Realization of finite Abelian groups by nets in $\mathbb{P}^2$

Abstract

In this 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 equal 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 $\mathbb{P}^2$ realize finite Abelian groups. We study the question of which groups can be realized in this way. Our main result is that, except for groups with all invariant factors below 10, the 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.