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Hypersurfaces that are not stably rational


Author: Burt Totaro
Journal: J. Amer. Math. Soc. 29 (2016), 883-891
MSC (2010): Primary 14E08; Secondary 14J45, 14J70
DOI: https://doi.org/10.1090/jams/840
Published electronically: July 13, 2015
MathSciNet review: 3486175
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Abstract: We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all $ d\geq 2\lceil (n+2)/3\rceil $ and $ n\geq 3$, a very general complex hypersurface of degree $ d$ in $ {\bf P}^{n+1}$ is not stably rational. The statement generalizes Colliot-Thélène and Pirutka's theorem that very general quartic 3-folds are not stably rational. The result covers all the degrees in which Kollár proved that a very general hypersurface is non-rational, and a bit more. For example, very general quartic 4-folds are not stably rational, whereas it was not even known whether these varieties are rational.


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Additional Information

Burt Totaro
Affiliation: Mathematics Department, UCLA, Box 951555, Los Angeles, California 90095-1555
Email: totaro@math.ucla.edu

DOI: https://doi.org/10.1090/jams/840
Received by editor(s): February 12, 2015
Received by editor(s) in revised form: February 26, 2015, and May 27, 2015
Published electronically: July 13, 2015
Article copyright: © Copyright 2015 American Mathematical Society