Marcelo Osnar Rodrigues de Abreu

arXiv2021

On the value set of $1$-forms for plane branches

The value semigroup $Γ$ and the value set $Λ$ of $1$-forms are, respectively, a topological and an analytical invariant of a plane branch. Giving a plane branch $\mathcal{C}$ with semigroup $Γ$ there are a finitely number of distinct possible sets $Λ_i$ according to the analytic class of $\mathcal{C}$. In this work we show that the value set of $1$-forms $Λ$ determines the semigroup $Γ$ and we present an effective method to recover $Γ$ by $Λ$. In particular, this allows us to decide if a subset of $\mathbb{N}$ is a value set of $1$-forms for a plane branch.

math.AG↗

arXiv2021

On the Analytic Invariants and Semiroots of Plane Branches

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

math.AG↗