On semistable degenerations of Fano varieties

Abstract

Consider a family of Fano varieties π : X −→ B ∋ o over a curve germ with a smooth total space X. Assume that the generic fiber is smooth and the special fiber F = π −1 (o) has simple normal crossings. Then F is called a semistable degeneration of Fano varieties. We show that the dual complex of F is a simplex of dimension ≤ dim F . Simplices of any admissible dimension can be realized for any dimension of the fiber. Using this result and the Minimal Model Program in dimension 3 we reproduce the classification of the semistable degenerations of del Pezzo surfaces obtained by Fujita. We also show that the maximal degeneration is unique and has trivial monodromy in dimension ≤ 3.