Hypersurfaces with defect

Abstract

A projective hypersurface X⊆Pn has defect if hi(X)≠hi(Pn) for some i∈{n,…,2n−2} in a suitable cohomology theory. This occurs for example when X⊆P4 is not Q-factorial. We show that hypersurfaces with defect tend to be very singular: In characteristic 0, we present a lower bound on the Tjurina number, where X is allowed to have arbitrary isolated singularities. For X with mild singularities, we prove a similar result in positive characteristic. As an application, we obtain an estimate on the asymptotic density of hypersurfaces without defect over a finite field.