Let $\pi: S\to C$ be a minimal conic bundle over a field $k$ of characteristic zero. That is, $S$ is a geometrically irreducible smooth surface with Picard rank $2$ and $C$ a geometrically irreducible smooth curve over $k$ and each fiber of $\pi$ is isomorphic to a plane conic. Is there an explicit description of the nef and ample cone of $S$? If not in general, maybe when $C=\mathbb{P}^1_k$?

Positivity in algebraic geometry I, §1.5A. I'm not sure if that's exactly what you're looking for. $\endgroup$ – R. van Dobben de Bruyn May 20 at 1:01