A two-dimensional rationality problem and intersections of two quadrics

TL;DR

This paper investigates the fixed fields of certain automorphisms of function fields over non-closed fields, showing they correspond to intersections of two quadrics in projective space and providing criteria for their rationality.

Contribution

It establishes a geometric interpretation of fixed fields as intersections of quadrics and offers rationality criteria using the Hilbert symbol, extending understanding of rationality problems.

Findings

Fixed fields are isomorphic to intersections of two quadrics in P^4_k.

Provides criteria for k-rationality using the Hilbert symbol.

Includes an alternative geometric proof of key results.

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.