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

 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
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