Smooth points of the space of plane foliations with a center

TL;DR

This paper proves that a certain class of logarithmic plane foliations, associated with generic line arrangements, are smooth points in the space of all degree d plane foliations with centers, highlighting their generic stability.

Contribution

It establishes the smoothness of the center set at logarithmic foliations derived from generic line arrangements with specific residue conditions.

Findings

Logarithmic foliations from generic line arrangements are smooth points in the center set.

The result applies to arrangements with pairwise natural and co-prime residues.

Provides insight into the geometric structure of the space of plane foliations.

Abstract

We prove that a logarithmic foliation corresponding to a generic line arrangement of $d+1 \geq 3$ lines in the complex plane, with pairwise natural and co-prime residues, is a smooth point of the center set of plane foliations (vector fields) of degree $d$.