The diagonal of quartic fivefolds

Abstract

We show that a very general quartic hypersurface in ${\mathbb{P}}^6$ over a field of characteristic different from 2 does not admit a decomposition of the diagonal, hence is not retract rational. This generalizes a result of Nicaise--Ottem, who showed stable irrationality over fields of characteristic 0. To prove our result, we introduce a new cycle-theoretic obstruction that may be seen as an analogue of the motivic obstruction for rationality in characteristic zero, introduced by Nicaise--Shinder and Kontsevich--Tschinkel.

arXiv comment27 pages; mistake in Section 4 fixed; final version; to appear in Algebraic Geometry