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Published online by Cambridge University Press: 24 October 2008
As particular cases of the general problem of finding the most general limiting tangential form of a degenerate manifold, we discuss here the degeneration of a curve on a surface into one or more multiple components, and the degeneration of a surface on a threefold into a single surface counting multiply, the method used consisting essentially in finding the approximate form of the curve or surface as the limit is approached. The approach system is supposed in all cases to be contained in a linear system of curves or surfaces, on the surface or threefold considered.
* We shall use freely, throughout this paper, one and the same symbol to denote a curve itself, and a function whose vanishing determines the curve.
† Zeuthen employed systems of this kind in considering the possible limiting forms of plane cubic and quartic curves: cf. Kjöbenhavn Skrifter (5), 10 (1873), 287. The general approach system of this kind is characterized geometrically by the fact that, in the representation of the system |K| by the points of a space of ρ dimensions, it is represented by a curve which passes through some point O of the (normal or projected) Veronese manifold M representing r-fold curves of the system |C|, while its osculating [i] at O. (i = 1, 2,…, r − 1) lies in the space T (i) whose points represent all curves of the form C r−iS (i), this space containing, but not necessarily coinciding with, the (i+ 1)-tangential space of M at O. For a triple line in the plane, for example, an approach system of plane cubics, which is to give six distinct limiting tangential points, must be represented in [9] by a curve osculating the Veronesean F 9 representing all threefold lines of the plane.
* The distances from C i of the sheets of K (λ) approaching C i vanish, of course, at the actual intersections of these sheets with C i and become indeterminate at the intersections of C i with the other limit curves; and at an intersection T of C i and C i a curve in the direction of C i, for example, meets K (λ) in only r j points near T, instead of in r i + r j points.
* The tangents from T to the curve π (Γ) have all definite directions along which their points of contact tend to T; but of course they need not all be real even when T is real. Of the r 1 + r 2 branches of K (λ) which pass near T, 2r 2 turn, after an odd number of bends, from the direction of C 2 to that of C 1, while the remaining r 1 − r 2 come from the direction of C 1 and continue in the same direction after an even number of infinitesimal bends near T.
* Cf., for example, the projection of a surface in ordinary space on to a multiple plane, in which the approach system can be taken as the system of transforms of the given surface by homologies of varying moduli, having the vertex and plane of projection as fundamental elements.