Non-rigid quartic 3-folds

Let $X\subset {\mathbb{P}}^4$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is \emph{birationally rigid}, i.e. the classical MMP on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset {\mathbb{P}}^4$. A singular point on such a hypersurface is either of type $c{A}_n$ ($n\ge 1$), or of type $c{D}_m$ ($m\ge 4$), or of type $c{E}_6,c{E}_7$ or $c{E}_8$. We first show that if $(P\in X)$ is of type $c{A}_n$, $n$ is at most $7$, and if $(P\in X)$ is of type $c{D}_m$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $c{A}_n$ for $2\le n\le 7$ (b) of a single point of type $c{D}_m$ for $m=4$ or $5$ and © of a single point of type $c{E}_k$ for $k=6,7$ or $8$.