SPECTRUM OF LINE ARRANGEMENTS WITH MULTIPLICITIES: A COMBINATORIAL APPROACH VIA LATTICE POINT ENUMERATION

We introduce a combinatorial method for computing the spectrum of singularities using lattice point enumeration in regions determined by the defining equation. For line arrangements with multiplicities, our approach reveals the interplay between global and local contributions to the spectrum through explicit counting of lattice points. A series of transformations of these regions preserves the spectral data while simplifying the counting process. We highlight the case of f(x, y) = xm1ym2 as an illustrative example, where the method reduces the problem to counting lattice points on two line segments and naturally explains the role of gcd(m1,m2). This work provides a new perspective on the spectra of line arrangements with multiplicities, with potential applications to general non-isolated singularities.