On the diagonal of low bidegree hypersurfaces

TL;DR

This paper investigates the decomposition of the diagonal for bidegree hypersurfaces in product spaces, developing an inductive cycle-theoretic method to analyze rationality properties across degrees and dimensions.

Contribution

It introduces a new inductive technique to study diagonal decompositions, extending known results to higher degrees and dimensions, and identifies specific hypersurfaces that are not retract rational.

Findings

A general (3,2) complete intersection in P^4 x P^3 does not admit a decomposition of the diagonal.

Bidegree hypersurfaces in certain ranges are not retract rational over fields of characteristic not two.

The method extends previous stable irrationality results to broader classes of hypersurfaces.

Abstract

We study the existence of a decomposition of the diagonal for bidegree hypersurfaces in a product of projective spaces. Using a cycle theoretic degeneration technique due to Lange, Pavic and Schreieder, we develop an inductive procedure that allows one to raise the degree and dimension starting from the quadric surface bundle of Hassett, Pirutka and Tschinkel. Furthermore, we are able to raise the dimension without raising the degree in a special case, showing that a very general $(3,2)$ complete intersection in $\mathbb P^4\times \mathbb P^3$ does not admit a decomposition of the diagonal. As a corollary of these theorems, we show that in a certain range, bidegree hypersurfaces which were previously only known to be stably irrational over fields of characteristic zero by results of Moe, Nicaise and Ottem, are not retract rational over fields of characteristic different from two.