A Greedy Algorithm to Compute Arrangements of Lines in the Projective Plane

We introduce a greedy algorithm optimizing arrangements of lines with respect to a property. We apply this algorithm to the case of simpliciality: it recovers all known simplicial arrangements of lines in a very short time and also produces a yet unknown simplicial arrangement with 35 lines. We compute a (certainly incomplete) database of combinatorially simplicial complex arrangements of hyperplanes with up to 50 lines. Surprisingly, it contains several examples whose matroids have an infinite space of realizations up to projectivities.

A rigid, not infinitesimally rigid surface with K ample

We produce an example of a rigid, but not infinitesimally rigid smooth compact complex surface with ample canonical bundle using results about arrangements of lines inspired by work of Hirzebruch, Kapovich & Millson, Manetti and Vakil.

Double Points of Free Projective Line Arrangements

We prove the Anzis–Tohăneanu conjecture, that is, the Dirac–Motzkin conjecture for supersolvable line arrangements in the projective plane over an arbitrary field of characteristic zero. Moreover, we show that a divisionally free arrangements of lines contain at least one double point that can be regarded as the Sylvester–Gallai theorem for some free arrangements. This is a corollary of a general result that if you add a line to a free projective line arrangement, then that line has to contain at least one double point. Also, we prove some conjectures and one open problems related to supersolvable line arrangements and the number of double points.

Special arrangements of lines: Codimension 2 ACM varieties in ℙ1 × ℙ1 × ℙ1

In this paper, we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimension 2 arithmetically Cohen–Macaulay (ACM) varieties in [Formula: see text], called varieties of lines. We also describe their ACM property from a combinatorial algebra point of view.

Splitting types of bundles of logarithmic vector fields along plane curves

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao’s Conjecture for arrangements having a line with at most four intersection points.

Arrangements of lines and monodromy of associated Milnor fibers

Consider an arrangement [Formula: see text] of homogeneous hyperplanes in [Formula: see text] with complement [Formula: see text]. The (co)homology of [Formula: see text] with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto [Formula: see text] endowed with the geometric monodromy. It is still an open problem to understand in general the cohomology of the Milnor fiber, even for dimension 1. In Sec. 1, we show that all questions about the first homology group are detected by a precise group, which is a quotient ot the commutator subgroup of [Formula: see text] by the commutator of its length zero subgroup, which didn’t appear in the literature before. In Sec. 2, we state a conjecture of [Formula: see text]-monodromicity for the first homology, which is of a different nature with respect to the known results. Let [Formula: see text] be the graph of double points of [Formula: see text] we conjecture that if [Formula: see text] is connected, then the geometric monodromy acts trivially on the first homology of the Milnor fiber (so the first Betti number is combinatorially determined in this case). This conjecture depends only on the combinatorics of [Formula: see text]. We show the truth of the conjecture under some stronger hypotheses.