Towards boundedness of minimal log discrepancies by Riemann--Roch theorem

TL;DR

This paper applies the Riemann--Roch theorem to study the boundedness of minimal log discrepancies in fixed dimensions, providing new bounds and characterizations for certain Gorenstein terminal singularities.

Contribution

It introduces a novel approach using Riemann--Roch to address boundedness of minimal log discrepancies and extends characterizations to specific Gorenstein terminal singularities.

Findings

Bounded minimal log discrepancy when multiplicity or embedding dimension is bounded.

Reconstruction of Gorenstein terminal three-fold singularity characterization.

Determination of the boundary for a special four-fold singularity.

Abstract

We introduce an approach of Riemann--Roch theorem to the boundedness problem of minimal log discrepancies in fixed dimension. After reducing it to the case of a Gorenstein terminal singularity, firstly we prove that its minimal log discrepancy is bounded if either multiplicity or embedding dimension is bounded. Secondly we recover the characterisation of a Gorenstein terminal three-fold singularity by Reid, and the precise boundary of its minimal log discrepancy by Markushevich, without explicit classification. Finally we provide the precise boundary for a special four-fold singularity, whose general hyperplane section has a terminal piece.