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Divisors computing minimal log discrepancies on lc surfaces

Part of: Birational geometry Local theory

Published online by Cambridge University Press:  14 February 2023

JIHAO LIU  and
LINGYAO XIE
JIHAO LIU
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Rd, Evanston, IL 60208, U.S.A. e-mail: jliu@northwestern.edu
LINGYAO XIE
Affiliation:
Department of Mathematics, The University of Utah, Salt Lake City, UT 84112, U.S.A. e-mail: lingyao@math.utah.edu
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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.

MSC classification

Primary: 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14B05: Singularities
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 175 , Issue 1 , July 2023 , pp. 107 - 128
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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