Kodaira dimension of moduli of special cubic fourfolds

TL;DR

This paper investigates the Kodaira dimension of moduli spaces of special cubic fourfolds, using modular forms to determine when these spaces are of general type or have nonnegative Kodaira dimension, narrowing down unknown cases.

Contribution

It applies the low-weight cusp form technique to classify the Kodaira dimension of Noether-Lefschetz divisors in the moduli space of cubic fourfolds, extending previous results.

Findings

C_d is of general type for n > 18, except for n in {20,21,25}

C_d has nonnegative Kodaira dimension for n > 13, excluding n=15

Only 20 values of d remain with unknown Kodaira dimension status

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.