On infinitesimal triangles

Varieties representing certain systems of Schubert triangles

“…All twelve of these varieties are simple on U with the single exception of ij/, which Semple [2, §3] has shown to be a double threefold, the nodal cone of U at each point 0 of ij/ being a proper [3]-cone with the tangent [3] to ^ at 0 as its vertex. This nodal cone is generated by a doubly-infinite system of [4]'s, and also by each of two simply-infinite systems of [5]'s. The tangent [5]'s at O to B A , B B , B c belong to one system, and the tangent [5]'s at 0 to s and f belong to the other.…”

“…Then the coordinates (omitting zeros, coefficients and repetitions) of points on F (4) are of the form u f v 9 w 18~f~9 (/, g = 0,1, ..., 9), so that the surface admits of a plane representation by the system of curves C IS (U 9 , V 9 ). The required model, which is the transform of a quadric surface by the complete system of novenic surfaces in its ambient space, is thus a rational surface 64 F 162 [99].…”

“…(i) the ^-triangles of E (4) , in which the contact points A and C coincide, are mapped on F (4) by a rational curve of order 18; and…”

“…The order of F (4) can be checked independently. By repeated applications of the method of degenerating collineations, we have found a minimal base for surfaces on SI, and we know how to obtain an equivalence relation for F (4) in terms of the elements of this base [3, Chapter III and Appendix I].…”

“…The order of F (4) can be checked independently. By repeated applications of the method of degenerating collineations, we have found a minimal base for surfaces on SI, and we know how to obtain an equivalence relation for F (4) in terms of the elements of this base [3, Chapter III and Appendix I]. Let us denote by A the fivefold on Q, corresponding to the condition that the vertex A should lie on a fixed straight line, by A 2 the fourfold corresponding to the condition that A should lie at a fixed point, etc.…”