On semistable degenerations of Fano varieties

TL;DR

This paper studies semistable degenerations of Fano varieties, showing the dual complex is a simplex of bounded dimension, and classifies such degenerations for del Pezzo surfaces, highlighting uniqueness and trivial monodromy in low dimensions.

Contribution

It proves the dual complex of semistable Fano degenerations is a simplex of bounded dimension and classifies degenerations of del Pezzo surfaces using the Minimal Model Program.

Findings

Dual complex of Fano degeneration is a simplex of dimension ≤ dim F.

Any admissible simplex dimension can be realized for any fiber dimension.

Maximal degeneration is unique with trivial monodromy in dimension ≤ 3.

Abstract

Consider a family of Fano varieties $\pi: X \longrightarrow B\ni o$ over a curve germ with a smooth total space $X$. Assume that the generic fiber is smooth and the special fiber $F=\pi^{-1}(o)$ 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 $\leq \mathrm{dim}\ F$. 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 $\leq3$.