By Brian Osserman

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Applying ‘Adjunction’, ˜ t , Ct ) is also log canonical. 3 implies that Xt is we conclude that (X therefore semilog canonical. References [Abr02] Dan Abramovich, Canonical models and stable reduction for pluriﬁbered varieties. Unpublished, arXiv:math/0207004, 2002. [ACV03] Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers. Comm. Algebra 31 (2003), 3547–3618. Special issue in honor of Steven L. Kleiman. [AGV08] Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks.

2, we have µ∗ OS = OW . In particular, S → C1 ∪ C2 has connected ﬁbers. Therefore there are prime components E1 , E2 of S such that µ(Ei ) = Ci and E1 ∩ E2 ∩ π −1 (x) = ∅. Let Z be a connected component of E1 ∩ E2 which intersects π −1 (x). Then C3 = µ(Z) is a log canonical center with x ∈ C3 ⊂ C1 ∪ C2 . (3) Let W be the union of some log canonical centers. 2 hold. 2, we have µ∗ OS = OW . 5] that W has seminormal singularities. (4) Fix x ∈ LCS(X, B) and consider (X, B) as a germ near x. By (1) and (3), there exists a unique log canonical center x ∈ C which is minimal with respect to inclusion.

8. We deﬁne Kλ = KL Gm , where Gm acts by scalars on L. We deﬁne Kω ⊂ Kλ as the locally closed subcategory corresponding to canonically ◦ polarized twisted varieties (f : X → B, λ) where λ is given by ωX /B . ω ω We deﬁne Kgor ⊂ K to be the open substack where the ﬁbers of f are CohenMacaulay. Since the dualizing sheaf is invertible–it is the polarizing line bundle–the ﬁbers are automatically Gorenstein. 3 guarantees that the second condition is locally closed. 9. We can interpret Kω in terms of Koll´ar families (f¯ : X → B, F ) ◦ ω with isomorphism F → ωX /B .

### A Galois theory example by Brian Osserman

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