Jumping of the nef cone for Fano varieties

TL;DR

This paper constructs examples of Fano varieties where the nef cone changes under deformation, demonstrating the optimality of previous results and exploring deformation invariance properties of Fano varieties.

Contribution

It provides explicit examples of Fano varieties with jumping nef cones and extends deformation invariance results for divisor class groups and Cox rings.

Findings

Nef cone jumps in certain Fano varieties under deformation

Deformation invariance of divisor class group for specific Fano varieties

Cox ring deforms flatly in families of terminal Q-factorial Fano varieties

Abstract

We construct Q-factorial terminal Fano varieties, starting in dimension 4, whose nef cone jumps when the variety is deformed. It follows that de Fernex and Hacon's results on deformations of 3-dimensional Fanos are optimal. The examples are based on the existence of high-dimensional flips which deform to isomorphisms, generalizing the Mukai flop. We also extend de Fernex and Hacon's positive results on deformations of Fano varieties. Toric Fano varieties which are smooth in codimension 2 and Q-factorial in codimension 3 are rigid. The divisor class group is deformation-invariant for klt Fanos which are smooth in codimension 2 and Q-factorial in codimension 3. The Cox ring deforms in a flat family when a terminal Fano which is Q-factorial in codimension 3 is deformed. A side result which seems to be new is that the divisor class group of a klt Fano variety maps isomorphically to ordinary homology.