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Abstract Given an NQC log canonical generalized pair$$(X,B+M)$$ whose underlying varietyXis not necessarily$$\mathbb {Q}$$ -factorial, we show that one may run a$$(K_X+B+M)$$ -MMP with scaling of an ample divisor which terminates, provided that$$(X,B+M)$$ has a minimal model in a weaker sense or that$$K_X+B+M$$ is not pseudo-effective. We also prove the existence of minimal models of pseudo-effective NQC log canonical generalized pairs under various additional assumptions, for instance when the boundary contains an ample divisor.
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Abstract We show that defines a birational map and has no fixed part for some bounded positive integermfor any ‐lc surfaceXsuch that is big and nef. For every positive integer , we construct a sequence of projective surfaces , such that is ample, for everyi, , and for any positive integerm, there existsisuch that has nonzero fixed part. These results answer the surface case of a question of Xu.more » « less
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Abstract Let$$(X\ni x,B)$$be an lc surface germ. If$$X\ni x$$is klt, we show that there exists a divisor computing the minimal log discrepancy of$$(X\ni x,B)$$that is a Kollár component of$$X\ni x$$. If$$B\not=0$$or$$X\ni x$$is not Du Val, we show that any divisor computing the minimal log discrepancy of$$(X\ni x,B)$$is a potential lc place of$$X\ni x$$. This extends a result of Blum and Kawakita who independently showed that any divisor computing the minimal log discrepancy on a smooth surface is a potential lc place.more » « less
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