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Aug 9, 2016 at 11:39 vote accept Daniel Loughran
Aug 9, 2016 at 9:34 comment added Sasha Conic bundles on $S$ come from degenerate quadrics (by projecting from their vertices) in the pencil of quadrics passing through $S$ in its anticanonical embedding. Under the above condition only one such quadric is defined over $k$, hence only two conic bundles are defined over $k$, and for these two the bases are just the bases of conic bundles on $S_0$. And yes, $S_0$ is $C \times C$.
Aug 9, 2016 at 7:34 comment added Daniel Loughran Where do the conic bundles on $S$ come from? Are you taking something like $S_0 = C \times C$?
Aug 8, 2016 at 22:05 history answered Sasha CC BY-SA 3.0