Spectrum of cones of projective hypersurfaces with singularities isolated

Abstract

Let $Z$ be a projective hypersurface such that its underlying reduced variety has only isolated singularities. In case its irreducible components have constant multiplicities, for instance if $\dim Z>1$, we show that the spectrum of its cone can be described by using the spectral numbers at singular points of the reduced hypersurface and the global degree. In the non-reduced plane curve case, assuming that the underlying reduced curve has only semi-weighted-homogeneous singularities, we express the spectrum of the cone in terms of the local weights and the weighted degrees and multiplicities of local irreducible components together with the degrees and multiplicities of global ones. These generalize a formula for reduced line arrangements. In the non-reduced ordinary (that is, semi-homogeneous) singularity case, the second formula is essentially equivalent to the one obtained by the third-named author.