Rationality problem of conic bundles

TL;DR

This paper investigates the rationality of a class of affine surfaces defined by quadratic equations over fields with characteristic not 2, providing algebraic criteria based on polynomial properties and extending geometric results.

Contribution

It offers an algebraic characterization of the rationality problem for conic bundle surfaces, complementing Iskovskikh's geometric approach with explicit polynomial-based conditions.

Findings

Rationality depends on the sum s of specific polynomial-related counts being at most 3.

Provides necessary and sufficient algebraic conditions for the rationality of the surface.

Extends geometric results to an algebraic framework for conic bundle rationality.

Abstract

Let $k$ be a field with char $k \not= 2$, $X$ be an affine surface defined by the equation $z^2=P(x)y^2+Q(x)$ where $P(x), Q(x) \in k[x]$ are separable polynomials. We will investigate the rationality problem of $X$ in terms of the polynomials $P(x)$ and $Q(x)$. The necessary and sufficient condition is $s \leq 3$ with minor exceptions, where $s=s_1+s_2+s_3+s_4$, $s_1$ (resp. $s_2$, resp. $s_3$) being the number of $c \in \overline{k}$ such that $P(c)=0$ and $Q(c) \not\in k(c)^2$ (resp. $Q(c)=0$ and $P(c) \not\in k(c)^2$, resp. $P(c)=Q(c)=0$ and $-\frac{Q}{P}(c) \not\in k(c)^2$). $s_4=0$ or $1$ according to the behavior at $x=\infty$. $X$ is a conic bundle over $\mathbb{P}_k^1$, whose rationality was studied by Iskovskikh. Iskovskikh formulated his results in geometric language. This paper aims to give an algebraic counterpart.