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Does there exist a smooth proper variety $X$ over $\operatorname{Spec} \mathbb Z$ that is not toric?

By Fontaine, we know that there is no Abelian scheme over $\operatorname{Spec} \mathbb Z$. Also by Fontaine, we know that any such $X$ has $H^p(X,\Omega^q) = 0$ for $p+q \leq 3, p\neq q$ (but I might be misquoting this theorem).

Moreover, I think it's easy to find examples of toric varieties that are smooth proper over $\mathbb Z$. For instance, I believe this includes $\mathbb P^n$, projective bundles over it, blow ups of such and the Hilbert scheme of points on a surface of this kind. Are there any other examples known?

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    $\begingroup$ Yes, quadrics . $\endgroup$
    – abx
    Dec 2, 2021 at 5:40
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    $\begingroup$ Or more generally flag varieties $G/P$. (I think they are toric iff they are isomorphic to a projective space). $\endgroup$
    – Nicolas Hemelsoet
    Dec 2, 2021 at 6:18

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