Moduli spaces for topologically quasi-homogeneous functions

Abstract

We consider the topological class of a germ of 2-variables quasi-homogeneous complex analytic function. Each element f in this class induces a germ of foliation (f = constants) and a germ of curve (f = 0). We first describe the moduli space of the foliations in this class and give analytic normal forms. The classification of curves induces a distribution on this moduli space. By studying the infinitesimal generators of this distribution, we can compute the generic dimension of the moduli space for the curves, and we obtain the corresponding generic normal forms.

The Analytic Classification of Plane Curves

“…. , ϕ r ] with ϕ i = (t n i , j≥m a ij t j i ) where a ij ∈ C are generic, 0 = a n im = a n lm = 0 for every i, l ∈ I and i = l. The generic component for the moduli space in this topological class was also considered in [GP2] by other methods.…”

“…Later, the authors in a joint work with Hefez (in 2015, see [HHR]), generalized such method to obtain the analytic classification of plane curves with two irreducible components. Genzmer and Paul (in 2016, see [GP2]), using tools of Foliation theory, described the moduli space for generic plane curves such that every branch admits semigroup n, m and they presented a method to obtain the normal form for the generic case. In 2019, Genzmer in [Ge], obtained a formula for the dimension of M g for any irreducible plane curve.…”

The Saito module and the moduli of a germ of curve in $\left(\mathbb{C}^{2},0\right)$

“…In 2010 and 2011, in [10,11], Emmanuel Paul and the author described the moduli space of a topologically quasi-homogeneous curve S as the spaces of leaves of an algebraic foliation defined on the moduli space of a foliation whose analytic invariant curve is precisely S. These works initiated an approach based upon the theory of foliations, which is at stake here. In 2019, in [8], the author gave an explicit formula for the number of moduli for a curve S, generic in its topological class : this formula involves only very elementary topological invariants of S, such as, the topological class of its desingularization.…”

“…It can be seen that the quotient reduces to a point. As a matter of fact, the curve (1.12) has no moduli [11].…”

Moduli spaces of a family of topologically non quasi-homogeneous functions

“…If f is quasi-homogeneous, then there exist coordinates (x, y) and positive coprime integers k and l such that the quasi-radial vector field R = kx ∂ ∂x + ly ∂ ∂y satisfies R(f ) = d · f , where the integer d is the quasi-homogeneous (k, l)-degree of f [6]. In [2], Genzmer and Paul constructed analytic normal forms of topologically quasi-homogeneous functions, the holomorphic functions topologically equivalent to a quasi-homogeneous function.…”