Birational rigidity of quartic three-folds with a double point of rank 3

Abstract Let V be a general three-dimensional quartic in the complex projective space ℙ 4 such that the only singularity of V is a double point of rank 3. We prove that V is a birationally rigid variety. Its group of birational self-maps is, up to the finite subgroup of biregular automorphisms, a free product of 25 cyclic groups of order 2. It follows that the complement to the set of birationally rigid factorial quartics with terminal singularities is of codimension at least 3 in the natural parameter space.