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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.

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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

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  • $\begingroup$ Thank you! I should definitely read this section. $\endgroup$ – aglearner May 25 '19 at 11:06

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