The Enumerative Geometry of Hyperplane Arrangements

TL;DR

This paper investigates the enumerative geometry of hyperplane arrangements, focusing on moduli space dimensions and degrees, using Schubert calculus to compute these invariants for specific arrangements and configurations.

Contribution

It introduces methods to compute the dimension and degree of moduli spaces of hyperplane arrangements with given intersection lattices, expanding understanding of their geometric complexity.

Findings

Computed the dimension and degree of moduli spaces for specific arrangements.

Derived combinatorial formulas for the number of arrangements passing through general points.

Calculated characteristic numbers for arrangements in projective planes with 3 and 4 lines.

Abstract

We study enumerative questions on the moduli space $\mathcal{M}(L)$ of hyperplane arrangements with a given intersection lattice $L$. Mn\"ev's universality theorem suggests that these moduli spaces can be arbitrarily complicated; indeed it is even difficult to compute the dimension $D =\dim \mathcal{M}(L)$. Embedding $\mathcal{M}(L)$ in a product of projective spaces, we study the degree $N=\mathrm{deg} \mathcal{M}(L)$, which can be interpreted as the number of arrangements in $\mathcal{M}(L)$ 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 $\mathbb{P}^2$ with 3 and 4 lines.