Some low-dimensional hypersurfaces that are not stably rational

TL;DR

This paper demonstrates that very general hypersurfaces of degree at least 4 in complex projective spaces of dimensions 6 to 9 are not stably rational, highlighting new examples of low-dimensional hypersurfaces with this property.

Contribution

It applies Voisin's method to establish non-stable rationality of certain high-dimensional hypersurfaces and their double covers, expanding the class of known non-stably rational varieties.

Findings

Hypersurfaces of degree ≥4 in dimensions 6-9 are not stably rational.

Double covers branched along quartic hypersurfaces are not stably rational.

Such hypersurfaces are known to be unirational.

Abstract

Using Voisin's method we prove that a very general hypersurface of degree at least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably rational and so, in particular, not rational. We obtain the same conclusion for the double covering of projective space of dimension 6, 7, 8 or 9, branched along a very general quartic hypersurface. On the other hand, such double coverings as well as general quartic hypersurfaces of dimension at least 5 are known to be unirational.