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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 $ℂ{ℙ}^{1}$ modulo projective automorphisms. Since a projective automorphism of $ℂ{ℙ}^{1}$ is uniquely defined by the images of three points, ${M}_{0,n+1}$ can be regarded as the complement in $ℂ{ℙ}^{n-2}$ to the projective hyperplane arrangement given by the polynomial ${z}_{1},\cdots ,{z}_{n-1}{\prod }_{i. More symmetrically, one can interpret ${M}_{0,n+1}×ℂ$ as the complement in $ℂ{ℙ}^{n-1}$ to the projectivization of the $n$-braid arrangement ${\prod }_{i. The space ${M}_{0,n+1}$ has a canonical compactification ${\overline{M}}_{0,n+1}$ that is a closed $2n-4$-dimensional complex manifold whose cohomology ring $R\left(n\right)={H}^{*}\left({\overline{M}}_{0,n+1}\right)$ plays an important part in algebraic geometry, field theory, and theory of operads. A presentation of $R\left(n\right)$ 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 $\overline{M}$ of the complement $M$ of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring ${H}^{*}\left(\overline{M}\right)$.

The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring $R={H}^{*}\left(\overline{M}\right)$ as a free $ℤ$-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 ${B}_{n}\left(={C}_{n}\right)$ 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., $\overline{M}={\overline{M}}_{0,n+1}$, 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 ${\Delta }$ 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 ${\Delta }$ for reflection arrangements of classical types. In section 5 we use this description to compute the Hilbert series of $R$ for these arrangements.

##### MSC:
 14F25 Classical real and complex cohomology 52C35 Arrangements of points, flats, hyperplanes 14D20 Algebraic moduli problems, moduli of vector bundles 14N10 Enumerative problems (algebraic geometry)