A Eudoxian study of discriminant curves associated to normal surface singularities

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.