Stable rationality of hypersurfaces in sch\"{o}n affine varieties

TL;DR

This paper extends motivic methods for proving non-stable rationality of hypersurfaces from toric varieties to schön affine varieties, including applications to hypersurfaces in complex Grassmannians.

Contribution

It generalizes previous results on hypersurfaces in toric varieties to schön affine varieties, demonstrating non-stable rationality using combinatorial and motivic techniques.

Findings

Proves non-stable rationality of hypersurfaces in schön affine varieties.

Shows irrationality of certain hypersurfaces in Grassmannian varieties.

Extends combinatorial degeneration methods beyond toric cases.

Abstract

In recent years, there has been a development in approaching rationality problems through the motivic methods (cf. [Kontsevich--Tschinkel'19], [Nicaise--Shinder'19], [Nicaise--Ottem'21]). This method requires the explicit construction of degeneration families of curves with favorable properties. While the specific construction is generally difficult, [Nicaise--Ottem'22] combines combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and shows the non-stable rationality of a very general hypersurface in projective spaces. In this paper, we extend the result of [Nicaise--Ottem'22] not only for hypersurfaces in algebraic tori but also to those in sch\"{o}n affine varieties. In application, we show the irrationality of certain hypersurfaces in the complex Grassmannian variety Gr(2, n) using the motivic method, which coincides with the result obtained by the same author in the previous research.