Modular Ideals and Factorizations of Affine Hyperplane Arrangements

Abstract

Let $\mathcal{A}$ be an affine hyperplane arrangement, let $L(\mathcal{A})$ be its intersection semilattice, and let ${\chi }_{\mathcal{A}}(t)$ be its characteristic polynomial. We introduce ideal decompositions of finite ranked meet-semilattices and prove that they induce purely combinatorial factorizations of characteristic polynomials. This framework recovers both Stanley's factorization associated with modular elements and Terao's factorization associated with nice partitions. For simple semimatroids, every join-closed ideal decomposition comes from a direct-sum decomposition. The two-factor case defines modular ideals, extending the role of modular elements from central to affine arrangements. Over an infinite field, every modular ideal of an arrangement intersection semilattice is realized by a suitable translated restriction of a subarrangement. We exhibit a nonempty Zariski-open set of valid translations, yielding essential realizations of the given modular ideal. We further prove that modular subarrangements correspond under coning to modular elements on the hyperplane at infinity and hence to M-ideals in the affine setting. Finally, every modular ideal gives an Orlik-Solomon graded vector-space decomposition, which becomes a graded-algebra decomposition when the two factors come from subarrangements.