A finiteness property of graded sequences of ideals

TL;DR

This paper proves that for a graded sequence of ideals with finite log canonical threshold, if the associated divisors have bounded discrepancies, then only finitely many such divisors exist.

Contribution

It establishes a finiteness property of divisors computing log canonical thresholds in graded sequences of ideals under bounded discrepancy conditions.

Findings

Finiteness of divisors computing thresholds under bounded discrepancies

Bounded discrepancies imply a finite set of divisors

Applicable to smooth varieties with finite log canonical thresholds

Abstract

Given a graded sequence of ideals (a_m) on a smooth variety $X$ having finite log canonical threshold, suppose that for every m we have a divisor E_m over X that computes the log canonical threshold of a_m, and such that the log discrepancies of the divisors E_m are bounded. We show that in this case the set of divisors E_m is finite.