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 $M\geq 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the parameter space. 
Thomas Eckl, Aleksandr Pukhlikov
math.AG