Explicit Rationality of Some Special Fano Fourfolds

Abstract

Recent results of Hassett, Kuznetsov and others pointed out countably many divisors $\mathcal{C}_{d}$ in the open subset of $\mathbb{P}^{55}= \mathbb{P}(\textit{H}^{o}(\mathcal{O}_{\mathbb{P}^{5}}(3)))$ 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 $\mathcal{C}_{14}$, in [RS19a] for the divisors $\mathcal{C}_{26}$ and $\mathcal{C}_{38}$, and in [RS19b] for $\mathcal{C}_{42}$. In this note we describe explicit birational maps from a general cubic fourfold in $\mathcal{C}_{14}$, in $\mathcal{C}_{26}$ and in $\mathcal{C}_{38}$ 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 $\mathbb{P}^{39}= \mathbb{P}(\textit{H}^{o}(\mathcal{O}_{Y}(2)))$ parametrizing smooth quadratic sections of a del Pezzo fivefold $\textit{Y}=\mathbb{G}(1,4)\cap\mathbb{P}^{8}\subset\mathbb{P}^{8}$, the so-called Gushel–Mukai fourfolds.