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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Hypersurfaces that are not stably rational
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by Burt Totaro PDF
J. Amer. Math. Soc. 29 (2016), 883-891 Request permission

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 $\textbf {P}^{n+1}$ 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.
References
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Additional Information
  • Burt Totaro
  • Affiliation: Mathematics Department, UCLA, Box 951555, Los Angeles, California 90095-1555
  • MR Author ID: 272212
  • Email: totaro@math.ucla.edu
  • 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
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 883-891
  • MSC (2010): Primary 14E08; Secondary 14J45, 14J70
  • DOI: https://doi.org/10.1090/jams/840
  • MathSciNet review: 3486175