Formalized proof, computation, and the construction problem in algebraic geometry

An informal discussion of how the construction problem in algebraic geometry motivates the search for formal proof methods. Also includes a brief discussion of my own progress up to now, which concerns the formalization of category theory within a ZFC-like environment

On supersolvable and nearly supersolvable line arrangements

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.Comment: v.3, a version of the Slope Problem, valid over the real and the complex numbers as well, is obtained, see Thm. 1.1 and Thm. 4.