The Spectrum of Hypersurface Singularities

TL;DR

This paper discusses the asymptotic mixed Hodge theory of isolated hypersurface singularities and applies semi-continuity of the spectrum to bound singularities on projective hypersurfaces.

Contribution

It provides a detailed exposition of the spectrum of hypersurface singularities and applies classical semi-continuity results to geometric bounding problems.

Findings

Semi-continuity of the spectrum constrains hypersurface singularities.

Bounds on singularities on projective hypersurfaces are derived.

Lecture notes on asymptotic mixed Hodge theory are presented.

Abstract

This text is the write-up of a series of lectures on the asymptotic mixed Hodge theory of isolated hypersurface singularities, held at the Third Latin American school on Algebraic Geometry and its applications (ELGA 3) in Guanajuato, Mexico, in august 2017. Its focus is on the classical application of the semi-continuity of the spectrum due to Varchenko and Steenbrink to the problem of bounding the possible singularities on a projective hypersurface.