Birational automorphisms of nodal quartic threefolds

Abstract

It is well-known that a nonsingular minimal cubic surface is birationally rigid; the group of its birational selfmaps is generated by biregular selfmaps and birational involutions such that all relations between the latter are implied by standard relations between reflections on an elliptic curve. It is also known that a factorial nodal quartic threefold is birationally rigid and its group of birational selfmaps is generated by biregular ones and certain birational involutions. We prove that all relations between these involutions are implied by standard relations on elliptic curves, complete the proof of birational rigidity over a non-closed field and describe the situations when some of the birational involutions in question become regular (and, in particular, complete the proof of the initial theorem on birational rigidity, since some details were not established in the original paper of M.Mella).