Algebra & Number Theory
- Algebra Number Theory
- Volume 6, Number 7 (2012), 1459-1482.
Log canonical thresholds, $F$-pure thresholds, and nonstandard extensions
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.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.ant/1513729891