On equivalent conjectures for minimal log discrepancies on smooth threefolds

TL;DR

This paper explores the equivalence of conjectures related to minimal log discrepancies on smooth threefolds, reducing the problem to specific boundary cases and proving key assertions under certain threshold conditions.

Contribution

It establishes the equivalence of the ACC for minimal log discrepancies with boundedness conditions and proves the assertion for particular threshold ranges.

Findings

ACC for minimal log discrepancies is equivalent to boundedness of certain divisors.

Reduction to boundaries as products of canonical parts and powers of maximal ideals.

Proved the assertion when the log canonical threshold is ≤ 1/2 or ≥ 1.

Abstract

On smooth threefolds, the ACC for minimal log discrepancies is equivalent to the boundedness of the log discrepancy of some divisor which computes the minimal log discrepancy. We reduce it to the case when the boundary is the product of a canonical part and the maximal ideal to some power. We prove the reduced assertion when the log canonical threshold of the maximal ideal is either at most one-half or at least one.