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A Eudoxian study of discriminant curves associated to normal surface singularities

Evelia Rosa García Barroso, Patrick Popescu-Pampu

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, Gwoździewicz and Parusiński, which is the special case in which $(S,s)$ is smooth. Our proof uses a common generalization of formulas of Lê, Casas-Alvero and Némethi 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 Gwoździewicz 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.

A Eudoxian study of discriminant curves associated to normal surface singularities

Abstract

Let be a finite morphism from a germ of normal complex analytic surface to the germ of at the origin. We show that the affine algebraic curve in defined by the initial Newton polynomial of a defining series of the discriminant germ of depends up to toric automorphisms only on the germs of curves defined by and . This result generalizes a theorem of Gryszka, Gwoździewicz and Parusiński, which is the special case in which is smooth. Our proof uses a common generalization of formulas of Lê, Casas-Alvero and Némethi 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 , for varying positive integral exponents , following a strategy initiated by Gwoździewicz 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.

Paper Structure

This paper contains 11 sections, 29 theorems, 79 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background on normal surface singularities and their intersection theory
  3. A theorem of Delgado and Maugendre on special members of pencils
  4. A generalization of formulas of Casas-Alvero, Lê and Némethi
  5. The Eudoxian strategy
  6. Relations between the discriminants of a family of finite morphisms
  7. Expressing tropical functions of divisors using intersection numbers
  8. Application to the tropical functions of discriminant divisors
  9. Combinatorial invariance of jacobian Newton polygons
  10. Comparison of divisorial and Fitting critical loci and discriminants
  11. Invariance of the initial Newton curve of a finite morphism

Key Result

Proposition 2.2

A morphism $\psi : (X,x) \to (Y,y)$ is finite if and only if it admits a representative such that $\psi^{-1}(y) = \{x\}$.

Figures (3)

  • Figure 1: The orientations of the compact edges of a Newton polygon (see Definition \ref{['def:Newtpol']})
  • Figure 2: The notations of Definition \ref{['def:asspol']}
  • Figure 3: The Newton polygon of the strict transform of $Z(f^b - \lambda g^a)$ when one has relation \ref{['eq:factorpb']}.

Theorems & Definitions (81)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Proposition 2.7
  • Proposition 2.8
  • proof
  • Definition 3.1
  • ...and 71 more