Realization of finite Abelian groups by nets in P^2

TL;DR

This paper investigates special line and point configurations called k-nets in the complex projective plane, revealing restrictions on their types and linking 3-nets to finite Abelian groups, with most realizable groups being subgroups of a 2-torus.

Contribution

It establishes new restrictions on the possible values of k for nets and characterizes the groups realizable by 3-nets in P^2, connecting geometric configurations to algebraic structures.

Findings

k can only be 3, 4, or 5

All known 3-net examples realize finite Abelian groups

Most realizable groups are subgroups of a 2-torus

Abstract

In the paper, we study special configurations of lines and points in the complex projective plane, so called k-nets. We describe the role of these configurations in studies of cohomology on arrangement complements. Our most general result is the restriction on k - it can be only 3,4, or 5. The most interesting class of nets is formed by 3-nets that relate to finite geometries, latin squares, loops, etc. All known examples of 3-nets in P^2 realize finite Abelian groups. We study the problem what groups can be so realized. Our main result is that, except for groups with all invariant factors under 10, realizable groups are isomorphic to subgroups of a 2-torus. This follows from the `algebraization' result asserting that in the dual plane, the points dual to lines of a net lie on a plane cubic.