Rationality problem of conic bundles

Abstract

Let $k$ be a field with char $k\ne 2$, $X$ 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 $s\le 3$ 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 $Q(c)\notin k(c{)}^2$ (resp. $Q(c)=0$ and $P(c)\notin k(c{)}^2$, resp. $P(c)=Q(c)=0$ and $-\frac{Q}{P}(c)\notin k(c{)}^2$). $s_4=0$ 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.

arXiv commentincorporates all of the content of arXiv:1308.0909