On the rationality problem for low degree hypersurfaces

TL;DR

This paper proves that very general low-degree hypersurfaces of certain dimensions are not stably rational or retract rational, using decomposition of the diagonal, thereby advancing understanding of their rationality properties.

Contribution

It establishes new bounds on the degree and dimension for which hypersurfaces are not rational, improving previous results in the field.

Findings

Hypersurfaces of degree at least 4 and certain dimensions are not stably rational.

The results extend to characteristic 2 with weaker degree bounds.

The work refines criteria for non-rationality of algebraic varieties.

Abstract

We show that a very general hypersurface of degree d at least 4 and dimension at most $(d+1)2^{d-4}$ over a field of characteristic different from 2 does not admit a decomposition of the diagonal; hence, it is neither stably nor retract rational, nor $\mathbb{A}^1$-connected. Similar results hold in characteristic 2 under a slightly weaker degree bound. This improves earlier results by the second named author and Moe.