Smoothing of semistable Fano varieties

Abstract

In this paper we study the smoothing of a semistable Fano variety over a perfect field $k$. 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 geometric criterion for the existence of a log structure of semistable type is given.