Hypersurfaces that are not stably rational

Abstract

We show that a wide class of hypersurfaces in all dimensions are not stably rational. Namely, for all d ≥ 2 ⌈ ( n + 2 ) / 3 ⌉ d\geq 2\lceil (n+2)/3\rceil and n ≥ 3 n\geq 3 , a very general complex hypersurface of degree d d in P n + 1 \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.