Realization of finite Abelian groups by nets in P²

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.