The nodal cubic surfaces and the surfaces from which they are derived by projection

Abstract

The cubic surfaces have been classified according to the character of their singularities by Schlafli and by Cayley, who find that there are 21 types in addition to ruled surfaces. In their treatment of the matter each case is considered separately by algebraical methods, and there is a marked lack of any simple unifying principle, which it is the object of this paper to supply. A means whereby this can be done is suggested by the theorem that every surface is the projection of a non-singular surface in higher space. The considerations employed in the proof of this result are somewhat abstruse, and the purely geometrical significance is obscure, so that the more detailed examination of particular cases is of genuine interest. Accordingly, the subject of this paper is the generation of the various nodal cubic surfaces by the projection of non-singular surfaces, specifically the non-ruled surfaces of order n in space of n dimensions (denoted throughout by F n ); it will be shown that these arise by the projection of the same surface, F 9 .