Analytic semiroots for plane branches and singular foliations

TL;DR

This paper introduces the concept of analytic semiroots for plane branches, showing how enlarged standard bases of differential forms can be used to understand equisingular deformations and their invariants.

Contribution

It establishes a structured approach to describe the semimodules of differential values for equisingular plane branches using enlarged standard bases and their geometric properties.

Findings

Enlarged standard bases define packages of equisingular branches.

Semimodules of differential values are described by truncation of initial generators.

All branches in a package share the same semimodule of differential values.

Abstract

The analytic moduli of equisingular plane branches has the semimodule of differential values as the most relevant system of discrete invariants. Focusing in the case of cusps, the minimal system of generators of this semimodule is reached by the differential values attached to the differential $1$-forms of the so-called standard bases. We can complete a standard basis to an enlarged one by adding a last differential $1$-form that has the considered cusp as invariant branch and the ``correct'' divisorial order. The elements of such enlarged standard bases have the ``cuspidal'' divisor as a ``totally dicritical divisor'' and hence they define packages of plane branches that are equisingular to the initial one. These are the analytic semiroots. In this paper we prove that the enlarged standard bases are well structured from this geometrical and foliated viewpoint, in the sense that the semimodules of differential values of the branches in the dicritical packages are described just by a truncation of the list of generators of the initial semimodule at the corresponding differential value. In particular they have all the same semimodule of differential values.