A Eudoxian study of discriminant curves associated to normal surface singularities

TL;DR

This paper studies the discriminant curves of finite morphisms from normal surface singularities to the plane, showing their dependence on the initial Newton polynomial up to toric automorphisms, generalizing previous smooth cases.

Contribution

It extends known results about discriminant curves from smooth to normal surface singularities, using a unified approach involving intersection formulas and pencils of curves.

Findings

Discriminant curves depend only on the germs of the defining curves up to toric automorphisms.

Generalization of Gryszka, Gwoździewicz, and Parusiński's theorem to normal surface singularities.

Application of the method to pencils generated by pairs of powers of the defining functions.

Abstract

Let $(f,g): (S,s) \to (\mathbb{C}^2, 0)$ be a finite morphism from a germ of normal complex analytic surface to the germ of $\mathbb{C}^2$ at the origin. We show that the affine algebraic curve in $\mathbb{C}^2$ defined by the initial Newton polynomial of a defining series of the discriminant germ of $(f,g)$ depends up to toric automorphisms only on the germs of curves defined by $f$ and $g$. This result generalizes a theorem of Gryszka, Gwo\'zdziewicz and Parusi\'nski, which is the special case in which $(S,s)$ is smooth. Our proof uses a common generalization of formulas of L\^e, Casas-Alvero and N\'emethi for the intersection number of the discriminant with a germ of plane curve. It uses also a theorem of Delgado and Maugendre characterizing the special members of pencils of curves on normal surface singularities. We apply it to the pencils generated by all pairs $(f^b, g^a)$, for varying positive integral exponents $a, b$, following a strategy initiated by Gwo\'zdziewicz and by Delgado and Maugendre. This is similar to the Eudoxian method of comparison of magnitudes by comparing the sizes of their positive integral multiples.