# Gaku Liu: Semistable reduction in characteristic 0

Time: Tue 2020-02-11 11.00 - 11.50

Lecturer: Gaku Liu, KTH Royal Institute of Technology

### 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.

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