A two-dimensional rationality problem and intersections of two quadrics
A two-dimensional rationality problem and intersections of two quadrics
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二维有理性问题和两个二次曲线的交集
DOI：
10.1007/s00229-021-01313-7
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发表时间：
2021
影响因子：
0.6
通讯作者：
Aiichi Yamasaki
中科院分区：
文献类型：
--
作者：
Akinari Hoshi;Ming-Chang Kang;Hidetaka Kitayama;Aiichi Yamasaki
Letkbe a field with charandkbe not algebraically closed. Letandbe a field extension ofkwherex,yare algebraically independent overk. Assume thatis ak-automorphism onLdefined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma : \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c\big (x+\frac{b}{x}\big )+d}{y} \end{aligned}$$\end{document}where,and at least one ofc,dis non-zero. Letbe the fixed subfield ofL. We show thatis isomorphic to the function field of a certain surface inwhich is given as the intersection of two quadrics. We give criteria for thek-rationality ofby using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thélène.
Letkbe a field with charandkbe not algebraically closed. Letandbe a field extension ofkwherex,yare algebraically independent overk. Assume thatis ak-automorphism onLdefined by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sigma : \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c\big (x+\frac{b}{x}\big )+d}{y} \end{aligned}$$\end{document}where,and at least one ofc,dis non-zero. Letbe the fixed subfield ofL. We show thatis isomorphic to the function field of a certain surface inwhich is given as the intersection of two quadrics. We give criteria for thek-rationality ofby using the Hilbert symbol. As an appendix of the paper, we also give an alternative geometric proof of a part of the result which is provided to the authors by J.-L. Colliot-Thélène.