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Two two-dimensional terminations. (English) Zbl 0791.14006
Varieties with log-terminal and log-canonical singularities are considered in the minimal model program, it is conjectured that most of the “interesting” sets associated with these varieties satisfy the ascending-chain condition. In fact, one of the main properties of flips is that log-discrepancies do not decrease [cf. V. V. Shokurov, Math. USSR, Izv. 24, 193-198 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No. 1, 203-208 (1984; Zbl 0565.14025)], so, the set of the minimal discrepancies is “interesting”.
The author proves the conjecture for 2 “interesting” sets:
(i) The set of minimal log-discrepancies for \(K_ X+B\) where \(X\) is a surface with log-canonical singularities and \(B\) is from a set satisfying the descending chain condition.
(ii) the set of sequences of numbers \((b_ 1,\dots,b_ n)\) (ordered in a special way) such that there is a surface \(X\) with log-canonical and numerically trivial \(K_ X+\sum b_ jB_ j\).

MSC:
14E30 Minimal model program (Mori theory, extremal rays)
14B05 Singularities in algebraic geometry
14J45 Fano varieties
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