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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.

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  • $\begingroup$ Do you know any examples for which the discriminant is not a divisor? $\endgroup$ – Daniel Loughran Jan 6 '15 at 10:47
  • $\begingroup$ What precisely are you assuming? Does the morphism $\pi$ have relative Picard number $1$? For the relative dualizing sheaf $\omega_{\pi}$, is $\omega_{\pi}^\vee$ $\pi$-ample? $\endgroup$ – Jason Starr Jan 6 '15 at 14:38
  • $\begingroup$ @DanielLoughran I don't know examples with $\Delta$ not a divisor, that why I was wondering if it is the common case. Do you know such examples ? Jason: Yes, I'm sorry I didn't say it, $\pi$ has relative Picard number 1 (it isn't always the case when you contract a extremal ray which is a Fano-Mori fibration? not a face in 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
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    $\begingroup$ I am still confused about "conic bundle". Are you assuming that $\pi$ is flat? Not every fiber-type extremal contraction is flat. If there is no flatness hypothesis, certainly $\Delta$ may not be a divisor. For instance, Proj of the symmetric algebra of a reflexive, non-locally-free sheaf of rank $2$ will have "discriminant locus" equal to a subset of codimension at least $2$. $\endgroup$ – Jason Starr Jan 6 '15 at 18:23
  • $\begingroup$ @JasonStarr Thank you very much, Jason! In fact flatness it wasn't an assumption, so the answer should be "no", by your example. $\endgroup$ – Pedro Montero Jan 7 '15 at 10:00
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The OP clarified that there should be no flatness hypothesis in this question. Without any flatness hypothesis, certainly there are examples where $\Delta$ is not a divisor. (There may be flat examples as well.) First of all, for a coherent sheaf $\mathcal{F}$ on $Y$ that is reflexive of rank $2$, yet not locally free, then for the relative Proj of the symmetric algebra of $\mathcal{F}$ over $Y$, $\Delta$ equals the non-locally free locus of $\mathcal{F}$, which has codimension $2$.

For instance, let $Y$ be $\mathbb{P}^3$ with one point specified $q$. Projection away from $q$ defines a morphism $\mathbb{P}^3\setminus\{q\} \to \mathbb{P}^2$. The pullback of the tangent sheaf of $\mathbb{P}^2$ extends (uniquely) to a coherent sheaf $\mathcal{F}$ that is reflexive of generic rank $2$, yet whose rank at $q$ equals $3$. Then $X=\text{Proj}_Y \text{Sym}^\bullet_{\mathcal{O}_Y} \mathcal{F}$ is a normal, local complete intersection scheme that is factorial. The projection $\pi$ is projective and surjective, the generic fiber is isomorphic to $\mathbb{P}^1$, yet the fiber over $\{q\}$ is isomorphic to $\mathbb{P}^2$. Thus $\Delta$ equals $\{q\}$, which has codimension $3$ in $Y$.

There are also examples where $\pi$ is equidimensional. For each integer $n>0$ and for each integer $e>0$, denote by $\overline{M}_{0,0}(\mathbb{P}^n,e[\text{line}])$, the coarse moduli space of genus $0$ (unpointed) stable maps to $\mathbb{P}^n$ with curve class $e[\text{line}]$. Denote by $Y$ the dense open subset that is the complement of the boundary divisor. For good measure, also remove the closed subset where the stable map has any nontrivial automorphisms, i.e., remove the "stacky locus". Denote by $$\rho:C\to Y$$ the (restriction to $Y$ of the) universal curve, and denote by $$h:C\to \mathbb{P}^n$$ the (restriction to $Y$ of the) universal stable map. Together these define a projective, finite morphism $$(\rho,h): C \to Y \times \mathbb{P}^n.$$ Define $X$ to be the closed image. Define $\pi$ to be the natural projection. If $n>2$ and $e>2$, then $\Delta$ is a nonempty subset of codimension $>1$.

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