by  S.Yu. Orevkov

Abstract:

Let   p   be a singular point of a complex analytic surface. It is called (by Abhyankar) quasirational if each irreducible component of the resolution is a rational curve.

We prove that the singularity of the form $z^g + f(x,y) = 0$ is quasirational if at least one of the numbers   m, n   is coprime with   g   where   (m,n)   is the first Puiseux pair of the curve   f(x,y) = 0 . The proof is topological and it is based on a study of roots of the Alexander polynomial of the link of the singularity.

In the case when   f = 0   has only one Puiseux pair   (m,n)   and all the numbers   m, n, g   are pairwise coprime, this fact was proven by Abhyankar.

Also we formulate (without proofs) a topological description of the resolution of $z^g + f(x,y) = 0$ in general case (without any assumption on gcd(n,g) and gcd(m,g)).