2015/09/04 by Sho Tanimoto, Anthony Várilly-Alvarado, Tanimoto, Sho +1
Mathematics · #14J15 #32M15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Primary 14J35 #Secondary 32N15 #math.AG #math.NT #msc:14J15 #msc:14J35 #msc:32M15 #msc:32N15
paper · pdf · doi:10.48550/arxiv.1509.01562
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
arxiv created 2016/08/14 · arxiv updated 2016/08/16
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 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 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 Cd is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.