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

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.