Varieties representing certain systems of Schubert triangles

Triangle models and relations between them

“…The model of the complete system of triangles lying in a plane is a sixfold, whose nature of course depends on the chosen definition of the geometric variable. An earlier paper [1] concentrated on the variety associated with the Schubert triangle, although it contained brief descriptions of the corresponding models for other types. We now study the relation between these sixfolds, paying particular attention to the partial dilatation which transforms the model of ordered triangles into the Schubert variety.…”

“…The model SI* of the ordered line-triads (a, b, c) in a plane n is the Segre product of three planes, which is a non-singulai sixfold F 6 9O [26]. If (« 0 (r) , uf\ w 2 (r) ) (r = 1,2,3) are the coordinates of the lines a, b, c respectively, the equations of SI* can be conveniently written in the form y 9i+3J+k = «, (1) «/ 2) «* (3) ft j, * = o, 1, 2). (i)…”

“…and the equations (7) and (8) are those of the sixfold H, which is the natural model of the complete system of ordered triangles (A, B, C, a, b, c) in n. We now list the degeneration varieties on Q, indicating in each case the system of ordered triangles represented: To exhibit H as a transform of Q*, we obtain the X-coordinates as functions of the ^-coordinates. The incidence relations (7) lead to x 0 (1) = w 1 (2) w 2 <3) -M 2 (2> WI (3) > etc., and we substitute expressions of this type in x, (1) x/ 2) x fc (3) . Because the general term in the expanded product can be expressed in four different ways as the product of two ^-coordinates, it is possible in many ways to write x t a) x/ 2) x k (3) as a quadratic function of the j-coordinates.…”