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I am desperately trying to understand what is a conic bundle. It seems like this is a completely standard term in algebraic geometry, there is even a page on wiki about it, but this doesn't really help. For example, I have the following test question.

Test question. Let $S$ be a smooth complex ruled surface $S\to C$ over a curve $C$. Suppose we blow up $S$ twice in the same fibre, so that the preimage of one point in $C$ is a union of three lines. Is this a conic bundle or not?

My problem is the following. According to some definitions that I saw, in a conic bundle the preimage of a point should be a conic. And a union of three lines is not a conic. However, I tried to trace back the definition of conic bundles, and one of the earliest versions that I found is an article of Sarkisov 1980 (in Russian): http://www.mathnet.ru/links/b10a1373601dacdd9b7debba2b3e1c8f/im1862.pdf

Sarkisov is just asking that the preimage of a generic point be a rational curve.

I fear that my main question (what is a conic bundle) is a complicated one, for example judging by the fact that the answer to the following question was not given by the mathoverflow community: References about conic bundles

Question 2. Why Sarkisov calls his bunldes conic bundles? Is there some relatively pedagogical place where on can read about this?

(I fear that I can't understand this because I misinterpret the expression "generic fiber")

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I think the reasonable definition is to ask for a flat morphism $f$ whose generic fiber is a rational curve. Then you may put more conditions according to your needs (for instance, there is a more strict notion of standard conic bundle). But the answer to Question 2 is, I think, quite simple. A rational curve $C$ over a field $k$ is not necessarily isomorphic to $\mathbb{P}^1_{k}$, because there will usually exist no line bundle on $C$ of degree $1$. However, there is always a line bundle of degree $2$, namely the tangent bundle $T_C$. The global sections of $T_C$ define an embedding $C\hookrightarrow \mathbb{P}^2_{k}$, whose image is a conic. Thus the generic fiber (and the general fibers as well) of $f$ are conics, hence the name.

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  • $\begingroup$ Thanks a lot! I would like to clarify two points. Do I understand correctly that if one only requires the morphism to be flat, then we can blow up as many points in the ruled surface as we want? But if we consider standard conic bundles we can only blow up one point in a ruled surface (I deduce this from maths.ed.ac.uk/cheltsov/pdf/ivan17e.pdf). Next is it easy to see that (generic fiber is a conic) implies (general fibres are conics)? And the last question is whether you are aware of some place where this is explained in a relatively clear way? $\endgroup$ – aglearner May 19 at 10:03
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    $\begingroup$ @aglearner: You can try to look into this survey: Prokhorov, Yu. G. The rationality problem for conic bundles. (Russian) Uspekhi Mat. Nauk 73 (2018), no. 3(441), 3--88; ; translation in Russian Math. Surveys 73 (2018), no. 3, 375–456. $\endgroup$ – Sasha May 19 at 11:42
  • $\begingroup$ O, this looks as what I was looking for! It should answer many of my questions. I like in particular Theorem 3.2. Thanks a lot! $\endgroup$ – aglearner May 19 at 12:05

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