What can be the dimension of a pointless smooth proper Z-scheme? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:00:46Z http://mathoverflow.net/feeds/question/111708 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111708/what-can-be-the-dimension-of-a-pointless-smooth-proper-z-scheme What can be the dimension of a pointless smooth proper Z-scheme? Will Sawin 2012-11-07T07:08:37Z 2012-11-07T07:08:37Z <blockquote> <p>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?</p> </blockquote> <p>Of course, there must also be such a morphism in every larger dimension - just take $X \times \mathbb P^n$.</p> <p>As described in <a href="http://mathoverflow.net/questions/9576/smooth-proper-scheme-over-z" rel="nofollow">this excellent question</a>, $d\geq2$. 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.</p> <p>But that still leaves a lot of uncertainty! Can anyone clarify?</p> <p>Here is an auxiliary question, which I think might prove easier to answer:</p> <blockquote> <p>What is the smallest dimension of an $X$ satisfying those conditions that is also the flag variety of a reductive group?</p> </blockquote>