On supersolvable and nearly supersolvable line arrangements

TL;DR

This paper introduces nearly supersolvable line arrangements in the projective plane, showing they are either free or nearly free, and explores their algebraic and geometric properties, including implications for the Slope Problem.

Contribution

It defines nearly supersolvable arrangements and proves their minimal Jacobian syzygy degree is determined by combinatorics, linking algebraic invariants to geometric configurations.

Findings

Nearly supersolvable arrangements are either free or nearly free.

The minimal Jacobian syzygy degree is combinatorially determined.

Derived a version of the Slope Problem for points in the plane.

Abstract

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.