On the parameter dependence of the real cubic surfaces in arnold’s form

Abstract

We consider the topological classification of cubic surfaces which are obtained as intersection of the sphere\(\mathbb{S}^3 \) with the algebraic variety defined by the zeroes of a homogeneous cubic polynomial in Arnold’s normal form. This classification is based on the parameters appearing in this normal form, obtaining a correspondence between the parameters of the surface and its topological type. General classifications of cubic surfaces are made in the projective space ℙ3(ℝ), but our method, based on a very simple combinatorial procedure is easier to implement in\(\mathbb{S}^3 \). We split the cubic surfaces parameter space into ten equivalence classes.