#### Vol. 6, No. 7, 2012

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Log canonical thresholds, $F$-pure thresholds, and nonstandard extensions

### Bhargav Bhatt, Daniel J. Hernández, Lance Edward Miller and Mircea Mustaţă

Vol. 6 (2012), No. 7, 1459–1482
##### Abstract

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 $F$-pure threshold). We show that the set of limit points of sequences of the form $\left({c}_{p}\right)$, where ${c}_{p}$ is the $F$-pure threshold of an ideal on an $n$-dimensional smooth variety in characteristic $p$, coincides with the set of log canonical thresholds of ideals on $n$-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
##### Mathematical Subject Classification 2010
Primary: 13A35
Secondary: 13L05, 14B05, 14F18
##### Milestones
Received: 1 June 2011
Revised: 16 November 2011
Accepted: 20 December 2011
Published: 4 December 2012
##### 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 Lance Edward Miller Department of Mathematics University of Utah Salt Lake City, UT 84112 United States Mircea Mustaţă Department of Mathematics University of Michigan Ann Arbor, MI 48109 United States