In Mori program in dimension $$3$$ there is a class of Mori contractions $$\phi: X\to C$$ called quadric bundles, where $$X$$ is a three-dimensional manifold and $$C$$ is a curve. As far as I understand, such contractions have the property that the morphism $$\phi$$ is flat and $$\phi^{-1}(C)$$ is a smooth quadric surface for a generic point $$x\in C$$ . Let's assume that we work over $$\mathbb C$$.

Questions. 1) What is the definition of quadric bundles?

2) Suppose that $$X$$ and $$C$$ are smooth and complex. Is it true that there is a projective $$\mathbb CP^3$$ bundle over $$C$$ into which $$X$$ naturally embeds - so that in each $$\mathbb CP^3$$ a quadric surface sits?

3) If there a classification of quadric bundles over $$\mathbb CP^1$$ with at most two singular fibres?

4) Is there some local model for neighborhoods of singular fibres?