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 $C=\mathbb{P}^1_k$?
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1$\begingroup$ 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. $\endgroup$– R. van Dobben de BruynCommented May 20, 2020 at 1:01
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$\begingroup$ 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 $\mathbb{P}^1$. $\endgroup$– HansCommented May 20, 2020 at 4:39
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1$\begingroup$ Note that if the Picard rank of $S$ is 2, then $\pi$ cannot have any reducible fibres. $\endgroup$– PopCommented May 20, 2020 at 10:48
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1$\begingroup$ 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). $\endgroup$– HansCommented May 20, 2020 at 13:23
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