Introduction: Let be the moduli space of -tuples of distinct points on modulo projective automorphisms. Since a projective automorphism of is uniquely defined by the images of three points, can be regarded as the complement in to the projective hyperplane arrangement given by the polynomial . More symmetrically, one can interpret as the complement in to the projectivization of the -braid arrangement . The space has a canonical compactification that is a closed -dimensional complex manifold whose cohomology ring plays an important part in algebraic geometry, field theory, and theory of operads. A presentation of was found by S. Keel [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)].
Recently C. de Concini and C. Procesi [Sel. Math., New Ser. 1, No. 3, 459-494 (1995; Zbl 0842.14038) and 495-535 (1995; Zbl 0848.18004)] generalized the construction. They defined a compactification of the complement of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring .
The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring as a free -module. We also study and use the Poincaré pairing on the basis. Then we use this basis to compute the Hilbert series of for the reflection arrangements of classical types and . We reduce the computation to summation over trees and apply the method of Yu. I. Manin [Prog. Math. 129, 401-417 (1995; Zbl 0871.14022)]. Arrangement of type is a braid arrangement, i.e., , and the Hilbert series has been computed by Manin (loc. cit.). We recover this result in our computation.
In section 2 we construct a set of monomials in and prove that it generates . In section 3 we use the Poincaré duality to prove that it is linearly independent. In section 4 we give a combinatorial description of for reflection arrangements of classical types. In section 5 we use this description to compute the Hilbert series of for these arrangements.