This is a subtype of the HodgeSpecialFourfold type.
The object IntersectionOfThreeQuadricsInP7 is a type, with ancestor classes HodgeSpecialFourfold < EmbeddedProjectiveVariety < MultiprojectiveVariety < MutableHashTable < HashTable < Thing.
The source of this document is in SpecialFanoFourfolds.m2:3654:0.