# What is a conic bundle and why is it called so?

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

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.