Jumping of the nef cone for Fano varieties

Abstract

We construct Q \textbf {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 improve earlier results on deformations of Fano varieties. Toric Fano varieties which are smooth in codimension 2 and Q \textbf {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 \textbf {Q} -factorial in codimension 3. The Cox ring deforms in a flat family under deformation of a terminal Fano which is Q \textbf {Q} -factorial in codimension 3. A side result which seems to be new is that the divisor class group of a klt Fano variety maps isomorphically to ordinary homology.