Log-canonical threshold for curves on a smooth surface

TL;DR

This paper demonstrates that the log-canonical threshold of a curve with an isolated singularity on a smooth surface can be computed using the term ideal in a suitable local system, relying on the Enriques diagram.

Contribution

It introduces a method to compute the log-canonical threshold via the term ideal and Enriques diagram, highlighting its dependence on a specific non-degenerate path.

Findings

The log-canonical threshold depends only on a non-degenerate path of the Enriques diagram.

The threshold can be computed using the term ideal in a suitable local parameter system.

The approach simplifies the calculation of singularity invariants for curves on smooth surfaces.

Abstract

It is shown that the log-canonical threshold of a curve with an isolated singularity is computed by the term ideal of the curve in a suitable system of local parameters at the singularity. The proof uses the Enriques diagram of the singularity and shows that the log-canonical threshold depends only on a non-degenerate path of that diagram.