Stable rationality of index one Fano hypersurfaces containing a linear space

TL;DR

This paper proves that most high-degree Fano hypersurfaces in projective space containing a linear space are not stably rational, and also examines cases with multiple double points, advancing understanding of rationality properties.

Contribution

It establishes stable irrationality for a broad class of hypersurfaces with linear spaces and singularities, extending previous rationality results in algebraic geometry.

Findings

Very general hypersurfaces containing an r-plane are not stably rational.

Hypersurfaces with multiple isolated double points also fail stable rationality.

Results apply to hypersurfaces of degree n+1 in projective space for n ≥ 3.

Abstract

We prove that a very general complex hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ containing an $r$-plane with multiplicity $m$ is not stably rational for $n \ge 3$, $m, r > 0$ and $n \ge m+r$. We also investigate failure of stable rationality of a very general hypersurface of degree $n+1$ in $\mathbb{P}^{n+1}$ admitting several isolated ordinary double points.