Bounding Projective Hypersurface Singularities

TL;DR

This paper compares various Hodge-theoretic methods to constrain the number and type of isolated singularities on projective hypersurfaces, introducing a new spectrum-based approach related to Varchenko's bound.

Contribution

It introduces a novel spectrum-based method for analyzing singularities on hypersurfaces, connecting affine cone spectra to existing bounds.

Findings

New spectrum method constrains singularities effectively

Explicit formulas relate to Varchenko's bound

Improves understanding of hypersurface singularities

Abstract

We compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on projective hypersurfaces of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko's bound.