The value semigroup of a plane curve singularity with several branches

TL;DR

This paper introduces a constructive method using Apéry sets to determine the value semigroup of plane curve singularities from their blow-up, enabling reconstruction and classification of all possible semigroups.

Contribution

It provides a novel, constructive approach to compute and classify value semigroups of plane curve singularities via blow-up and blow-down procedures.

Findings

A procedure to derive the value semigroup from the blow-up semigroup.

A method to reconstruct a plane curve from its blow-up.

A complete characterization of possible multiplicity trees.

Abstract

We present a constructive procedure, based on the notion of Ap\'ery set, to obtain the value semigroup of a plane curve singularity from the value semigroup of its blow-up and viceversa. In particular we give a blow-down process that allows to reconstruct a plane algebroid curve form its blow-up, even if it is not local. Then we characterize numerically all the possible multiplicity trees of plane curve singularities, obtaining in this way a constructive description of all their value semigroups.