Maximally algebraic potentially irrational cubic fourfolds

Laza, Radu

doi:10.1090/proc/15430

Maximally algebraic potentially irrational cubic fourfolds
HTML articles powered by AMS MathViewer

by Radu Laza PDF

Proc. Amer. Math. Soc. 149 (2021), 3209-3220 Request permission

Abstract:

A well known conjecture due to Hassett asserts that a cubic fourfold $X$ whose transcendental cohomology $T_X$ cannot be realized as the transcendental cohomology of a $K3$ surface is irrational. Since the geometry of cubic fourfolds is intricately related to the existence of algebraic $2$-cycles on them, it is natural to ask for the most algebraic cubic fourfolds $X$ to which this conjecture is still applicable. In this paper, we show that for an appropriate “algebraicity index” $\kappa _X\in \mathbb {Q}_+$, there exists a unique class of cubics maximizing $\kappa _X$, not having an associated $K3$ surface; namely, the cubic fourfolds with an Eckardt point (previously investigated in by Laza, Pearlstein, and Zhang [Adv. Math. 340 (2018), pp. 684-722]). Arguably, they are the most algebraic conjecturally irrational cubic fourfolds, and thus a good testing ground for Hassett’s irrationality conjecture for cubic fourfolds.

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
A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry, and P. Stellari, Stability conditions in families, arXiv:1902.08184, 2019.
C. Herbert Clemens and Phillip A. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2) 95 (1972), 281–356. MR 302652, DOI 10.2307/1970801
I. V. Dolgachev and D. Markushevich, Lagrangian tens of planes, Enriques surfaces and holomorphic symplectic fourfolds, arXiv:1906.01445, 2019.
Igor Dolgachev, Integral quadratic forms: applications to algebraic geometry (after V. Nikulin), Bourbaki seminar, Vol. 1982/83, Astérisque, vol. 105, Soc. Math. France, Paris, 1983, pp. 251–278. MR 728992
I. Dolgachev, Classical algebraic geometry, Cambridge University Press, Cambridge, 2012, A modern view.
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, Alena Pirutka, and Yuri Tschinkel, A very general quartic double fourfold is not stably rational, Algebr. Geom. 6 (2019), no. 1, 64–75. MR 3904799, DOI 10.14231/ag-2019-004
Radu Laza, Deformations of singularities and variation of GIT quotients, Trans. Amer. Math. Soc. 361 (2009), no. 4, 2109–2161. MR 2465831, DOI 10.1090/S0002-9947-08-04660-6
Radu Laza, The moduli space of cubic fourfolds via the period map, Ann. of Math. (2) 172 (2010), no. 1, 673–711. MR 2680429, DOI 10.4007/annals.2010.172.673
Eduard Looijenga, The period map for cubic fourfolds, Invent. Math. 177 (2009), no. 1, 213–233. MR 2507640, DOI 10.1007/s00222-009-0178-6
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
Giovanni Mongardi, Symplectic involutions on deformations of $\textrm {K}3^{[2]}$, Cent. Eur. J. Math. 10 (2012), no. 4, 1472–1485. MR 2925616, DOI 10.2478/s11533-012-0073-z
D. R. Morrison, On $K3$ surfaces with large Picard number, Invent. Math. 75 (1984), no. 1, 105–121. MR 728142, DOI 10.1007/BF01403093
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
Kieran G. O’Grady, Pairwise incident planes and hyperkähler four-folds, A celebration of algebraic geometry, Clay Math. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 2013, pp. 553–566. MR 3114956, DOI 10.4169/mathhorizons.22.3.34
Burt Totaro, Hypersurfaces that are not stably rational, J. Amer. Math. Soc. 29 (2016), no. 3, 883–891. MR 3486175, DOI 10.1090/jams/840
È. B. Vinberg, The two most algebraic $K3$ surfaces, Math. Ann. 265 (1983), no. 1, 1–21. MR 719348, DOI 10.1007/BF01456933
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, Some aspects of the Hodge conjecture, Jpn. J. Math. 2 (2007), no. 2, 261–296. MR 2342587, DOI 10.1007/s11537-007-0639-x