Generally rational polynomials in two variables

Abstract

Let $\mathbf{k}$ be an algebraically closed field. A polynomial $F \in \mathbf{k}[X, Y]$ is said to be generally rational if, for almost all $\lambda \in \mathbf{k}$, the curve ``$F=\lambda$'' is rational. It is well known that, if $\mathop{\mathrm{char}}\mathbf{k}=0$, $F$ is generally rational iff there exists $G \in \mathbf{k}(X, Y)$ such that $\mathbf{k}(F, G)=\mathbf{k}(X, Y)$. We give analogous results valid in arbitrary characteristic.