Severi varieties on Enriques surfaces of base change type and on rational elliptic surfaces

If an irreducible curve on the very general Enriques surface splits in the K3 cover, its preimage consists of two linearly equivalent irreducible curves. We prove the nonemptiness of countable families of Severi varieties of curves of any genus on Enriques surfaces of base change type, whose members split in nonlinearly equivalent curves in the K3 cover. Our machinery leads us to provide examples of special superabundant logarithmic Severi varieties of curves of any genus on rational elliptic surfaces.