Milnor numbers of projective hypersurfaces with isolated singularities

TL;DR

This paper investigates the relationship between the sum of Milnor numbers and the multiplicity of singular points on projective hypersurfaces, proving that large Milnor sums restrict the hypersurface's geometric structure.

Contribution

It establishes a new link between Milnor numbers and multiplicity constraints, confirming a conjecture by Dimca and Papadima.

Findings

Large Milnor sums imply the hypersurface cannot have high multiplicity points unless it is a cone.

Confirmed a conjecture of Dimca and Papadima regarding hypersurface singularities.

Provides conditions under which hypersurfaces with isolated singularities are constrained in structure.

Abstract

Let V be a projective hypersurface of fixed degree and dimension which has only isolated singular points. We show that, if the sum of the Milnor numbers at the singular points of V is large, then V cannot have a point of large multiplicity, unless V is a cone. As an application, we give an affirmative answer to a conjecture of Dimca and Papadima.