模拟环面簇超曲面的稳定有理性 II
Stable rationality of hypersurfaces of mock toric variety II
AI总结：
本文用模拟环面簇替代环面簇构造退化族，借助母题方法证明了射影空间中非常一般超曲面的稳定非有理性可推出格拉斯曼簇中相应超曲面的稳定非有理性。
摘要：
近年来，通过母题方法研究有理性问题取得了进展（参见 [Kontsevich--Tschinkel'19]、[Nicaise--Shinder'19]、[Nicaise--Ottem'21]）。该方法需要显式构造具有良好性质的曲线退化族。虽然具体构造通常较为困难，[Nicaise--Ottem'22] 结合组合方法构造了环面簇中超曲面的退化族，并提及了射影空间中非常一般超曲面的稳定有理性。本文用模拟环面簇（mock toric varieties）替代环面簇，并利用母题方法证明如下定理：若 $\mathbb{P}^{2n-5}_\mathbb{C}$ 中次数为 $d$ 的非常一般超曲面不是稳定有理的，则 $\mathrm{Gr}_\mathbb{C}(2, n)$ 中次数为 $d$ 的非常一般超曲面也不是稳定有理的。
英文摘要：
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.
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