There are several MO-hard problems about smooth and proper schemes over $\mathbb Z$, including:
Smooth proper schemes over $\mathbb Z$ with points everywhere locallySmooth proper schemes over $\mathbb Z$ with points everywhere locally
Non-simply-connected smooth proper scheme over $\mathbb Z$Non-simply-connected smooth proper scheme over $\mathbb Z$
What can be the dimension of a pointless smooth proper $\mathbb Z$-scheme?What can be the dimension of a pointless smooth proper $\mathbb Z$-scheme?
All these were inspired by Poonen's (solved) question, asking whether such a scheme necessarily has a section:
Smooth proper scheme over $\mathbb Z$Smooth proper scheme over $\mathbb Z$
