Existence des diviseurs dicritiques, d’après S.S. Abhyankar

Abstract

In this article, there are new proofs of the existence and unicity of dicritical divisors of a pencil of plane curves of $⟨F,G⟩$ Incidentally, we prove the equivalence between dicritical divisors and Rees valuations. Furthermore, in the case where $G_{\mathrm{red}}$ is regular at the base points of $⟨F,G⟩$, we have that $F/G$ is residually a polynomial along any dicritical divisor; this reproves geometrically [2, Theorem (7.1)]. As a corollary of the latter proof, we get a generalization of the connectedness theorem of [8].