On the Blow-analytic Equivalence of Tribranched Plane Curves

TL;DR

This paper establishes the finiteness of blow-analytic equivalence classes for plane curve germs with a fixed number of branches and a fixed invariant, introducing standard forms of dual graphs to classify these classes.

Contribution

It develops the concept of standard forms of dual graphs and proves finiteness of equivalence classes, providing explicit bounds and classifications for tribranched curves.

Findings

Finite number of standard forms for fixed μ'

Explicit upper bounds for tribranched cases

Complete classification for μ' ≤ 2

Abstract

We prove the finiteness of the number of blow-analytic equivalence classes of embedded plane curve germs for any fixed number of branches and for any fixed value of $\mu'$ ---a combinatorial invariant coming from the dual graphs of good resolutions of embedded plane curve singularities. In order to do so, we develop the concept of standard form of a dual graph. We show that, fixed $\mu'$ in $\mathbb{N}$, there are only a finite number of standard forms, and to each one of them correspond a finite number of blow-analytic equivalence classes. In the tribranched case, we are able to give an explicit upper bound to the number of graph standard forms. For $\mu'\leq 2$, we also provide a complete list of standard forms.