Monotonic invariants under blowups

TL;DR

This paper introduces a new monotonic invariant for plane curve singularities, demonstrating its non-negativity and behavior under blowups, offering insights into singularity theory and invariant properties.

Contribution

It proves that the invariant 3μ-4τ is non-negative, non-decreasing under blowups, and strictly increasing unless the curve is smooth, providing a new perspective on singularity invariants.

Findings

The invariant 3μ-4τ is non-negative for singularities.

The invariant is non-decreasing under blowups.

It strictly increases unless the curve is smooth.

Abstract

We prove that the numerical invariant $3\mu-4\tau$ of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by A. Dimca and G.-M. Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.