Homology of Singular Cubic Surfaces in P(3,C)

An 1871 paper by Diekmann contains all the information needed to compute the integer singular homology for all classes of not ruled singular cubic surfaces. Three linearly independent quadrics in projective threespace deene a cubic Cremona transformation that associates to any point x the intersection of the polar planes of x with respect to the three quadrics. That intersection is a point y except if x is a vertex of a common polar tetrahedron of two quadrics. In that case, the image of x is a projective line. The locus of the exceptional points is a sextic space curve. Any cubic surface in space that is not ruled (i.e., is neither Cayley's surface nor a cone over an elliptic curve) can be obtained as the image of some projective plane in the map just described, known as Grassmann's generation of cubic surfaces 3], 11], 8]. If the six points do not lie on a conic and no three of them are collinear then the surface is regular; the images of the six conics through any ve points and the fteen diagonals give the 27 straight lines on the surface. Since a cubic surface with a singular line is ruled, the classiication of the singular cubic surfaces with isolated singularities is reduced to a study of the admissible conngurations of six points in the complex projective plane. There are much more than 21 combinatorially distinct conngurations; several conngurations can give surfaces of the same Schll aai class. A study of these equivalences and the determination of the nature of the singularities from the plane connguration in the case of one singular point has been given by Diekmann 2]. Diekmann's work leads immediately to a method for the determination of the singular homology over Z for all singular cubic surfaces that are not ruled. The method used was developed by the author long ago 4], 5], 6]. The connection with the counting formulas of algebraic geometry is not obvious in all cases. A modern classiication of cubic surfaces is given by J.W. Bruce and C.T.C. Wall 1]; they give the classiication in terms of Arnol'd`s classiication of singularities. In addition, they very carefully determine the moduli of the normal forms for the singularities; these are missing in most of the normal forms given by 10], pp. 324-325. The geometry of the singular cubic surfaces has been studied in detail by G. Salmon …