Arrangements of lines and monodromy of associated Milnor fibers

Consider an arrangement (Formula presented.) of homogeneous hyperplanes in (Formula presented.) with complement (Formula presented.). The (co)homology of (Formula presented.) with twisted coefficients is strictly related to the cohomology of the Milnor fiber associated to the natural fibration onto (Formula presented.) 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 presented.) 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 presented.)-monodromicity for the first homology, which is of a different nature with respect to the known results. Let (Formula presented.) be the graph of double points of (Formula presented.) we conjecture that if (Formula presented.) 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 presented.). We show the truth of the conjecture under some stronger hypotheses.