1. A two-dimensional rationality problem and intersections of two quadrics
- Author
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Aiichi Yamasaki, Ming-chang Kang, Hidetaka Kitayama, and Akinari Hoshi
- Subjects
General Mathematics ,High Energy Physics::Phenomenology ,010102 general mathematics ,Zero (complex analysis) ,Field (mathematics) ,Algebraic geometry ,01 natural sciences ,Hilbert symbol ,Combinatorics ,Mathematics - Algebraic Geometry ,Number theory ,Field extension ,12F20, 13A50, 14E08 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Algebraic Geometry (math.AG) ,Function field ,Mathematics - Abstract
Let $k$ be a field with char $k\neq 2$ and $k$ be not algebraically closed. Let $a\in k\setminus k^2$ and $L=k(\sqrt{a})(x,y)$ be a field extension of $k$ where $x,y$ are algebraically independent over $k$. Assume that $\sigma$ is a $k$-automorphism on $L$ defined by \[ \sigma: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d \in k$, $b\neq 0$ and at least one of $c,d$ is non-zero. Let $L^{\langle\sigma\rangle}=\{u\in L:\sigma(u)=u\}$ be the fixed subfield of $L$. We show that $L^{\langle\sigma\rangle}$ is isomorphic to the function field of a certain surface in $P^4_k$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{\langle\sigma\rangle}$ by 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-Th\'el\`ene., Comment: To appear in Manuscripta Math. The main theorems (old Theorem 1.7 and Theorem 1.8) incorporated into (new) Theorem 1.8. Section 3 and Section 4 interchanged
- Published
- 2021