# On line arrangements with applications to 3-nets

## TL;DR

This paper establishes a correspondence between line arrangements and lines in projective space, classifies certain 3-net configurations over complex numbers, and introduces new properties and constructions for these nets.

## Contribution

It introduces a novel correspondence linking line arrangements to projective lines, enabling classification of (3,q)-nets and revealing new properties and realizability conditions.

## Key findings

- Nine realizable (3,6)-nets out of twelve possible cases.
- New properties for 3-nets, including moduli dimensions and field realization constraints.
- Construction of a (3,8)-net family related to the Quaternion group.

## Abstract

We show a one-to-one correspondence between arrangements of d lines in the projective plane, and lines in P^{d-2}. We apply this correspondence to classify (3,q)-nets over the complex numbers for all q<=6. When q=6, we have twelve possible combinatorial cases, but we prove that only nine of them are realizable. This new case shows several new properties for 3-nets: different dimensions for moduli, strict realization over certain fields, etc. We also construct a three dimensional family of (3,8)-nets corresponding to the Quaternion group.