GENERALIZED POINCARE-SERIES FOR MODELS OF THE BRAID ARRANGEMENTS

Let A(n-1) subset of Cn-1 be the complexified Coxeter arrangement of h yperplanes of type A(n-1) (n greater than or equal to 3). It is well k nown that the ''minimal'' projective De Concini-Procesi model YAn-1 of A(n-1) is isomorphic to the moduli space (M) over bar(0,) (n+1) of st able n + 1-pointed curves of genus 0. In this paper we study, from the point of view of models of arrangements, the action of the symmetric group Sigma(n) on the integer cohomology ring R(A(n-1)) of YAn-1. In f act we find a formula for the generalized Poincare series which encode s all the information about this representation of Sigma(n). This form ula, which is obtained by using the elementary combinatorial propertie s of a Z-basis of R(A(n-1)) and turns out to be very direct, should be compared with a more general result due to Getzler (see [5]).