Existence of log canonical closures


Let f:Xβ†’Uf:X\to U be a projective morphism of normal varieties and (X,Ξ”)(X,\Delta) a dlt pair. We prove that if there is an open set U0βŠ‚UU^0\subset U, such that (X,Ξ”)Γ—UU0(X,\Delta)\times_U U^0 has a good minimal model over U0U^0 and the images of all the non-klt centers intersect U0U^0, then (X,Ξ”)(X,\Delta) has a good minimal model over UU. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness

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