Non-rigid quartic 3-folds

Abstract

Let X C 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 c P4. A singular point on such a hypersurface is either of type cAn (n > or equal 1), or of type cDm (m> or equal 4), or of type cE6, cE7 or cE8. We first show that if (P e X) is of type cAn, n is at most 7, and if (P \in X) is of type cDm, 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 cAn for 2\leq n\leq 7 (b) of a single point of type cDm for m= 4 or 5 and (c) of a single point of type cEk for k=6,7 or 8.