A Modular Compactification of the General Linear Group

In this paper we give a modular description of a certain compactification KGln of the general linear group Gln. The variety KGln is constructed as follows: First one embeds Gln in the obvious way in the projective space which contains the affine space of n× n matrices as a standard open set. Then one successively blows up the closed subschemes defined by the vanishing of the r×r subminors (1 ≤ r ≤ n), along with the intersection of these subschemes with the hyperplane at infinity. We were led to the problem of finding a modular description of KGln in the course of our research on the degeneration of moduli spaces of vector bundles. Let me explain in some detail the relevance of compactifications of Gln in this context. Let B be a regular integral one-dimensional base scheme and b0 ∈ B a closed point. Let C → B be a proper flat familly of curves over B which is smooth outside b0 and whose fibre C0 at b0 is irreducible with one ordinary double point p0 ∈ C0. Let C̃0 → C0 be the normalization of C0 and let p1, p2 ∈ C̃0 the two points lying above the singular point p0.