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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 I am hoping that there is 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).

3 deleted 117 characters in body

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

(i) Every every fibre is a (possibly degenerate) conic
(ii) For every irreducible curve $C \subset B$, the fibre above $C$ is an irreducible surface.

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 generic fibre of morphism $\pi$ restricted to the generic fibre is exactly given by the morphism $p$. I am pretty sure that I can assume the appropriate analogue of condition (i), I'm not entirely sure about condition (ii) yet but I am not worrying too much about it at the momentp$and every fibre is a conic. Does an analogue of the above result hold in my case? If so, does anyone have a reference to where it has been worked out in the literature? I hope it is clear, but just to clarify I am hoping that there is a reduced normal crossings divisor$\Delta \subset \mathbb{P}^1_{\mathbb{Z}}$which satisfies the appropriate analogues of conditions (a), (b), (c) and (d). 2 added 91 characters in body I am looking for an "arithmetic" analogue of a well known result on threefolds with a conic bundle structure. The following result can be found in [Iskovskikh - On the rationality problem for conic bundles, Lemma 1]. Let$X$be a smooth irreducible threefold over$\mathbb{C}$with a morphism$\pi:X \to B$to a smooth rational surface$B$such that (i) Every fibre is a (possibly degenerate) conic (ii) For every irreducible curve$C \subset B$, the fibre above$C$is an irreducible surface. Then, then there exists a reduced normal crossings divisor (the "discriminant curve")$\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}}$, i.e. the generic fibre of$\pi$is exactly given by the morphism$p$. I am pretty sure that I can assume the appropriate analogues analogue of conditions condition (i) and i), I'm not entirely sure about condition (ii).ii) yet but I am not worrying too much about it at the moment. Does an analogue of the above result hold in my case? If so, does anyone have a reference to where it has been worked out in the literature? I hope it is clear, but just to clarify I am hoping that there is a reduced normal crossings divisor$\Delta \subset \mathbb{P}^1_{\mathbb{Z}}\$ which satisfies the appropriate analogues of conditions (a), (b), (c) and (d).

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