Kodaira dimension of moduli of special cubic fourfolds
Abstract
A special cubic fourfold is a smooth hypersurface of degree three and dimension four that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether-Lefschetz divisors C_d 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 C_d. For example, if d = 6n + 2, then we show that C_d is of general type for n > 18, n not in {20,21,25}, it has nonnegative Kodaira dimension if n > 13 and if n is not equal to 15. In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only 20 values of d for which no information on the Kodaira dimension of C_d is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.
Keywords
Cite
@article{arxiv.1509.01562,
title = {Kodaira dimension of moduli of special cubic fourfolds},
author = {Sho Tanimoto and Anthony Várilly-Alvarado},
journal= {arXiv preprint arXiv:1509.01562},
year = {2016}
}
Comments
36 pages, Minor expository changes (e.g. Lemma 4.3 and Remark 4.4). References added. Magma scripts included as ancillary files in the arxiv distribution