[Paper Review] Puiseux coefficients and parametric deformation of plane curve singularities

Maciej Borodzik|arXiv (Cornell University)|Sep 24, 2009

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper investigates parametric deformations of cuspidal plane curve singularities by analyzing how Puiseux coefficients—derived from power series expansions of parametrized curves—evolve as deformation parameters vary. It establishes that higher-order derivatives of Puiseux coefficients depend only on sums of indices, leading to a WDVV-like equation and revealing unexpected functional dependencies, with key results on adjacency in singularity theory and the role of essential/inessential terms in topological type preservation.

ABSTRACT

We study deformations of plane curve singularities from an analytic point of view and obtain some new concrete results. We show some rather unexpected properties of Puiseux coefficients treated as functions on a suitably defined parameter space. The methods used in paper are very elementary.

Motivation & Objective

• To understand how Puiseux coefficients of plane curve singularities change under parametric deformations.
• To clarify the role of essential and inessential terms in determining the topological type of singularities.
• To address the adjacency problem in singularity theory by analyzing coefficient evolution under deformation.
• To uncover unexpected functional properties of Puiseux coefficients as functions on the parameter space.
• To establish a WDVV-like equation satisfied by certain generating functions derived from Puiseux coefficients.

Proposed method

• Models deformations by treating Puiseux coefficients as functions of a complex parameter s, with coefficients a_j(s), b_j(s) varying analytically.
• Expresses Puiseux coefficients c_j(s) via rational functions of initial coefficients a_p, a_{p+1}, ..., b_q, b_{q+1}, ..., using polynomial expressions γ_j(s).
• Introduces a generating function G for the coefficients g_k, derived from the inverse of the parametrization, to study higher-order derivatives.
• Applies the chain rule and implicit differentiation to compute partial derivatives of z with respect to c_k, revealing dependence on i+k in second derivatives.
• Demonstrates that mixed third derivatives ∂³g_k/∂c_i∂c_j∂c_m depend only on i+j+m, enabling the construction of a WDVV-like equation.
• Uses matrix η with η^{ab}=1 iff a+b=N+1 to define a structure on g_{N+3} satisfying the WDVV equation for any N>2.

Experimental results

Research questions

• RQ1 How do Puiseux coefficients behave as functions of a deformation parameter in parametric families of plane curve singularities?
• RQ2 Which terms in the Puiseux expansion are essential for preserving the topological type of a singularity under deformation?
• RQ3 Can the adjacency problem for singularities be resolved by tracking coefficient evolution in parametric deformations?
• RQ4 What functional dependencies emerge when higher-order derivatives of Puiseux coefficients are analyzed?
• RQ5 Does the generating function of Puiseux coefficients satisfy a WDVV-type equation, and if so, under what conditions?

Key findings

• Higher-order partial derivatives of the generating function g_k depend only on the sum of indices i+j+k, not on individual values.
• The third derivatives ∂³g_k/∂c_i∂c_j∂c_m depend only on i+j+m, leading to a WDVV-like equation for g_{N+3} on C^N.
• The WDVV equation is satisfied for any N>2, with the structure defined by a matrix η that is 1 when a+b=N+1 and 0 otherwise.
• The resulting multiplication on C^N is associative but degenerate, indicating a 'massless' Frobenius structure.
• The functional dependence of Puiseux coefficients on the deformation parameter reveals non-trivial, symmetric behavior across index sums.
• The paper confirms that changing an inessential coefficient (e.g., c_7 in a Puiseux expansion) does not alter the characteristic sequence, while changing an essential one does.