Degenerate singularities of one dimensional Foliations

TL;DR

This paper derives formulas for the degrees of spaces of one-dimensional foliations in the projective plane with specific singularities, analyzing their geometric properties after blow-ups and contact conditions.

Contribution

It provides explicit formulas for the degrees of foliation spaces with dicritical singularities and studies the geometric behavior after blowing up such singularities.

Findings

Formulas for degrees of foliations with dicritical singularities

Analysis of foliation behavior after blowing up singularities

Degree of locus with leaves of total contact with exceptional line

Abstract

We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the exceptional line; we also find the degree of the locus defined by imposing a leaf of total contact with the exceptional line.