A bound of the number of weighted blow-ups to compute the minimal log discrepancy for smooth 3-folds

TL;DR

This paper establishes a bound on the number of weighted blow-ups needed to compute the minimal log discrepancy for smooth 3-folds, providing a weighted blow-up version of the Mustata-Nakamura Conjecture and implications for ACC.

Contribution

It proves that the minimal log discrepancy for smooth 3-folds can be computed with at most two weighted blow-ups, advancing understanding of singularity invariants.

Findings

Minimal log discrepancy is computed by at most two weighted blow-ups.

If mld ≥ 1, it is computed by a single weighted blow-up.

The result implies the ACC conjecture for such pairs.

Abstract

We study a pair consisting of a smooth 3-fold defined over an algebraically closed field and a general real ideal. We show that the minimal log discrepancy of every such a pair is computed by a prime divisor obtained by at most two weighted blow-ups. This bound is regarded as a weighted blow-up version of Mustata-Nakamura Conjecture. We also show that if the mld of such a pair is not less than 1, then it is computed by at most one weighted blow-up. As a consequence, ACC of mld holds for such pairs.