An arithmetic analogue of the discriminant curve of a conic bundle threefold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:32:35Z http://mathoverflow.net/feeds/question/93859 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93859/an-arithmetic-analogue-of-the-discriminant-curve-of-a-conic-bundle-threefold An arithmetic analogue of the discriminant curve of a conic bundle threefold Daniel Loughran 2012-04-12T11:59:31Z 2012-04-13T12:01:02Z <p>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.</p> <hr> <p>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.</p> <p>Then, then there exists a reduced normal crossings divisor (the "discriminant curve") $\Delta \subset B$ such that for any $b \in B$ we have:</p> <p>(a) $\pi^{-1}(b) \cong \mathbb{P}^1$, if $b \not \in \Delta$<br> (b) $\pi^{-1}(b)$ is two intersecting lines if $b \in \Delta \backslash Sing (\Delta)$<br> (c) $\pi^{-1}(b)$ is a non-reduced line if $b \in Sing(\Delta)$<br> (d) In particular, there are only finitely many non-reduced fibres.</p> <hr> <p>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.</p> <blockquote> <p>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?</p> </blockquote> <p>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).</p>