THE NON-SINGULAR CUBIC SURFACES

Abstract

THIS is a very remarkable monograph ; it is a direct product of war circumstances. After a curt verification of the existence (Cayley, Salmon, 1849) of a symmetrical system of twenty-seven lines, each met by five pairs of mutually intersecting lines, the author turns to a diagrammatic representation of the lines, by the joining segments of nine points which lie in threes on three coplanar concurrent lines. This is reached by considering how the lines would vary in a continuous deformation of the surface into three planes, and, beautifully executed as they are, the various diagrams serve the author's purpose well. Actually, the representation is the dual of one considered by Bennett, in which the lines of the surface are represented by the points in which the rays of three coplanar pencils, α, β, γ, each of three rays, meet one another ; in this representation a line of the (Steiner) system of nine lines, represented by the intersection of a ray b of the pencil β, with a ray c of the pencil γ meets the four lines indicated by the intersections of the other rays of these two pencils, and meets the six lines indicated by the intersections of b and c with the rays of the pencil α. This law of intersection is unaltered by the interchange of the rays of a pencil among themselves, or by the interchange of the pencils.

The Non-Singular Cubic Surfaces

A New Method of Investigation with Special Reference to Questions of Reality. By B. Segre. Pp. xi + 180. (Oxford : Clarendon Press ; London : Oxford University Press, 1942.) 15s. net.