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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Jumping of the nef cone for Fano varieties

Author: Burt Totaro
Journal: J. Algebraic Geom. 21 (2012), 375-396
Published electronically: June 29, 2011
MathSciNet review: 2877439
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Abstract | Additional Information


We construct $\textbf {Q}$-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 $\textbf {Q}$-factorial in codimension 3 are rigid. The divisor class group is deformation-invariant for klt Fanos which are smooth in codimension 2 and $\textbf {Q}$-factorial in codimension 3. The Cox ring deforms in a flat family under deformation of a terminal Fano which is $\textbf {Q}$-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.

Additional Information

Burt Totaro
Affiliation: DPMMS, Wilberforce Road, Cambridge CB3 0WB, England
MR Author ID: 272212

Received by editor(s): July 16, 2009
Received by editor(s) in revised form: January 8, 2010
Published electronically: June 29, 2011