Kodaira dimension of moduli of special cubic fourfolds
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Kodaira dimension of moduli of special cubic fourfolds. / Tanimoto, Sho; Varilly-Alvarado, Anthony.
In: Journal fuer die Reine und Angewandte Mathematik, Vol. 2019, No. 752, 2019, p. 265-300.Research output: Contribution to journal › Journal article › Research › peer-review
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TY - JOUR
T1 - Kodaira dimension of moduli of special cubic fourfolds
AU - Tanimoto, Sho
AU - Varilly-Alvarado, Anthony
PY - 2019
Y1 - 2019
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
VL - 2019
SP - 265
EP - 300
JO - Journal fuer die Reine und Angewandte Mathematik
JF - Journal fuer die Reine und Angewandte Mathematik
SN - 0075-4102
IS - 752
ER -
ID: 142942429