ON A MODULAR COMPACTIFICATION OF Gln

In this paper we give a modular description of a certain compactification KGln of the general linear group Gln. As a corollary we obtain a modular description of the so-called “wonderful compactification” (cf. [CP]) of the homogenous space PGln = (PGln × PGln)/PGln. 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 moduli spaces of vectorbundles on singular curves. Let me explain briefly the relevance of compactifications of Gln in this context. Let C0 be an irreducible projective algebraic curve (over C, say) with one ordinary double point and let C̃0 be the normalization of C0 and p1, p2 ∈ C̃0 the two points lying above the singular point of C0. To give a vector bundle E on C0 is the same as to give a vector bundle Ẽ on C̃0 plus an isomorphism from its fiber at p1 to its fiber at p2. This indicates that the moduli space of vector bundles of rank n on C0 is a principal Gln-bundle over the space of vector bundles of rank n on C̃0. In particular, this moduli space is not compact. Thus to obtain a compact moduli space, one has to look for more general objects, than just vectorbundles on C0. To my knowledge there are currently two approaches to this problem. One is, to consider torsion-free sheaves on C0 (cf. [S], [F]). The other is by Gieseker (cf. [G]),