2015/01/16 by Anzis, Benjamin, Tohaneanu, Stefan · 1 citation
Engineering · Mathematics · #05B35 #06C10 (Secondary) #52C30 (Primary) #52C35 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1501.04039
Given a rank 3 real arrangement \mathcal A of n lines in the projective plane, the Dirac-Motzkin conjecture (proved by Green and Tao in 2013) states that for n sufficiently large, the number of simple intersection points of \mathcal A is greater than or equal to n/2. With a much simpler proof we show that if \mathcal A is supersolvable, then the conjecture is true for any n (a small improvement of original conjecture). The Slope problem (proved by Ungar in 1982) states that n non-collinear points in the real plane determine at least n-1 slopes; we show that this is equivalent to providing a lower bound on the multiplicity of a modular point in any (real) supersolvable arrangement. In the second part we find connections between the number of simple points of a supersolvable line arrangement, over any field of characteristic 0, and the degree of the reduced Jacobian scheme of the arrangement. Over the complex numbers even though the Sylvester-Gallai theorem fails to be true, we conjecture that the supersolvable version of the Dirac-Motzkin conjecture is true.