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Regular version of the site

Working paper

On semistable degenerations of Fano varieties

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.