An object of the class HodgeSpecialFourfold is basically represented by a couple (S,X), where $X$ is a fourfold and $S$ is a surface contained in $X$. Such objects are created by the function specialFourfold.
The object HodgeSpecialFourfold is a type, with ancestor classes EmbeddedProjectiveVariety < MultiprojectiveVariety < MutableHashTable < HashTable < Thing.