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What is the smallest dimension $d$ such that there is a smooth proper morphism $X \to \operatorname{Spec} \mathbb Z$ of relative dimension $d$, with $X$ nonempty, without a section?

Of course, there must also be such a morphism in every larger dimension - just take $X \times \mathbb P^n$.

As described in this excellent question, $d\geq2$. As described in the accepted answer, $d\leq 6$. We can improve that to $d \leq 5$ by noting that the E7 lattice also produces a nonsingular hypersurface, because the unique potential singular point over $\mathbb F_2$ fails to lie on the hypersurface.

But that still leaves a lot of uncertainty! Can anyone clarify?

Here is an auxiliary question, which I think might prove easier to answer:

What is the smallest dimension of an $X$ satisfying those conditions that is also the flag variety of a reductive group?

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Will Sawin
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  • What if you twist a surface over $\mathbf{Q}$ with good reduction? Does this give a possible strategy? Say, for example, you write down a K3 surface over $\mathbf{Q}$ with everywhere good reduction (if it exists) and you twist it such that it still has good reduction but no more $\mathbf{Q}$-points. – Jan Hendrik Nov 07 '12 at 08:58
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    @Jan Hendrik: No such K3 surface exists (by a theorem of Fontaine). – naf Nov 07 '12 at 11:37
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    I have one observation regarding Question 2. There are speculations (perhaps even conjectures) of Colliot-Thélène that for a geometrically rational variety, or even geometrically rationally connected variety, over a number field, the only obstruction to the Hasse principle is the Brauer-Manin obstruction. For a smooth, projective, geometrically rational scheme over $\mathbb{Z}$, the Brauer-Manin obstruction vanishes. Also there are local points at all non-Archimedean places. So, according to this speculation, there is a rational point if and only if there is a real point. – Jason Starr Nov 07 '12 at 15:24
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    My vague argument for question $2$ goes like this: A flag variety of a reductive group is a twist of a $G/P$, $G$ a split reductive group. The automorphism group of that is $G/Z(G)$, so the twist is classified by $H^1(G/Z(G))$. The twist must be nontrivial, else there is a point. There is an exact sequence $H^1(G) \to H^1(G/Z(G)) \to H^2(Z(G))$. For $SL_n$ and $SP_n$, $Z(G)$ is a group of roots of unity, so its cohomology measures ramification. Thus if the variety is unramified, the image in $H^2(Z(G))$ should be trivial. So $H^1(G)$ must be nontrivial. – Will Sawin Nov 08 '12 at 00:21
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    But that's also trivial for $SL_n$ and $SP_n$ by a version of Hilbert 90. So the group must be something else - an exceptional lie group or $Spin(n)$, $n\geq 7$. The smallest dimension of a flag variety of one of these groups is $5$. There's a lot of steps there I do not know how to check but perhaps this is correct. – Will Sawin Nov 08 '12 at 00:22
  • As a first step, one could try to investigate twists under $Gal(C/R)$. What are the real forms of flag varieties which have no real points ? Note that real forms of semi-simple Lie groups have been classified (by É. Cartan). I don't know whether real forms of flag varieties have been classified. – François Brunault Nov 08 '12 at 12:27
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    I think every semisimple group has an empty flag variety - just take a compact real form and look at the set of all Borel subgroups contained in it. – Will Sawin Nov 08 '12 at 15:38

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