Jumping of the nef cone for Fano varieties

Totaro, Burt

doi:10.1090/S1056-3911-2011-00557-2

Author: Burt Totaro
Journal: J. Algebraic Geom. 21 (2012), 375-396
DOI: https://doi.org/10.1090/S1056-3911-2011-00557-2
Published electronically: June 29, 2011
MathSciNet review: 2877439
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Abstract | References | Additional Information

Abstract: 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.

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Additional Information

Burt Totaro
Affiliation: DPMMS, Wilberforce Road, Cambridge CB3 0WB, England
Email: b.totaro@dpmms.cam.ac.uk

DOI: https://doi.org/10.1090/S1056-3911-2011-00557-2
Received by editor(s): July 16, 2009
Received by editor(s) in revised form: January 8, 2010
Published electronically: June 29, 2011