Milnor numbers of projective hypersurfaces with isolated singularities

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.