A (2,3)-complete intersection fourfold with no decomposition of the diagonal

Abstract

We apply the specialization technique based on the decomposition of the diagonal to the intersection of a quadric and cubic hypersurface in P6. We find an explicit example defined over Q that is smooth, and does not admit a decomposition of the diagonal, and is therefore not retract rational. The proof uses the specialization of Nicaise and Ottem (Tropical degenerations and stable rationality, 2020), who proved that the very general complete intersection of this type is stably irrational using the motivic volume.