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