A Stably Irrational (2,3)-Complete Intersection Fourfold over $\mathbb{Q}$

TL;DR

This paper constructs an explicit example of a smooth four-dimensional complete intersection over the rationals that is stably irrational, advancing understanding of rationality properties in algebraic geometry.

Contribution

It provides the first explicit example of a stably irrational (2,3)-complete intersection fourfold over , , using specialization techniques and decomposition of the diagonal.

Findings

Explicit example of a stably irrational fourfold over .

Application of specialization and decomposition of the diagonal techniques.

Demonstration that very general such intersections are stably irrational.

Abstract

We apply the specialization technique based on the decomposition of the diagonal to find an explicit example over $\mathbb{Q}$ of a quadric and cubic hypersurface in $\mathbb{P}^6$ such that their intersection is a smooth stably irrational fourfold. Using the same degeneration, Nicaise and Ottem have already proven that the the very general complete intersection of this type is stably irrational using the motivic volume.