Kodaira dimension of moduli of special cubic fourfolds

Abstract

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 𝒞 d {{\mathcal{C}}_{d}} in the moduli space 𝒞 {{\mathcal{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 {{\mathcal{C}}_{d}} . For example, if d = 6 ⁢ n + 2 {d=6n+2} , then we show that 𝒞 d {{\mathcal{C}}_{d}} is of general type for n > 18 {n>18} , n ∉ { 20 , 21 , 25 } {n\notin\{20,21,25\}} ; it has nonnegative Kodaira dimension if n > 13 {n>13} and n ≠ 15 {n\neq 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 𝒞 d {{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.