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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 \(\mathbb{C}\mathbb{P}^1\) modulo projective automorphisms. Since a projective automorphism of \(\mathbb{C}\mathbb{P}^1\) is uniquely defined by the images of three points, \(M_{0,n+1}\) can be regarded as the complement in \(\mathbb{C} \mathbb{P}^{n-2}\) to the projective hyperplane arrangement given by the polynomial \(z_1,\dots, z_{n-1}\prod_{i<j} (z_i-z_j)\). More symmetrically, one can interpret \(M_{0,n+1} \times\mathbb{C}\) as the complement in \(\mathbb{C}\mathbb{P}^{n-1}\) to the projectivization of the \(n\)-braid arrangement \(\prod_{i<j}(z_i-z_j)\). 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 \(R(n)=H^* (\overline M_{0,n+1})\) 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 \(\overline M\) of the complement \(M\) of any complex projective hyperplane (even subspace) arrangement. They also gave a presentation for the cohomology ring \(H^* (\overline M)\).

The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring \(R=H^*(\overline M)\) as a free \(\mathbb{Z}\)-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 \(B_n(=C_n)\) 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., \(\overline M=\overline M_{0,n+1}\), 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.

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 \(H^* (\overline M)\).

The goal of this paper is to use this presentation in order to construct a monomial basis of the graded ring \(R=H^*(\overline M)\) as a free \(\mathbb{Z}\)-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 \(B_n(=C_n)\) 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., \(\overline M=\overline M_{0,n+1}\), 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 (co)homology in algebraic geometry |

52C35 | Arrangements of points, flats, hyperplanes (aspects of discrete geometry) |

14D20 | Algebraic moduli problems, moduli of vector bundles |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |