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Stable rationality of hypersurfaces in schön affine varieties

Taro Yoshino

TL;DR

The paper extends the motivic method for rationality from hypersurfaces in tori to schön affine varieties by constructing strictly toroidal models via tropical compactifications and computing stable birational volumes. It develops valuations, tropical fans, and linear-system machinery on schön varieties to control degenerations and strata, enabling a Grassmannian Gr$(2,n)$ application. Under known non-stable-rationality results in projective space, it deduces non-stable rationality for very general hypersurfaces in Gr$(2,n)$ for suitable degrees, yielding explicit degree bounds and broadening the scope of motivic techniques in rationality problems. The work connects to mock toric varieties and previous Y24a/Y24b developments, providing a unified framework for irrationality results across a wider class of rational ambient spaces.

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 schön 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.

Stable rationality of hypersurfaces in schön affine varieties

TL;DR

The paper extends the motivic method for rationality from hypersurfaces in tori to schön affine varieties by constructing strictly toroidal models via tropical compactifications and computing stable birational volumes. It develops valuations, tropical fans, and linear-system machinery on schön varieties to control degenerations and strata, enabling a Grassmannian Gr application. Under known non-stable-rationality results in projective space, it deduces non-stable rationality for very general hypersurfaces in Gr for suitable degrees, yielding explicit degree bounds and broadening the scope of motivic techniques in rationality problems. The work connects to mock toric varieties and previous Y24a/Y24b developments, providing a unified framework for irrationality results across a wider class of rational ambient spaces.

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 schön 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.

Paper Structure

This paper contains 25 sections, 45 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The motivic method for the rationality problem
  3. The difficulty of the motivic method
  4. The significance of this paper
  5. The rationality problem for hypersurfaces in Grassmannian varieties
  6. Outline of the paper
  7. Relationship with mock toric varieties
  8. Acknowledgment
  9. Notation
  10. Schön affine varieties
  11. Definition of tropical compactification
  12. Properties of tropical compactification
  13. Example of schön affine varieties
  14. Stable birational volume of a schön variety
  15. The definition of some properties of fans
  16. ...and 10 more sections

Key Result

Proposition 1.1

S19 Let $k$ be an uncountable field of $\operatorname{char}(k)\neq 2$, and $n \geq 3$ and $d \geq 2 + \log_2(n)$ be integers. Then a very general hypersurface $H_d\subset\mathbb P^{n+1}_k$ of degree $d$ is not stably rational, i.e., $H_d\times\mathbb P^m_k$ is not rational for any $m\in \mathbb Z_{\

Theorems & Definitions (91)

  • Proposition 1.1
  • Proposition 1.2
  • Theorem 1.3: See Theorem \ref{['thm: d in Grassmannian']}
  • Corollary 1.4: See Theorem \ref{['cor: log bound']}
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 81 more