Kodaira dimension of moduli of special cubic fourfolds

Sho Tanimoto; Anthony Varilly-Alvarado

doi:10.1515/crelle-2016-0053

Kodaira dimension of moduli of special cubic fourfolds

Sho Tanimoto, Anthony Varilly-Alvarado

Institut for Matematiske Fag

2 Citationer (Scopus)

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 Cd in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the ?low-weight cusp form trick? of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of Cd . For example, if d = 6n + 2, then we show that Cd is of general type for n > 18, n {20, 21, 25}; it has nonnegative Kodaira dimension if n > 13 and n ≠ 15. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of Cd is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

Originalsprog	Engelsk
Tidsskrift	Journal fuer die Reine und Angewandte Mathematik
Vol/bind	2019
Udgave nummer	752
Sider (fra-til)	265-300
ISSN	0075-4102
DOI	https://doi.org/10.1515/crelle-2016-0053
Status	Udgivet - 1 jul. 2019

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10.1515/crelle-2016-0053

Citationsformater

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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 Cd in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the ?low-weight cusp form trick? of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of Cd . For example, if d = 6n + 2, then we show that Cd is of general type for n > 18, n {20, 21, 25}; it has nonnegative Kodaira dimension if n > 13 and n ≠ 15. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of Cd is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.",

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T1 - Kodaira dimension of moduli of special cubic fourfolds

AU - Tanimoto, Sho

AU - Varilly-Alvarado, Anthony

PY - 2019/7/1

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N2 - 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 Cd in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the ?low-weight cusp form trick? of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of Cd . For example, if d = 6n + 2, then we show that Cd is of general type for n > 18, n {20, 21, 25}; it has nonnegative Kodaira dimension if n > 13 and n ≠ 15. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of Cd is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

AB - 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 Cd in the moduli space C of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the ?low-weight cusp form trick? of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of Cd . For example, if d = 6n + 2, then we show that Cd is of general type for n > 18, n {20, 21, 25}; it has nonnegative Kodaira dimension if n > 13 and n ≠ 15. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of Cd is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

U2 - 10.1515/crelle-2016-0053

DO - 10.1515/crelle-2016-0053

M3 - Journal article

SN - 0075-4102

VL - 2019

SP - 265

EP - 300

JO - Journal fuer die Reine und Angewandte Mathematik

JF - Journal fuer die Reine und Angewandte Mathematik

IS - 752

ER -

Kodaira dimension of moduli of special cubic fourfolds

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