Abstract
Let X = X (3, 6) be the configuration space of six lines in general position in the projective plane; the space can be thought of the quotient space
X = GL (3) ? M*(3, 6; C)/(C*)6,
where M*(3, 6; C) is the set of 3 × 6 complex matrices such that any 3 by 3 minor does not vanish. In the previous paper [MSY], we constructed a projective compactification X of X and showed that it is the Satake compactification of the quotient of the 4-dimensional Hermitian symmetric domain H (2, 2) = {z∈M (2, 2; C)|(z − z*)/2i > 0} by an arithmetic group, say Λ acting on H (2, 2).
In this paper, we make a detailed study of combinatorial properties of the smooth affine variety X, the singular projective variety X and the parabolic parts of the arithmetic group Λ, and make, as an application of these, a toroidal compactification of X, which gives a non-singular model of X. We also treat several other arithmetic subgroups commensurable with Λ. As an appendix, we study the variety X as a toric variety.
