On the geometry of real or complex supersolvable line arrangements

Abstract

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.

13 pages, 3 figures