The moduli functor M_{n,v}  of stable varieties of dimension n is a higher-dimensional generalization proposed by Kollár and Shepherd-Barron to find a geometric compactification of moduli spaces of varieties with ample canonical bundle, similar to the Deligne--Mumford compactification for smooth curves. In the past decades, work of various birational geometers showed that M_{n,v,C} is a proper DM stack of finite type, admitting a coarse projective moduli space.

Despite the satisfactory answer over C, the theory of moduli of stable varietes presents further additional difficulties in positive and mixed characteristic and many basic questions are still unsolved. However, recent progress on the MMP allowed to show that in the case of surfaces M_{2.v} exists as a separated Artin stack of finite type over Z[1/30].

I will report on a joint work in progress with E. Arvidsson and Zs, Patakfalvi where, assuming the existence of semi-stable reduction, we conclude that M_{2,v} is proper. To achieve this, we give a geometric characterisation of the failure of the S_3-condition for 3-dimensional log canonical singularities.