Non-rigid quartic -folds

Let X⊂ P⁴ be a terminal factorial quartic $3$ -fold. If X is non-singular, X is birationally rigid , i.e. the classical minimal model program on any terminal 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⊂ P⁴ . A singular point on such a hypersurface is of type cAₙ ( n 1 ), or of type cDₘ ( m 4 ) or of type cE₆,cE₇ or cE₈ . We first show that if (P∈ X) is of type cAₙ , n is at most $7$ and, if (P∈ X) is of type cDₘ