On the locus of non-rigid hypersurfaces

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in ${\mathbb{P}}^M$, where $M\ge 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\left(\genfrac{}{}{0px}{}{M-3}{2}\right)+1$ in the parameter space.