Rational cubic fourfolds with a symplectic group of automorphisms

A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G. Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has an associated K3 surface S in the derived sense. This is equivalent to have a cohomologically associated K3 surface and therefore X is conjecturally rational. In this note we prove that cubic fourfolds with a cyclic group of symplectic automorphisms whose order is not a power of 2, are rational and belong to the Hassett divisor Cd, with d = 14, 42. We also describe rational cubic fourfolds X with a symplectic group of automorphisms G, such that (G, SG (X), is a Lech pair where th rank of SG equals 19 or 20.