Who succeed Lucky Luciano as head of the Luciano-family (later:genovese-family)? Was is Vito Genovese or Frank Costello?

Genovese was Luciano's underboss before Lucky was deported, but Vito left USA for Italy around the same time Luciano had gone to prison...

So between Luciano had gone to the can and Costello became boss, was Genovese the boss for a short period?

'This was just another Bronx tale.'