Kodaira dimension of moduli of special cubic fourfolds
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Sho Tanimoto
und
Anthony Várilly-Alvarado
Abstract
A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors
Funding statement: Tanimoto was partially supported by Lars Hesselholt’s Niels Bohr Professorship. Várilly-Alvarado was supported by NSF grant DMS-1103659 and NSF CAREER grant DMS-1352291.
Acknowledgements
We are grateful to Brendan Hassett for many conversations where he patiently answered our questions, and for his constant encouragement. We thank Klaus Hulek for an illuminating discussion at the Simons Symposium “Geometry over Non-Closed Fields” in March of 2015. We also thank an anonymous referee for their careful reading of the manuscript, and for pertinent suggestions that improved the exposition of the paper. Computer calculations were carried out in Magma [5].
References
[1] N. Addington and R. Thomas, Hodge theory and derived categories of cubic fourfolds, Duke Math. J. 163 (2014), no. 10, 1885–1927. 10.1215/00127094-2738639Suche in Google Scholar
[2] W. L. Baily, Jr. and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. (2) 84 (1966), 442–528. 10.2307/1970457Suche in Google Scholar
[3] A. Beauville and R. 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. Suche in Google Scholar
[4]
R. E. Borcherds,
Automorphic forms on
[5] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system. I: The user language, J. Symbolic Comput. 24 (1997), no. 3–4, 235–265. 10.1006/jsco.1996.0125Suche in Google Scholar
[6] J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. I. Quadratic forms of small determinant, Proc. R. Soc. Lond. Ser. A 418 (1988), no. 1854, 17–41. 10.1098/rspa.1988.0072Suche in Google Scholar
[7] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer, New York 1999. 10.1007/978-1-4757-6568-7Suche in Google Scholar
[8]
G. Fano,
Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del
[9]
V. Gritsenko,
Modular forms and moduli spaces of abelian and
[10]
V. A. Gritsenko, K. Hulek and G. K. Sankaran,
The Kodaira dimension of the moduli of
[11] V. A. Gritsenko, K. Hulek and G. K. Sankaran, Moduli spaces of irreducible symplectic manifolds, Compos. Math. 146 (2010), no. 2, 404–434. 10.1112/S0010437X0900445XSuche in Google Scholar
[12] V. A. Gritsenko, K. Hulek and G. K. Sankaran, Moduli spaces of polarized symplectic O’Grady varieties and Borcherds products, J. Differential Geom. 88 (2011), no. 1, 61–85. 10.4310/jdg/1317758869Suche in Google Scholar
[13] V. A. Gritsenko, K. Hulek and G. K. Sankaran, Moduli of K3 surfaces and irreducible symplectic manifolds, Handbook of moduli. Vol. I, Adv. Lect. Math. (ALM) 1, International Press, Boston (2013), 459–526. Suche in Google Scholar
[14] E. Grosswald, Representations of integers as sums of squares, Springer, New York 1985. 10.1007/978-1-4613-8566-0Suche in Google Scholar
[15] B. Hassett, Special cubic fourfolds, Ph.D. thesis (revised), Harvard University, 1996; available at http://www.math.brown.edu/~bhassett/papers/cubics/cubiclong.pdf. Suche in Google Scholar
[16] B. Hassett, Special cubic fourfolds, Compos. Math. 120 (2000), no. 1, 1–23. 10.1023/A:1001706324425Suche in Google Scholar
[17] D. Huybrechts, The K3 category of a cubic fourfold, preprint (2015), https://arxiv.org/abs/1505.01775; to appear in Compos. Math. 10.1112/S0010437X16008137Suche in Google Scholar
[18] H. Iwaniec, Topics in classical automorphic forms, Grad. Stud. Math. 17, American Mathematical Society, Providence 1997. 10.1090/gsm/017Suche in Google Scholar
[19]
S. Kondō,
On the Kodaira dimension of the moduli space of
[20] K.-W. Lai, New cubic fourfolds with odd degree unirational parametrizations, preprint (2016), https://arxiv.org/abs/1606.03853. 10.2140/ant.2017.11.1597Suche in Google Scholar
[21] R. Laza, The moduli space of cubic fourfolds via the period map, Ann. of Math. (2) 172 (2010), no. 1, 673–711. 10.4007/annals.2010.172.673Suche in Google Scholar
[22] Z. Li and L. Zhang, Modular forms and special cubic fourfolds, Adv. Math. 245 (2013), 315–326. 10.1016/j.aim.2013.06.003Suche in Google Scholar
[23] E. Looijenga, The period map for cubic fourfolds, Invent. Math. 177 (2009), no. 1, 213–233. 10.1007/s00222-009-0178-6Suche in Google Scholar
[24] S. Ma, Finiteness of stable orthogonal modular varieties of non-general type, preprint (2013), https://arxiv.org/abs/1309.7121. Suche in Google Scholar
[25]
K. McKinnie, J. Sawon, S. Tanimoto and A. Várilly-Alvarado,
Brauer groups on
[26]
S. Mukai,
On the moduli space of bundles on
[27] V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111–177, 238. Suche in Google Scholar
[28] H. Nuer, Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces, preprint (2015), https://arxiv.org/abs/1503.05256; to appear in Algebr. Geom. 10.1007/978-3-319-46209-7_5Suche in Google Scholar
[29] S. L. Tregub, Three constructions of rationality of a cubic fourfold (Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (1984), 8–14. Suche in Google Scholar
[30]
C. Voisin,
Théorème de Torelli pour les cubiques de
[31] C. Voisin, Some aspects of the Hodge conjecture, Jpn. J. Math. 2 (2007), no. 2, 261–296. 10.1007/s11537-007-0639-xSuche in Google Scholar
[32] T. Yang, An explicit formula for local densities of quadratic forms, J. Number Theory 72 (1998), no. 2, 309–356. 10.1006/jnth.1998.2258Suche in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Weak approximation for isotrivial families
- Genus of abstract modular curves with level-ℓ structures
- Strong shift equivalence and algebraic K-theory
- The solvable length of groups of local diffeomorphisms
- Positivity properties of metrics and delta-forms
- Iwasawa theory for the symmetric square of a modular form
- Cohomology and torsion cycles over the maximal cyclotomic extension
- Integrability of continuous bundles
- Kodaira dimension of moduli of special cubic fourfolds
Artikel in diesem Heft
- Frontmatter
- Weak approximation for isotrivial families
- Genus of abstract modular curves with level-ℓ structures
- Strong shift equivalence and algebraic K-theory
- The solvable length of groups of local diffeomorphisms
- Positivity properties of metrics and delta-forms
- Iwasawa theory for the symmetric square of a modular form
- Cohomology and torsion cycles over the maximal cyclotomic extension
- Integrability of continuous bundles
- Kodaira dimension of moduli of special cubic fourfolds
