  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.
