In the context of the Minimal Model Program it can arise that we need to deal with contractions of extremal rays that are conic bundles $\pi:X\to Y$ with relative Picard number 1 (with possibly degenerate fibers). So we can consider the discriminant of $\pi$:

$$\Delta=\{y\in Y\;|\;\pi^{-1}(y)\;\text{is not a smooth conic} \}. $$

As far as I read this discriminant is only used in the case that $X$ and $Y$ are smooth projective varieties, in which case it is proved that $\Delta$ is an effective divisor (A. Beauville, "Variété de Prym et Jacobiennes Intermédiaires", for instance).

So I think that it is a natural to ask, **what can we say about $\Delta$ in the singular case ?**

By singular I mean when $X$ and $Y$ are normal projective varieties (let's say $\mathbb{Q}$-factorial) with the singularities arising from the MMP. Maybe we can say something in the case of terminal singularities by using that the singular locus is of codimension at least 3?

I found some articles dealing with this kind of conic bundles:

- S. Cutkosky, "Elementary contractions of Gorenstein threefolds".
- S. Mori and Y. Prokhorov, "On $\mathbb{Q}$-conic bundles".
- G. Della Noce, "On the Picard number of singular Fano varities".

But it seems to me that they don't use the discriminant in their analysis (at least not explicitely).

I was reading the proof of Beauville (page 321 in his paper) to the fact that $\Delta$ is a divisor, but he uses the fact that $X$ is smooth and the flatness of $\pi$.

Thanks a lot in advance for you help and comments.

raywhich is a Fano-Mori fibration? not afacein general I mean) and the generic fiber is smooth and isomorphic to $\mathbb{P}^1$ but it may happen that there are double lines or two lines crossing as fibers. I edited it. Thanks for your comments! $\endgroup$ – Pedro Montero Jan 6 '15 at 15:12