Contact exponent and the Milnor number of plane curve singularities

TL;DR

This paper studies the contact exponent of plane curve singularities over algebraically closed fields, proving its invariance, stability, and establishing bounds for the Milnor number without using Newton diagrams, utilizing logarithmic distance instead.

Contribution

It introduces a new proof of contact exponent invariance and stability, and provides bounds for the Milnor number using logarithmic distance, avoiding Newton diagrams.

Findings

Contact exponent is an equisingularity invariant.

Established a bound for the Milnor number.

Determined equisingularity classes where the bound is attained.

Abstract

We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].