Singularities with the highest Mather minimal log discrepancy

TL;DR

This paper classifies singularities with the highest Mather minimal log discrepancies, linking them to known classes and providing insights into a conjecture by Shokurov on minimal log discrepancies.

Contribution

It characterizes singularities with maximal Mather minimal log discrepancies and relates them to established classes, advancing understanding of log discrepancy conjectures.

Findings

Singularities with Mather minimal log discrepancy in (d-1, d) are classified as specific known types.

The work establishes a connection between Mather and usual minimal log discrepancies.

Implication of Shokurov's conjecture on minimal log discrepancies is derived.

Abstract

This paper characterizes singularities with Mather minimal log discrepancies in the highest unit interval, i.e., the interval between $d-1$ and $d$, where $d$ is the dimension of the scheme. The class of these singularities coincides with one of the classes of (1) compound Du Val singularities, (2) normal crossing double singularities, (3) pinch points, and (4) pairs of non-singular varieties and boundaries with multiplicities less than or equal to 1 at the point. As a corollary, we also obtain one implication of an equivalence conjectured by Shokurov for the usual minimal log discrepancies.