# Nef and effective cone of minimal conic bundle

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 $$C=\mathbb{P}^1_k$$?

• There is some discussion of nef and effective cones on ruled surfaces in Lazarsfeld's Positivity in algebraic geometry I, §1.5A. I'm not sure if that's exactly what you're looking for. – R. van Dobben de Bruyn May 20 at 1:01
• Thanks, but I this is not exactly what I need. Note that $\pi$ can have some reducible fibers whereas in Lazarsfeld all fibers are a $\mathbb{P}^1$. – Hans May 20 at 4:39
• Note that if the Picard rank of $S$ is 2, then $\pi$ cannot have any reducible fibres. – Pop May 20 at 10:48
• You are right. What I meant is that there can be fibers that are not geometrically irreducible (I am not working over an algebraically closed field). – Hans May 20 at 13:23