Existence des diviseurs dicritiques, d'apr\`es S.S.Abhyankar

TL;DR

This paper explores dicritical divisors in singular plane foliations, providing a geometric interpretation and new proofs of their existence based on Abhyankar's algebraic generalization.

Contribution

It offers a geometric perspective on dicritical divisors and establishes their existence using novel proofs inspired by Abhyankar's algebraic approach.

Findings

Geometric interpretation of dicritical divisors

New proofs of their existence

Extension of Abhyankar's algebraic definition

Abstract

In geometric terms, given a singular foliation of the plane, a dicritical divisor is (whenever it exists) an irreducible component of the exceptional divisor which is transverse to the foliation. Abhyankar gave recently a definition of the dicritical divisors which generalize and algebraicize the geometrical definition in the local case and the polynomial case. Following his work, we give a geometrical interpretation of these dicritical divisors and new proofs of their existence.