New examples of rational Gushel-Mukai fourfolds
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Mathematische Zeitschrift
New examples of rational Gushel-Mukai fourfolds Michael Hoff1 · Giovanni Staglianò2 Received: 4 November 2019 / Accepted: 15 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We construct new examples of rational Gushel-Mukai fourfolds, giving more evidence for the analog of the Kuznetsov Conjecture for cubic fourfolds: a Gushel–Mukai fourfold is rational if and only if it admits an associated K3 surface.
1 Introduction A Gushel-Mukai fourfold is a smooth prime Fano fourfold X ⊂ P8 of degree 10 and index 2 (see [21]). These fourfolds are parametrized by a coarse moduli space M4G M of dimension 24 (see [5, Theorem 5.15]), and the general fourfold [X ] ∈ M4G M is a smooth quadratic section of a smooth hyperplane section of the Grassmannian G(1, 4) ⊂ P9 of lines in P4 . In [3] (see also [4–6]), following Hassett’s analysis of cubic fourfolds (see [12,13]), the authors studied Gushel-Mukai fourfolds via Hodge theory andthe period map. In particular, they showed that inside M4G M there is a countable union d GMd of (not necessarily irreducible) hypersurfaces parametrizing Hodge-special Gushel-Mukai fourfolds, that is, fourfolds that contain a surface whose cohomology class does not come from the Grassmannian G(1, 4). The index d is called the discriminant of the fourfold and it runs over all positive integers congruent to 0, 2, or 4 modulo 8 (see [3]). However, as far as the authors know, explicit geometric descriptions of Hodge-special Gushel-Mukai fourfolds in GMd are unknown for d > 12. In Theorem 3.3, we shall provide such a description when d = 20. As in the case of cubic fourfolds, all Gushel-Mukai fourfolds are unirational. Some rational examples are classical and easy to construct, but no examples have yet been proved to be irrational. Furthermore, there are values of the discriminant d such that a fourfold in GMd admits an associated K3 surface of degree d. For instance, this occurs for d = 10 and d = 20. The hypersurface GM10 has two irreducible components, and the general fourfold in each of these two components is rational (see [3, Propositions 7.4 and 7.7] and Examples 2.1 and 2.2).
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Michael Hoff [email protected] Giovanni Staglianò [email protected]
1
Universität des Saarlandes, Campus E2 4, 66123 Saarbrücken, Germany
2
Dipartimento di Matematica e Informatica, Università degli Studi di Catania, Viale A. Doria 5, 95125 Catania, Italy
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M. Hoff, G. Staglianò
Some of these fourfolds were already studied by Roth in [22]. As far as the authors know, there were no other known examples of rational Gushel-Mukai fourfolds. In Theorem 3.4, we shall provide new examples of rational Gushel-Mukai fourfolds that belong to GM20 . A classical and still open question in algebraic geometry is the rationality of smooth cubic hypersurfaces in P5 (cubic fourfolds for short). An important conjecture, known as Kuznetsov’s Conjecture (see [1,14,18,19]) asserts that a cubic fourfold is rational if and only if it admits an associated K3 surface
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