Application of classical invariant theory to biholomorphic classification of plane curve singularities, and associated binary forms

TL;DR

This paper applies classical invariant theory to classify plane curve singularities up to biholomorphic equivalence, proposing a new approach to derive invariants from moduli algebras.

Contribution

It introduces a novel method using invariant theory to solve the classification problem for certain plane curve singularities and suggests a conjecture for extracting invariants from moduli algebras.

Findings

Solved biholomorphic equivalence for specific singularity families

Proposed a conjecture linking invariants to moduli algebras

Motivated a new approach for classifying plane curve singularities

Abstract

We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way of extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.