Smooth Points of the Space of Plane Foliations with a Center.

The article focuses on the structure of the center set \(\mathcal{M}(d)\) of complex polynomial plane vector fields (foliations) of degree \(d\), particularly proving that logarithmic foliations associated with generic line arrangements of \(d+1 \geq 3\) lines in the complex plane, having pairwise natural and coprime residues, are smooth points of \(\mathcal{M}(d)\). It establishes that the irreducible algebraic set \(\mathcal{L}(1^{d+1})\) of such logarithmic foliations forms an irreducible component of the center set, with the tangent space at these points characterized via Melnikov functions and cohomological conditions involving logarithmic differential forms. The paper develops the Picard–Lefschetz theory for fibrations defined by products of powers of linear polynomials, computes intersection indices of vanishing cycles, and analyzes the monodromy action on homology to support these results. Additionally, it reviews the known classification of irreducible components of \(\mathcal{M}(2)\) (quadratic case), describing their singularities and relating them to classical families such as Lotka–Volterra and Hamiltonian foliations.