Explicit Rationality of Some Special Fano Fourfolds

Recent results of Hassett, Kuznetsov and others pointed out countably many divisors in the open subset of parametrizing all cubic 4-folds and lead to the conjecture that the cubics corresponding to these divisors should be precisely the rational ones. Rationality has been proved by Fano for the first divisor, in [RS19a] for the divisors and, and in [RS19b] for. In this note we describe explicit birational maps from a general cubic fourfold in to P4, providing concrete geometric realizations of the more abstract constructions in [RS19a] and of the theoretical framework developed in [RS19b]. We also exhibit an explicit relationship between the divisor C14 and a certain divisor in the open subset of parametrizing smooth quadratic sections of a del Pezzo fivefold, the so-called Gushel–Mukai fourfolds.