Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds

The works of Hassett and Kuznetsov identify countably many divisors Cd in the open subset of P55=P(H0(OP5(3))) parameterizing all cubic fourfolds and conjecture that the cubics corresponding to these divisors are precisely the rational ones. Rationality has been known classically for the first family C14. We use congruences of 5-secant conics to prove rationality for the first three of the families Cd, corresponding to d=14,26,38 in Hassett’s notation.