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? 


5) Is there some relatively pedagogical place where I can read about this staff?


I would be grateful for answers to any of these questions. 



What are quadric bundles?
have you consulted chapter one of: 


Variétés de Prym et jacobiennes intermédiaires
Beauville, Arnaud
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 10 (1977) no. 3, p. 309-391


