A two-dimensional rationality problem and intersections of two quadrics

Abstract

Let $k$ be a field with char $k\ne 2$ 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 \[ σ: \sqrt{a}\mapsto -\sqrt{a},\ x\mapsto \frac{b}{x},\ y\mapsto \frac{c(x+\frac{b}{x})+d}{y} \] where $b,c,d\in k$, $b\ne 0$ and at least one of $c,d$ is non-zero. Let $L^{⟨\sigma ⟩}=\{u\in L:\sigma (u)=u\}$ be the fixed subfield of $L$. We show that $L^{⟨\sigma ⟩}$ is isomorphic to the function field of a certain surface in $P_k^4$ which is given as the intersection of two quadrics. We give criteria for the $k$-rationality of $L^{⟨\sigma ⟩}$ 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-Thélène.

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