Cubic fourfolds with an involution

Marquand, Lisa

doi:10.1090/tran/8811

Cubic fourfolds with an involution
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by Lisa Marquand;

Trans. Amer. Math. Soc. 376 (2023), 1373-1406

DOI: https://doi.org/10.1090/tran/8811

Published electronically: December 1, 2022

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Abstract:

There are three types of involutions on a cubic fourfold; two of anti-symplectic type, and one symplectic. Here we show that cubics with involutions exhibit the full range of behaviour in relation to rationality conjectures. Namely, we show a general cubic fourfold with a symplectic involution has no associated $K3$ surface and is conjecturely irrational. In contrast, a cubic fourfold with a particular anti-symplectic involution has an associated $K3$, and is in fact rational. We show such a cubic is contained in the intersection of all non-empty Hassett divisors; we call such a cubic Hassett maximal. We study the algebraic and transcendental lattices for cubics with an involution both lattice theoretically and geometrically.

References

Asher Auel, Marcello Bernardara, Michele Bolognesi, and Anthony Várilly-Alvarado, Cubic fourfolds containing a plane and a quintic del Pezzo surface, Algebr. Geom. 1 (2014), no. 2, 181–193. MR 3238111, DOI 10.14231/AG-2014-010
Nicolas Addington and Richard Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. J. 163 (2014), no. 10, 1885–1927. MR 3229044, DOI 10.1215/00127094-2738639
Asher Auel, Brill-Noether special cubic fourfolds of discriminant 14, Facets of algebraic geometry. Vol. I, London Math. Soc. Lecture Note Ser., vol. 472, Cambridge Univ. Press, Cambridge, 2022, pp. 29–53. MR 4381896
Arnaud Beauville and Ron Donagi, La variété des droites d’une hypersurface cubique de dimension $4$, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706 (French, with English summary). MR 818549
Arnaud Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39–64. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786479, DOI 10.1307/mmj/1030132707
Anita Buckley and Tomaž Košir, Determinantal representations of smooth cubic surfaces, Geom. Dedicata 125 (2007), 115–140. MR 2322544, DOI 10.1007/s10711-007-9144-x
Arend Bayer, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, and Paolo Stellari, Stability conditions in families, Publ. Math. Inst. Hautes Études Sci. 133 (2021), 157–325. MR 4292740, DOI 10.1007/s10240-021-00124-6
Emma Brakkee, Moduli spaces of twisted K3 surfaces and cubic fourfolds, Math. Ann. 377 (2020), no. 3-4, 1453–1479. MR 4126898, DOI 10.1007/s00208-020-01990-x
Michele Bolognesi, Francesco Russo, and Giovanni Staglianò, Some loci of rational cubic fourfolds, Math. Ann. 373 (2019), no. 1-2, 165–190. MR 3968870, DOI 10.1007/s00208-018-1707-7
Nils Bruin and Emre Can Sertöz, Prym varieties of genus four curves, Trans. Amer. Math. Soc. 373 (2020), no. 1, 149–183. MR 4042871, DOI 10.1090/tran/7902
Chiara Camere, Symplectic involutions of holomorphic symplectic four-folds, Bull. Lond. Math. Soc. 44 (2012), no. 4, 687–702. MR 2967237, DOI 10.1112/blms/bdr133
I. V. Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996), no. 3, 2599–2630. Algebraic geometry, 4. MR 1420220, DOI 10.1007/BF02362332
Igor V. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012. A modern view. MR 2964027, DOI 10.1017/CBO9781139084437
Lie Fu, Classification of polarized symplectic automorphisms of Fano varieties of cubic fourfolds, Glasg. Math. J. 58 (2016), no. 1, 17–37. MR 3426426, DOI 10.1017/S001708951500004X
Víctor González-Aguilera and Alvaro Liendo, Automorphisms of prime order of smooth cubic $n$-folds, Arch. Math. (Basel) 97 (2011), no. 1, 25–37. MR 2820585, DOI 10.1007/s00013-011-0247-0
Federica Galluzzi, Cubic fourfolds containing a plane and $K3$ surfaces of Picard rank two, Geom. Dedicata 186 (2017), 103–112. MR 3602887, DOI 10.1007/s10711-016-0181-1
Brendan Edward Hassett, Special cubic hypersurfaces of dimension four, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.)–Harvard University. MR 2694332
Brendan Hassett, Some rational cubic fourfolds, J. Algebraic Geom. 8 (1999), no. 1, 103–114. MR 1658216
Brendan Hassett, Special cubic fourfolds, Compositio Math. 120 (2000), no. 1, 1–23. MR 1738215, DOI 10.1023/A:1001706324425
Brendan Hassett, Cubic fourfolds, K3 surfaces, and rationality questions, Rationality problems in algebraic geometry, Lecture Notes in Math., vol. 2172, Springer, Cham, 2016, pp. 29–66. MR 3618665
Brendan Hassett and Yuri Tschinkel, Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces, J. Inst. Math. Jussieu 9 (2010), no. 1, 125–153. MR 2576800, DOI 10.1017/S1474748009000140
Daniel Huybrechts, Generalized Calabi-Yau structures, $K3$ surfaces, and $B$-fields, Internat. J. Math. 16 (2005), no. 1, 13–36. MR 2115675, DOI 10.1142/S0129167X05002734
Daniel Huybrechts, The global Torelli theorem: classical, derived, twisted, Algebraic geometry—Seattle 2005. Part 1, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 235–258. MR 2483937, DOI 10.1090/pspum/080.1/2483937
Daniel Huybrechts, The K3 category of a cubic fourfold, Compos. Math. 153 (2017), no. 3, 586–620. MR 3705236, DOI 10.1112/S0010437X16008137
E. Izadi, A Prym construction for the cohomology of a cubic hypersurface, Proc. London Math. Soc. (3) 79 (1999), no. 3, 535–568. MR 1710164, DOI 10.1112/S0024611599012046
A. Javanpeykar and D. Loughran, Complete intersections: moduli, Torelli, and good reduction, Math. Ann. 368 (2017), no. 3-4, 1191–1225. MR 3673652, DOI 10.1007/s00208-016-1455-5
Ljudmila Kamenova, Giovanni Mongardi, and Alexei Oblomkov, Symplectic involutions of $K3^{[n]}$ type and Kummer $n$ type manifolds, Bull. Lond. Math. Soc. 54 (2022), no. 3, 894–909. MR 4453747, DOI 10.1112/blms.12594
Shigeyuki Kond\B{o}, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of $K3$ surfaces, Duke Math. J. 92 (1998), no. 3, 593–603. With an appendix by Shigeru Mukai. MR 1620514, DOI 10.1215/S0012-7094-98-09217-1
Maxim Kontsevich and Yuri Tschinkel, Specialization of birational types, Invent. Math. 217 (2019), no. 2, 415–432. MR 3987175, DOI 10.1007/s00222-019-00870-9
Alexander Kuznetsov, Derived categories of cubic fourfolds, Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, Birkhäuser Boston, Boston, MA, 2010, pp. 219–243. MR 2605171, DOI 10.1007/978-0-8176-4934-0_{9}
Alexander Kuznetsov, Derived categories view on rationality problems, Rationality problems in algebraic geometry, Lecture Notes in Math., vol. 2172, Springer, Cham, 2016, pp. 67–104. MR 3618666
Radu Laza, The moduli space of cubic fourfolds, J. Algebraic Geom. 18 (2009), no. 3, 511–545. MR 2496456, DOI 10.1090/S1056-3911-08-00506-7
Radu Laza, Maximally algebraic potentially irrational cubic fourfolds, Proc. Amer. Math. Soc. 149 (2021), no. 8, 3209–3220. MR 4273129, DOI 10.1090/proc/15430
Christian Lehn, Manfred Lehn, Christoph Sorger, and Duco van Straten, Twisted cubics on cubic fourfolds, J. Reine Angew. Math. 731 (2017), 87–128. MR 3709061, DOI 10.1515/crelle-2014-0144
Radu Laza, Gregory Pearlstein, and Zheng Zhang, On the moduli space of pairs consisting of a cubic threefold and a hyperplane, Adv. Math. 340 (2018), 684–722. MR 3886178, DOI 10.1016/j.aim.2018.10.017
Radu Laza, Giulia Saccà, and Claire Voisin, A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold, Acta Math. 218 (2017), no. 1, 55–135. MR 3710794, DOI 10.4310/ACTA.2017.v218.n1.a2
Radu Laza and Zhiwei Zheng, Automorphisms and periods of cubic fourfolds, Math. Z. 300 (2022), no. 2, 1455–1507. MR 4363785, DOI 10.1007/s00209-021-02810-x
Lisa Marquand, On symplectic birational involutions of manifolds of OG10 type, 2022.
Hideyuki Matsumura and Paul Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963/64), 347–361. MR 168559, DOI 10.1215/kjm/1250524785
Giovanni Mongardi, Towards a classification of symplectic automorphisms on manifolds of $K3^{[n]}$ type, Math. Z. 282 (2016), no. 3-4, 651–662. MR 3473636, DOI 10.1007/s00209-015-1557-x
Shigeru Mukai, Finite groups of automorphisms of $K3$ surfaces and the Mathieu group, Invent. Math. 94 (1988), no. 1, 183–221. MR 958597, DOI 10.1007/BF01394352
V. V. Nikulin, Finite groups of automorphisms of Kählerian $K3$ surfaces, Trudy Moskov. Mat. Obshch. 38 (1979), 75–137 (Russian). MR 544937
V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238 (Russian). MR 525944
Genki Ouchi, Automorphism groups of cubic fourfolds and K3 categories, Algebr. Geom. 8 (2021), no. 2, 171–195. MR 4174288, DOI 10.14231/ag-2021-003
A. N. Rudakov and I. R. Shafarevich, Surfaces of type $K3$ over fields of finite characteristic, Current problems in mathematics, Vol. 18, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1981, pp. 115–207 (Russian). MR 633161
Francesco Russo and Giovanni Staglianò, Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds, Duke Math. J. 168 (2019), no. 5, 849–865. MR 3934590, DOI 10.1215/00127094-2018-0053
Claire Voisin, Théorème de Torelli pour les cubiques de $\textbf {P}^5$, Invent. Math. 86 (1986), no. 3, 577–601 (French). MR 860684, DOI 10.1007/BF01389270
Claire Voisin, Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal, J. Algebraic Geom. 22 (2013), no. 1, 141–174. MR 2993050, DOI 10.1090/S1056-3911-2012-00597-9
Song Yang and Xun Yu, On lattice polarizable cubic fourfolds, 2021.
Chenglong Yu and Zhiwei Zheng, Moduli spaces of symmetric cubic fourfolds and locally symmetric varieties, Algebra Number Theory 14 (2020), no. 10, 2647–2683. MR 4190414, DOI 10.2140/ant.2020.14.2647
Yuri G. Zarhin, Algebraic cycles over cubic fourfolds, Boll. Un. Mat. Ital. B (7) 4 (1990), no. 4, 833–847 (English, with Italian summary). MR 1086707
Zhiwei Zheng, Orbifold aspects of certain occult period maps, Nagoya Math. J. 243 (2021), 137–156. MR 4298656, DOI 10.1017/nmj.2019.36