Non-degeneracy of the discriminant

TL;DR

This paper investigates conditions under which the discriminant curve of a holomorphic map germ from 2 to 2 is non-degenerate in the Kouchnirenko sense, focusing on the geometric and algebraic properties of the map.

Contribution

It characterizes pairs of holomorphic map germs with smooth curves and their discriminant curves that are non-degenerate in the Kouchnirenko sense.

Findings

Identifies conditions for non-degeneracy of the discriminant curve.

Provides criteria linking the geometry of the map to algebraic non-degeneracy.

Enhances understanding of the discriminant's structure in holomorphic mappings.

Abstract

Let $(l,f):(C^2,0)\rightarrow (C^2,0)$ be the germ of a holomorphic mapping such that $l=0$ is a smooth curve which is not a branch of the singular curve $f=0$. The direct image of the critical locus of this mapping is called the discriminant curve. In this paper we study the pairs $(l,f)$ for which the discriminant curve is non-degenerate in the Kouchnirenko sense.