The 3-cuspidal quartic and braid monodromy of degree 4 coverings

TL;DR

This paper investigates the discriminant surface of degree 4 extensions defined by deformed bidouble covers, computes the braid monodromy of a related quartic curve, and clarifies classical geometric theorems involving cuspidal quartic surfaces.

Contribution

It introduces a detailed analysis of the discriminant of degree 4 covers, computes the braid monodromy of a specific quartic curve, and corrects a classical theorem regarding quartic surfaces with a twisted cubic cuspidal curve.

Findings

Discriminant surface is a cuspidal quartic on a twisted cubic.

Computed the braid monodromy of a 3-cuspidal affine quartic.

Corrected a classical proof about quartic surfaces with twisted cubic cuspidal curve.

Abstract

We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the discriminant of the general equation of degree 3. We then take a(u,v), b(u,v) and get a 3-cuspidal affine quartic curve whose braid monodromy we compute. This calculation of the local braid monodromy is a step towards the determination of global braid monodromies, e.g. for the (a,b,c) surfaces previously considered by the authors. In the revision we fill a gap (noticed by the referee) in the proof of the classical theorem that any quartic surface which has the twisted cubic as cuspidal curve is its tangential developable, and we changed a base point in order to make a picture correct.