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Log
canonical thresholds, $F$-pure thresholds, and nonstandard
extensions
Bhargav Bhatt, Daniel J. Hernández, Lance Edward Miller and Mircea Mustaţă |
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Vol. 6 (2012), No. 7, 1459–1482
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Abstract |
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We present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the -pure threshold). We show that the set of limit points of sequences of the form , where is the -pure threshold of an ideal on an -dimensional smooth variety in characteristic , coincides with the set of log canonical thresholds of ideals on -dimensional smooth varieties in characteristic zero. We prove this by combining results of Hara and Yoshida with nonstandard constructions. |
Keywords
$F$-pure threshold, log canonical threshold, ultrafilters,
multiplier ideals, test ideals
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Mathematical Subject Classification 2010
Primary: 13A35
Secondary: 13L05, 14B05, 14F18
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Milestones
Received: 1 June 2011
Revised: 16 November 2011
Accepted: 20 December 2011
Published: 4 December 2012
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Authors |
| Bhargav Bhatt | |
| Department of Mathematics University
of Michigan Ann Arbor, MI 48109 United States |
School of Mathematics Institute for Advanced Study Princeton, NJ 08540 United States |
| Daniel J. Hernández | |
| Department of Mathematics University of Minnesota Minneapolis, MN 55455 United States |
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| Lance Edward Miller | |
| Department of Mathematics University of Utah Salt Lake City, UT 84112 United States |
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| Mircea Mustaţă | |
| Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States |
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