The works of Hassett and Kuznetsov identify countably many divisors in the open subset of 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 . We use congruences of -secant conics to prove rationality for the first three of the families , corresponding to in Hassett’s notation.
"Congruences of -secant conics and the rationality of some admissible cubic fourfolds." Duke Math. J. 168 (5) 849 - 865, 1 April 2019. https://doi.org/10.1215/00127094-2018-0053