Hypersurfaces that are not stably rational

Totaro, Burt

doi:10.1090/jams/840

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.

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