On line arrangements with applications to 3-nets
Abstract
We show a one-to-one correspondence between arrangements of d lines in the projective plane, and lines in P^{d-2}. We apply this correspondence to classify (3,q)-nets over the complex numbers for all q<=6. When q=6, we have twelve possible combinatorial cases, but we prove that only nine of them are realizable. This new case shows several new properties for 3-nets: different dimensions for moduli, strict realization over certain fields, etc. We also construct a three dimensional family of (3,8)-nets corresponding to the Quaternion group.
Cite
@article{arxiv.0704.0469,
title = {On line arrangements with applications to 3-nets},
author = {Giancarlo Urzua},
journal= {arXiv preprint arXiv:0704.0469},
year = {2009}
}
Comments
19 pages, to appear in Advances in Geometry. Sections 5, 6 and 7 of old version are worked out in more detail at the beginning of "Arrangements of rational sections over curves and the varieties they define"