Intersection rings of spaces of triangles
摘要
In 1880 Schubert In 1954 Semple [13] gave a modern construction of this space, which we denote X, as an algebraic submanifold of a product of projective and Grassmann manifolds. Tyrrell [15] verified Schubert's prescription of the cycles and their relations in codimension one, and calculated a few other intersection products. The aim of this note is to complete this analysis. We give a formula for the Chow ring (or cohomology ring) of this space: it is generated by seven classes in codimension one, with an ideal of relations generated by twelve classes. In particular we verify that Schubert's basis is correct in all dimensions, and the intersections are as he specified. It is interesting, however, that one of the defining relations for the intersection ring is independent of those given by Schubert before he lists the basis.