On the geometry of real or complex supersolvable line arrangements

Abstract

Given a rank 3 real arrangement 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 A is greater than or equal to n/2. With a much simpler proof we show that if 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.