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Explicit Rationality of Some Special Fano Fourfolds

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Rationality of Varieties

Part of the book series: Progress in Mathematics ((PM,volume 342))

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.

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Correspondence to Francesco Russo .

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Russo, F., Staglianò, G. (2021). Explicit Rationality of Some Special Fano Fourfolds. In: Farkas, G., van der Geer, G., Shen, M., Taelman, L. (eds) Rationality of Varieties. Progress in Mathematics, vol 342. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-75421-1_12

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