On the geometry of real or complex supersolvable line arrangements

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.

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