Rationality problem of generalized Ch\^atelet surfaces

TL;DR

This paper provides an accessible formulation for the rationality problem of generalized Châtelet surfaces, giving necessary and sufficient conditions based on the parameters a and P(x) under certain field conditions.

Contribution

It offers a clear, algebraic criterion for the k-rationality of these surfaces, simplifying previous geometric conditions.

Findings

k-rationality depends on whether a is a square in k

Necessary and sufficient conditions involve properties of P(x)

Results assume characteristic of k is not 2 and P(x) is separable

Abstract

The surface $z^2=ay^2+P(x), \, a \in k, \, P(x) \in k[x]$ is not $k$-rational, if $a \not\in k^2$ and $P(x)$ satisfies some conditions. This result essentially due to Iskovskih but his statement is in terms of algebraic geometry, and not so easy to access for the researchers of the field extension. This paper aims to give a formulation accessible more easily. A necessary and sufficient condition for $k$-rationality is given in terms of $a$ and $P$, assuming that $\ch k \not= 2$ and every irreducible component of $P(x)$ is separable over $k$.