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

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

2004-11-10

arxiv.org/abs/math/0411235v3

Mathematics

Algebraic Geometry

To appear in the proceedings of the Conference "Projective varieties with unexpected properties", in honour of the 150-th anni

Scientific paper

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.

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