Degree of logarithmic foliations of type (1,1,1)

TL;DR

This paper computes the projective degree of a specific component of degree 1 holomorphic foliations in projective space, characterized by logarithmic differential forms with simple poles, using blow-up techniques.

Contribution

It introduces a method to determine the projective degree of a component of logarithmic foliations via resolution of rational maps with blow-ups.

Findings

Computed the projective degree of the component.

Resolved the rational parametrization map through blow-ups.

Characterized the general element as a logarithmic differential form with simple poles.

Abstract

The space of codimension one holomorphic foliations of degree 1 in a projective space has an irreducible component whose general element is a logarithmic differential 1-form with simple poles in three hyperplanes. We compute its projective degree by resolving its rational parametrization map through succesive blow-ups with smooth centers.