There are several MO-hard problems about smooth and proper schemes over $\mathbb Z$, including:

> [Smooth proper schemes over $\mathbb Z$ with points everywhere locally][2]


> [Non-simply-connected smooth proper scheme over $\mathbb Z$][3]



> [What can be the dimension of a pointless smooth proper $\mathbb Z$-scheme?][4]


All these were inspired by Poonen's (solved) question, asking whether such a scheme necessarily has a section:

> [Smooth proper scheme over $\mathbb Z$][1]




  [1]: http://mathoverflow.net/questions/9576/smooth-proper-scheme-over-z
  [2]: http://mathoverflow.net/questions/10569/smooth-proper-schemes-over-rings-of-integers-with-points-everywhere-locally?rq=1
  [3]: http://mathoverflow.net/questions/11066/non-simply-connected-smooth-proper-scheme-over-z?rq=1
  [4]: http://mathoverflow.net/questions/111708/what-can-be-the-dimension-of-a-pointless-smooth-proper-z-scheme