On Stable Rationality of Polytopes

TL;DR

This paper investigates the stable rationality of polytopes and hypersurfaces in toric varieties, providing new bounds and a combinatorial approach to determine non-stable rationality in algebraic geometry.

Contribution

It establishes that non-stable rationality is preserved under inclusions for a broad class of polytopes and introduces a combinatorial method to study stable rationality of hypersurfaces.

Findings

Proves preservation of non-stable rationality under inclusions for certain polytopes.

Provides improved bounds for non-stably rational hypersurfaces in projective space.

Extends bounds to double covers and new classes of varieties in products of projective space.

Abstract

Nicaise--Ottem introduced the notion of (stably) rational polytopes and studied this using a combinatorial description of the motivic volume. In this framework, we ask whether being non-stably rational is preserved under inclusions. We prove this holds for a large class of polytopes, leading to a combinatorial strategy for studying stable rationality of hypersurfaces in toric varieties. As a result, we obtain new bounds for non-stably rational hypersurface in projective space, improving the ones given by Schreieder when the field has characteristic 0. We also obtain similar bounds for double covers of projective space and some new classes of non-stably rational varieties in products of projective space.