The diagonal of quartic fivefolds

TL;DR

This paper proves that very general quartic fivefolds in projective 6-space are not retract rational by introducing a new cycle-theoretic obstruction, extending previous results on stable irrationality to positive characteristic fields.

Contribution

It introduces a novel cycle-theoretic obstruction for non-rationality applicable in positive characteristic, generalizing prior stable irrationality results.

Findings

Quartic fivefolds are not retract rational over certain fields.

A new cycle-theoretic obstruction is developed.

Extension of irrationality results to positive characteristic fields.

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.