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The works of Hassett and Kuznetsov identify countably many divisors $C_d$ inthe open subset of$\mathbb{P}^{55}=\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^5}(3)))$ parametrizingall cubic 4-folds and conjecture that the cubics corresponding to thesedivisors are precisely the rational ones. Rationality has been knownclassically for the first family $C_{14}$. We use congruences of 5-secantconics to prove rationality for the first three of the families $C_d$,corresponding to $d=14, 26, 38$ in Hassett's notation.

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