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")
Best Answer
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.