2014/09/22 by Paul, Thomas, Traves, Will, Wakefield, Max
#14N15 #14N20 #52C35 #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1409.6275
We study enumerative questions on the moduli space M(L) of hyperplane arrangements 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 dimension D =dim M(L). Embedding M(L) in a product of projective spaces, we study the degree N=deg M(L), which can be interpreted as the number of arrangements in M(L) that pass through D points in general position. For generic arrangements N 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.