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Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

Pedro Fortuny Ayuso, Javier Ribón

TL;DR

The paper provides an explicit, algorithmic framework to compute the generic analytic moduli of a plane branch by constructing a nice basis of Kähler differentials for a generic curve in a fixed equisingularity class ${\mathcal C}$, and extends the construction to configurations with a normal crossings divisor $E$. Central to the approach is the level-by-level assembly of an Apery basis using Puiseux data, approximate roots, and the notion of leading variables, together with a detailed analysis of how these bases transform under blow-up. The authors then connect these constructions to families of equisingular curves and desingularization, enabling a combinatorial description of the generic semimodule ${\Lambda}_{\Gamma}$ and its $E$-preserving counterpart ${\Lambda}_{\Gamma}^{E}$. As a major application, they give an algorithm for the generic semimodule in terms of the multiplicity $n$ and Puiseux exponents, and provide an alternative proof of Genzmer's dimension formula for the generic moduli, interpreted via flow-fanning exponents and sliding divisors. The results yield a constructive bridge between topological invariants (semigroup, exponents) and analytic moduli, with a clear computational pathway for determining the dimensions and bases of analytic invariants in plane curve singularities.

Abstract

Let ${\mathcal C}$ be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the ${\mathbb C}[[x]]$-module of Kähler differentials for generic $Γ\in {\mathcal C}$, algorithmically, and study its behaviour under blow-up. As a first application, we give an algorithm providing the generic semimodule in an equisingularity class in terms of its multiplicity and its Puiseux characteristic exponents. As another application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the moduli of analytic classes in the equisingularity class of $Γ$.

Explicit Computation of The Generic Component of the Analytic Moduli of a Plane Branch

TL;DR

The paper provides an explicit, algorithmic framework to compute the generic analytic moduli of a plane branch by constructing a nice basis of Kähler differentials for a generic curve in a fixed equisingularity class , and extends the construction to configurations with a normal crossings divisor . Central to the approach is the level-by-level assembly of an Apery basis using Puiseux data, approximate roots, and the notion of leading variables, together with a detailed analysis of how these bases transform under blow-up. The authors then connect these constructions to families of equisingular curves and desingularization, enabling a combinatorial description of the generic semimodule and its -preserving counterpart . As a major application, they give an algorithm for the generic semimodule in terms of the multiplicity and Puiseux exponents, and provide an alternative proof of Genzmer's dimension formula for the generic moduli, interpreted via flow-fanning exponents and sliding divisors. The results yield a constructive bridge between topological invariants (semigroup, exponents) and analytic moduli, with a clear computational pathway for determining the dimensions and bases of analytic invariants in plane curve singularities.

Abstract

Let be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the -module of Kähler differentials for generic , algorithmically, and study its behaviour under blow-up. As a first application, we give an algorithm providing the generic semimodule in an equisingularity class in terms of its multiplicity and its Puiseux characteristic exponents. As another application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the moduli of analytic classes in the equisingularity class of .
Paper Structure (43 sections, 33 theorems, 162 equations, 1 algorithm)

This paper contains 43 sections, 33 theorems, 162 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Nice bases and blow-up
  3. Equisingularity class of a pair $(\Gamma_{\circ}, E)$
  4. The set of exponents of an equisingularity class
  5. The semigroup and the semi-module
  6. Combinatorial invariants of $(\Gamma, E)$
  7. Approximate roots of $(\Gamma, E)$
  8. Parametrized $1$-forms
  9. Case $E=\emptyset$
  10. Distinguished modules of $1$-forms
  11. Ordered families
  12. The basic step --- leading variables
  13. Leading variables of products
  14. Terminal families
  15. Definition and first properties of terminal families
  16. ...and 28 more sections

Key Result

Theorem 1.1

There is a nice basis for ${\mathcal{C}}$: If $n$ is the multiplicity of $\Gamma_{\circ}$ at the origin, there exists an open subset $U$ of $\tilde{\mathcal{C}}$ and germs of $1$-forms $\Omega_{1,\Gamma}, \hdots, \Omega_{n, \Gamma}$, for $\Gamma \in U$, such that Moreover, $\Omega_{j, \Gamma}$ depends polynomially on $\Gamma \in U$ and $\nu_{\Gamma} (\Omega_{1, \Gamma}), \hdots, \nu_{\Gamma} (\Om

Theorems & Definitions (125)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem : Genzmer-moduli-2020
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Remark 2.2
  • ...and 115 more