Algebra & Number Theory

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

Bhargav Bhatt, Daniel Hernández, Lance Edward Miller, and Mircea Mustaţă

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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 F-pure threshold). We show that the set of limit points of sequences of the form (cp), where cp 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.

Article information

Algebra Number Theory, Volume 6, Number 7 (2012), 1459-1482.

Received: 1 June 2011
Revised: 16 November 2011
Accepted: 20 December 2011
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 13A35: Characteristic p methods (Frobenius endomorphism) and reduction to characteristic p; tight closure [See also 13B22]
Secondary: 13L05: Applications of logic to commutative algebra [See also 03Cxx, 03Hxx] 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx] 14F18: Multiplier ideals

$F$-pure threshold log canonical threshold ultrafilters multiplier ideals test ideals


Bhatt, Bhargav; Hernández, Daniel; Miller, Lance Edward; Mustaţă, Mircea. Log canonical thresholds, $F$-pure thresholds, and nonstandard extensions. Algebra Number Theory 6 (2012), no. 7, 1459--1482. doi:10.2140/ant.2012.6.1459.

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