The Enumerative Geometry of Hyperplane Arrangements

We study enumerative questions on the moduli space M(L) of hyperplane ar-rangements with a given intersection lattice L. Mnëv’s universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it is even difficult to compute the dimensionD = dimM(L). EmbeddingM(L) in a product of projective spaces, we study the degree N = degM(L), which can be interpreted as the number of arrangements inM(L) that pass through D points in general position. For generic arrangementsN can be computed combinatorially and this number also appears in the study of the Chow variety of zero dimensional cycles. We compute D and N using Schubert calculus in the case where L is the intersection lattice of the arrangement obtained by taking multiple cones over a generic arrangement. We also calculate the characteristic numbers for families of generic arrangements in P2 with 3 and 4 lines.