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Rationality of Mukai Varieties over Non-closed Fields

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Rationality of Varieties

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

Abstract

We discuss birational properties of Mukai varieties, i.e., of higherdimensional analogues of prime Fano threefolds of genusg ∈ {7, 8, 9, 10} over an arbitrary field k of zero characteristic. In the case of dimension n ≥ 4 we prove that these varieties are k-rational if and only if they have a k-point except for the case of genus 9, where the same holds for n ≥ 5. Furthermore, we prove that Mukai varieties of genus g ∈ {7, 8, 9, 10} and dimension n ≥ 5 contain cylinders if they have a k-point. Finally, we prove that the embedding X ↪ Gr(3, 7) for prime Fano threefolds of genus 12 is defined canonically over any field of zero characteristic and use this to give a new proof of the criterion of k-rationality for these threefolds.

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Correspondence to Alexander Kuznetsov .

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Kuznetsov, A., Prokhorov, Y. (2021). Rationality of Mukai Varieties over Non-closed Fields. 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_10

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