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Abstract Let be a submonoid of a free Abelian group of finite rank. We show that if is a field of prime characteristic such that the monoid ‐algebra is , then is a finitely generated ‐algebra, or equivalently, that is a finitely generated monoid. Split‐‐regular rings are possibly non‐Noetherian or non‐‐finite rings that satisfy the defining property of strongly ‐regular rings from the theories of tight closure and ‐singularities. Our finite generation result provides evidence in favor of the conjecture that rings in function fields over have to be Noetherian. The key tool is Diophantine approximation from convex geometry.
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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.more » « less
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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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