  We introduce a new class of line arrangements in the projective plane, called
nearly supersolvable, and show that any arrangement in this class is either
free or nearly free. More precisely, we show that the minimal degree of a
Jacobian syzygy for the defining equation of the line arrangement, which is a
subtle algebraic invariant, is determined in this case by the combinatorics.
When such a line arrangement is nearly free, we discuss the splitting types and
the jumping lines of the associated rank two vector bundle, as well as the
corresponding jumping points, introduced recently by S. Marchesi and J.
Vall\`es. As a by-product of our results, we get a version of the Slope
Problem, looking for lower bounds on the number of slopes of the lines
determined by $n$ points in the affine plane over the real or the complex
numbers as well.
