On line arrangements with applications to 3-nets

Details On line arrangements with applications to 3-nets On line arrangements with applications to 3-nets

2007-04-03

arxiv.org/abs/0704.0469v4

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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