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On the properness of the moduli space of stable surfaces over
$\mathbb{Z}$[1/30]
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We show the properness of the moduli stack of stable surfaces over
$\mathbb{Z}\left[ {1/30} \right]$, assuming the locally-stable reduction conjecture for stable surfaces. This relies on a local Kawamata–Viehweg vanishing theorem for 3-dimensional log canonical singularities at closed point of characteristic 
$p \ne 2,3$ and 
$5$, which are not log canonical centres.
