Spectrum of singularities via lattice point enumeration: The case of projective curve arrangements
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We introduce a combinatorial method for computing the spectrum of the singularity at the origin of a homogeneous polynomial defining a curve arrangement in the projective plane. For arrangements with ordinary singularities, our approach counts lattice points in specific regions of $\mathbb{Z}^3$, naturally separating global contributions determined by the total degree from local contributions at singular points. This geometric interpretation reveals that the spectrum depends only on the total degree and multiplicities at singular points, leading to efficient computations of the Milnor fiber's Euler characteristic and monodromy eigenspace dimensions. Our method suggests potential generalizations to more complex singularities through appropriate modifications of the local regions.
Abstract
We introduce a combinatorial method for computing the spectrum of the singularity at the origin of a homogeneous polynomial defining a curve arrangement in the projective plane. For arrangements with ordinary singularities, our approach counts lattice points in specific regions of $\mathbb{Z}^3$, naturally separating global contributions determined by the total degree from local contributions at singular points. This geometric interpretation reveals that the spectrum depends only on the total degree and multiplicities at singular points, leading to efficient computations of the Milnor fiber's Euler characteristic and monodromy eigenspace dimensions. Our method suggests potential generalizations to more complex singularities through appropriate modifications of the local regions.