On the Analytic Invariants and Semiroots of Plane Branches

TL;DR

This paper explores how the set of values of differentials of plane branches and their semiroots relate to topological invariants, providing bounds and relations for the Tjurina and Milnor numbers within fixed topological classes.

Contribution

It introduces a method to determine parts of the differential value set of a plane branch from its semiroots, linking analytical and topological invariants and generalizing previous results.

Findings

The set of differentials of semiroots determines part of the differential set of the original curve.

A bound on the Tjurina number in terms of the Milnor numbers and gcd of semigroup generators.

Equality of Tjurina and Milnor number difference for certain topological classes when gcd is 2.

Abstract

The value semigroup of a $k$-semiroot $C_k$ of a plane branch $C$ allow us to recover part of the value semigroup $\Gamma =\langle v_0,\ldots ,v_g\rangle$ of $C$, that is, it is related to topological invariants of $C$. In this paper we consider the set of values of differentials $\Lambda_k$ of $C_k$, that is an analytical invariant, and we show how it determine part of the set of values of differentials $\Lambda$ of $C$. As a consequence, in a fixed topological class, we relate the Tjurina number $\tau$ of $C$ with the Tjurina number of $C_k$. In particular, we show that $\tau\leq \mu-\frac{3n_g-2}{4}\mu_{g-1}$ where $n_g=gcd(v_0,\ldots ,v_{g-1})$, $\mu$ and $\mu_{g-1}$ denote the Milnor number of $C$ and $C_{g-1}$ respectively. If $n_g=2$, we have that $\tau=\mu-\mu_{g-1}$ for any curve in the topological class determined by $\Gamma$ that is a generalization of a result obtained by Luengo and Pfister.