Hypersurfaces with defect

A projective hypersurface $X\subseteq {\mathbb{P}}^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$forsome$i \in {n, \dots, 2n-2}$ in a suitable cohomology theory. This occurs for example when $X\subseteq {\mathbb{P}}^4$ is not $\mathbb{Q}$-factorial. We show that in characteristic 0, the Tjurina number of hypersurfaces with defect is large. For $X$ with mild singularities, there is a similar result in positive characteristic. As an application, we obtain a lower bound on the asymptotic density of hypersurfaces without defect over a finite field.