Finite abelian groups acting on rationally connected threefolds I: Groups of product type

Abstract: We initiate the study of finite abelian groups that faithfully act on 3-dimensional rationally connected varieties. We show that these groups can be naturally divided into three types: the groups of product type are finite abelian groups that are products of two groups which belong to the Cremona group of rank 1 and 2, respectively; the group of K3 type consists of cyclic extensions of finite abelian groups acting on a K3 surface; the third type consists of groups that act on terminal Fano threefolds with empty anti-canonical linear system. The classification of groups of product type follows from a result of J. Blanc. For the groups of K3 type, we show that there are only finitely many of them. We also formulate a conjecture regarding the groups of the third type.