Location of the real cubic surfaces in the parameter space

We consider the space C of non-singular cubic surfaces W(a, b, c, d) which are obtained as intersection of the sphere §3 with the algebraic variety x3 + y3 + z3 + w3 + (ax + by + cz+ dw)3 = 0. By using some symmetries, we determine ten different classes of equivalence of C. Through a simple geometrical method, based on the observation that the problem can be reduced to enumerating the connected components of the complement of a certain piecewise-linear set in the 4-dimensional space, we give a partition of R4 in accordance with the different equivalence classes. The novelty of our approach is the non-algebraic way with which the problem is treated. © 2008 Birkhäuser Verlag Basel/Switzerland.