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.

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"

References in corpus (4)

• The Hesse pencil of plane cubic curves
• Resonance varieties, admissible line combinatorics and combinatorial pencils
• Equivalence classes of Latin squares and nets in CP^2
• A new bound on the number of special fibers in a pencil of curves

Cited by in corpus (3)

• Arrangements of curves and algebraic surfaces
• Exceptional Loci on ${\bar{M}}_{0,n}$ and Hypergraph Curves
• Equivalence classes of Latin squares and nets in CP^2