Realization of finite Abelian groups by nets in P^2

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.

14 pages

Cited by in corpus (12)

• Completely reducible hypersurfaces in a pencil
• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Old and new examples of k-nets in P^2
• The Classification of Exceptional CDQL Webs on Compact Complex Surfaces
• Resonance varieties, admissible line combinatorics and combinatorial pencils
• Classification of k-nets
• Using Tropical Degenerations For Proving The Nonexistence Of Certain Nets
• Some computational results on small 3-nets embedded in a projective plane over a field
• Group-labeled light dual multinets in the projective plane (with Appendix)
• The Only Complex 4-Net Is the Hesse Configuration
• Probabilities of incidence between lines and a plane curve over finite fields
• 3-nets realizing a diassociative loop in a projective plane