Smoothing of semistable Fano varieties

TL;DR

This paper investigates the smoothing of semistable Fano varieties over arbitrary fields, classifying degenerations in characteristic 0 and establishing conditions for smoothing in positive and mixed characteristics.

Contribution

It provides a classification of reduced semistable Fano degenerations in characteristic 0 and proves smoothing results in positive and mixed characteristics under certain log structure assumptions.

Findings

Classification of semistable Fano degenerations in characteristic 0

Smoothing of semistable Fano varieties in positive characteristic with $W_2$ lifting

Smoothing results over mixed characteristic bases

Abstract

Given any field $k$ (not necessarily perfect), we study the smoothing of a semistable Fano variety over $k$. In characteristic 0, the reduced semistable Fano degenerate fibers of Mori fibrations are classified. In positive characteristic, under a suitable $W_2$ lifting assumption, we prove that a semistable Fano variety always appears as a degenerate fiber in a semistable family if it has a global log structure (in the sense of Fontaine-Illusie-Kato) of semistable type. A similar smoothing result over a mixed characteristic base is also obtained.