## Hypersurfaces that are not stably rational

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- by Burt Totaro
- J. Amer. Math. Soc.
**29**(2016), 883-891 - DOI: https://doi.org/10.1090/jams/840
- Published electronically: July 13, 2015
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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 $\textbf {P}^{n+1}$ 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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## Bibliographic 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