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Cohomology bases for the De Concini-Procesi models of hyperplane arrangements and sums over trees. (English) Zbl 0989.14006

Introduction: Let M 0,n+1 be the moduli space of n+1-tuples of distinct points on 1 modulo projective automorphisms. Since a projective automorphism of 1 is uniquely defined by the images of three points, M 0,n+1 can be regarded as the complement in n-2 to the projective hyperplane arrangement given by the polynomial z 1 ,,z n-1 i<j (z i -z j ). More symmetrically, one can interpret M 0,n+1 × as the complement in n-1 to the projectivization of the n-braid arrangement i<j (z i -z j ). The space M 0,n+1 has a canonical compactification M ¯ 0,n+1 that is a closed 2n-4-dimensional complex manifold whose cohomology ring R(n)=H * (M ¯ 0,n+1 ) plays an important part in algebraic geometry, field theory, and theory of operads. A presentation of R(n) 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 M ¯ of the complement M of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring H * (M ¯).

The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring R=H * (M ¯) 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 R for the reflection arrangements of classical types B n (=C n ) and D n . 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 A n-1 is a braid arrangement, i.e., M ¯=M ¯ 0,n+1 , 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 Δ of monomials in R and prove that it generates R. In section 3 we use the Poincaré duality to prove that it is linearly independent. In section 4 we give a combinatorial description of Δ for reflection arrangements of classical types. In section 5 we use this description to compute the Hilbert series of R for these arrangements.

14F25Classical real and complex cohomology
52C35Arrangements of points, flats, hyperplanes
14D20Algebraic moduli problems, moduli of vector bundles
14N10Enumerative problems (algebraic geometry)