Stable rationality of hypersurfaces of mock toric variety II

TL;DR

This paper extends motivic methods to study the stable rationality of hypersurfaces in mock toric varieties, establishing a link between hypersurfaces in projective spaces and Grassmannians.

Contribution

It proves a new theorem connecting the stable rationality of hypersurfaces in projective spaces to those in Grassmannians using mock toric varieties.

Findings

If a very general hypersurface in projective space is not stably rational, then the corresponding hypersurface in Grassmannian is also not stably rational.

The paper demonstrates the application of motivic methods to mock toric varieties.

It advances understanding of rationality problems in algebraic geometry.

Abstract

In recent years, there has been a development in approaching rationality problems through 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 mentions the stable rationality of a very general hypersurface in projective spaces. In this paper, we substitute mock toric varieties for toric varieties and we prove the following theorem from the motivic method: If a very general hypersurface of degree $d$ in $\mathbb{P}^{2n-5}_\mathbb{C}$ is not stably rational, then a very general hypersurface of degree $d$ in $\mathrm{Gr}_\mathbb{C}(2, n)$ is not stably rational.