An arithmetic analogue of the discriminant curve of a conic bundle threefold

2012-04-12T11:59:31Z

I am looking for an "arithmetic" analogue of a well known result on threefolds with a conic bundle structure. The following result can be found in [Iskovskikh - On the rationality problem for conic bundles, Lemma 1]. Note that Iskovskikh has some extra condition of relative minimality which I am pretty sure I don't need for the result I want.

Let $X$ be a smooth irreducible threefold over $\mathbb{C}$ with a morphism $\pi:X \to B$ to a smooth rational surface $B$ such that every fibre is a (possibly degenerate) conic.

Then, then there exists a reduced normal crossings divisor (the "discriminant curve") $\Delta \subset B$ such that for any $b \in B$ we have:

(a) $\pi^{-1}(b) \cong \mathbb{P}^1$, if $b \not \in \Delta$
(b) $\pi^{-1}(b)$ is two intersecting lines if $b \in \Delta \backslash Sing (\Delta) $
(c) $\pi^{-1}(b)$ is a non-reduced line if $b \in Sing(\Delta) $
(d) In particular, there are only finitely many non-reduced fibres.

In my situation, I have a smooth conic bundle surface $p:S \to \mathbb{P}^1$ defined over $\mathbb{Q}$, and I have chosen a regular model $\pi: X \to \mathbb{P}^1_{\mathbb{Z}}$, i.e. the morphism $\pi$ restricted to the generic fibre is exactly the morphism $p$ and every fibre is a conic.

Does an analogue of the above result hold in my case? If so, does anyone have a reference to where it has been worked out in the literature?

I hope it is clear, but just to clarify that I want a reduced normal crossings divisor $\Delta \subset \mathbb{P}^1_{\mathbb{Z}}$ which satisfies the appropriate analogues of conditions (a), (b), (c) and (d).

An arithmetic analogue of the discriminant curve of a conic bundle threefold - MathOverflow

An arithmetic analogue of the discriminant curve of a conic bundle threefold