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On the Zariski invariant of plane branches

Marcelo Escudeiro Hernandes, Mauro Fernando Hernández Iglesias

TL;DR

This work addresses the problem of identifying the Zariski invariant $\lambda_f$ of a plane branch from geometric data. It introduces a geometric characterization: $\lambda_f$ equals $n$ times the maximal contact with a special family $\mathcal{B}$ of curves (or, equivalently, the maximal intersection multiplicity with these curves), where $\mathcal{B}$ consists of branches analytically equivalent to $y^{n_1}-x^{m_1}=0$ with $n_1=n/e_1$, $m_1=m/e_1$ and $e_1=\gcd(n,m)$. The key result provides the exact formula $\lambda_f = n\cdot\max_{C\in\mathcal{B}}\{\text{cont}(C_f,C)\}=\max_{C\in\mathcal{B}}\{\text{I}(C_f,C)\}-(n_1-1)m$ and discusses its invariance under analytic coordinate changes, its behavior when $\lambda_f=\infty$, and its implications for comparing Zariski invariants via a triangular-like inequality. The paper also illustrates the approach with examples and outlines how this geometric interpretation facilitates computation of the invariant and analysis of analytic equivalence among branches.

Abstract

We show how to obtain the Zariski invariant of a plane branch employing the contact order or the intersection multiplicity with elements in a particular family of curves and we present some consequences of this result.

On the Zariski invariant of plane branches

TL;DR

This work addresses the problem of identifying the Zariski invariant of a plane branch from geometric data. It introduces a geometric characterization: equals times the maximal contact with a special family of curves (or, equivalently, the maximal intersection multiplicity with these curves), where consists of branches analytically equivalent to with , and . The key result provides the exact formula and discusses its invariance under analytic coordinate changes, its behavior when , and its implications for comparing Zariski invariants via a triangular-like inequality. The paper also illustrates the approach with examples and outlines how this geometric interpretation facilitates computation of the invariant and analysis of analytic equivalence among branches.

Abstract

We show how to obtain the Zariski invariant of a plane branch employing the contact order or the intersection multiplicity with elements in a particular family of curves and we present some consequences of this result.
Paper Structure (2 sections, 4 theorems, 42 equations)

This paper contains 2 sections, 4 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. The Zariski invariant, contact and intersection multiplicity of plane branches

Key Result

Proposition 2.2

(merle) Let $C_f\in K(n,m,\beta_2,\ldots,\beta_g)$, $\Gamma_f=\langle n,m,v_2,\ldots,v_g\rangle$ its values semigroup and $C_h$ be any plane branch. The following statements are equivalent:

Theorems & Definitions (12)

  • Example 1.1
  • Remark 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4
  • proof
  • Example 2.5
  • Proposition 2.6
  • proof
  • Example 2.7
  • ...and 2 more