Complete collineations revisited

This paper takes a new look at some old spaces. The old spaces are the moduli spaces of complete collineations, introduced and explored by many of the leading lights of 19th-century algebraic geometry, such as Chasles, Schubert, Hirst, and Giambelli. They are roughly compactifications of the spaces of linear maps of a fixed rank between two fixed vector spaces, in which the boundary added is a divisor with normal crossings. This renders them useful in solving many enumerative problems on linear maps, and they are famous as much for the intricacy of the resulting formulas as for the elegance and symmetry of the underlying geometry. The new look comes from some recent quotient constructions in algebraic geometry: the Chow quotient and inverse limit of Mumford quotients, introduced in 1991 by Kapranov, Sturmfels, and Zelevinsky, and the Hilbert quotient, essentially due to Bialynicki-Birula and Sommese in 1987. These were motivated partly by the search for a quotient more canonical than the Mumford or geometric invariant theory quotient, which depends on the choice of a linearization, and partly by the attractive polyhedral interpretations possessed by all three quotients in the setting of toric geometry. But for us, their chief interest arose later, in two papers of Kapranov from 1993. They showed, among many other things, that Chow quotients, and several related operations, could be used to give elegant new constructions of the moduli space M 0,n of stable punctured curves of genus zero. This paper will demonstrate that every one of Kapranov’s constructions has a counterpart for complete collineations. In fact, one piece of the puzzle was already in place: Vainsencher had already shown in 1984 that the complete collineations could be obtained by a blowup construction very similar to one given by Kapranov for M 0,n . But all the other pieces also fit neatly. For example, M 0,n is constructed by Kapranov as the Chow quotient of a Grassmannian by a (C) -action; likewise, the complete collineations are constructed here as the Chow quotient of a Grassmannian by a C -action.