Arrangements of lines and monodromy of associated Milnor fibers

Abstract

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.