**Lecturer: **
Gaku Liu, KTH Royal Institute of Technology

**Location: **
Institut Mittag-Leffler, Seminar Hall Kuskvillan

### Abstract

Semistable reduction is a relative generalization of the classical problem of resolution of singularities of varieties; the goal is, given a surjective morphism \(f : X \to B\) of varieties in characteristic 0, to change \(f\) so that it is "as nice as possible". The problem goes back to at least Kempf, Knudsen, Mumford, and Saint-Donat (1973), who proved a strongest possible version when \(B\) is a curve. The key ingredient in the proof is the following combinatorial result: Given any \(d\)-dimensional polytope \(P\) in \(\mathbb{R}^d\) with integer-coordinate vertices, there is a dilation of \(P\) which can be triangulated into simplices with integer-coordinate vertices each with volume \(1/d!\).

In 2000, Abramovich and Karu proved, for any base \(B\), that \(f\) can be made into a weakly semistable morphism \(f' : X' \to B'\). They conjectured further that \(f'\) can be made semistable, which amounts to making \(X'\) smooth. They explained why this is the best resolution of \(f\) one might hope for. In this talk I will outline a proof of this conjecture. They key ingredient is a relative generalization of the above combinatorial result of KKMS. I will also discuss some other consequences in combinatorics of our constructions. This is joint work with Karim Adiprasito and Michael Temkin.