Kodaira dimension of moduli of special cubic fourfolds

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  • Sho Tanimoto
  • Anthony Varilly-Alvarado
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.
TidsskriftJournal fuer die Reine und Angewandte Mathematik
Udgave nummer752
Sider (fra-til)265-300
StatusUdgivet - 2019

ID: 142942429