3-nets realizing a group in a projective plane
Algebraic Geometry
2011-06-20 v3
Abstract
In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and comprises 3-nets realizing cyclic and direct products of two cyclic groups. Another known infinite family, due to Pereira and Yuzvinsky, comprises 3-nets realizing dihedral groups. We prove that there is no further infinite family. Urzua's 3-nets realizing the quaternion group of order 8 are the unique sporadic examples. If p is larger than the order of the group, the above classification holds true in characteristic p>0 apart from three possible exceptions Alt_4, Sym_4 and Alt_5.
Cite
@article{arxiv.1104.4439,
title = {3-nets realizing a group in a projective plane},
author = {Gabor Korchmaros and Gabor Nagy and Nicola Pace},
journal= {arXiv preprint arXiv:1104.4439},
year = {2011}
}