On cubic fourfolds with an inductive structure

Koike, Kenji

doi:10.1090/proc/15932

On cubic fourfolds with an inductive structure
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Proc. Amer. Math. Soc. 150 (2022), 3757-3769 Request permission

Abstract:

We study the number of planes in four dimensional projective hypersurfaces which have so-called inductive structure. We also determine transcendental lattices of cubic fourfolds of this type.

References

Michela Artebani and Igor Dolgachev, The Hesse pencil of plane cubic curves, Enseign. Math. (2) 55 (2009), no. 3-4, 235–273. MR 2583779, DOI 10.4171/LEM/55-3-3
Arnaud Beauville, Some surfaces with maximal Picard number, J. Éc. polytech. Math. 1 (2014), 101–116 (English, with English and French summaries). MR 3322784, DOI 10.5802/jep.5
T. D. Browning and D. R. Heath-Brown, The density of rational points on non-singular hypersurfaces. II, Proc. London Math. Soc. (3) 93 (2006), no. 2, 273–303. With an appendix by J. M. Starr. MR 2251154, DOI 10.1112/S0024611506015784
Samuel Boissière and Alessandra Sarti, Counting lines on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 1, 39–52. MR 2341513
Araceli Bonifant and John Milnor, On real and complex cubic curves, Enseign. Math. 63 (2017), no. 1-2, 21–61. MR 3777131, DOI 10.4171/LEM/63-1/2-2
Lucia Caporaso, Joe Harris, and Barry Mazur, How many rational points can a curve have?, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 13–31. MR 1363052, DOI 10.1007/978-1-4612-4264-2_{2}
A. Degtyarev, I. Itenberg, and J. C. Ottem, Planes in cubic fourfolds, arXiv:2105.13951v1, 2021.
Alex Degtyarev and Ichiro Shimada, On the topology of projective subspaces in complex Fermat varieties, J. Math. Soc. Japan 68 (2016), no. 3, 975–996. MR 3523534, DOI 10.2969/jmsj/06830975
David Eisenbud and Joe Harris, 3264 and all that—a second course in algebraic geometry, Cambridge University Press, Cambridge, 2016. MR 3617981, DOI 10.1017/CBO9781139062046
D. Mumford, On the equations defining abelian varieties. I, Invent. Math. 1 (1966), 287–354. MR 204427, DOI 10.1007/BF01389737
Sławomir Rams and Matthias Schütt, 64 lines on smooth quartic surfaces, Math. Ann. 362 (2015), no. 1-2, 679–698. MR 3343894, DOI 10.1007/s00208-014-1139-y
B. Segre, On arithmetical properties of quartic surfaces, Proc. London Math. Soc. (2) 49 (1947), 353–395. MR 21952, DOI 10.1112/plms/s2-49.5.353
Tetsuji Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), no. 2, 175–184. MR 552586, DOI 10.1007/BF01428804
Tetsuji Shioda and Toshiyuki Katsura, On Fermat varieties, Tohoku Math. J. (2) 31 (1979), no. 1, 97–115. MR 526513, DOI 10.2748/tmj/1178229881
Tetsuji Shioda and Naoki Mitani, Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds, Lecture Notes in Math., Vol. 412, Springer, Berlin, 1974, pp. 259–287. MR 0382289
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