An (ordinary) Gushel-Mukai fourfold is the intersection of a smooth del Pezzo fivefold G(1,4)∩ℙ8⊂ℙ8 with a quadric hypersurface in ℙ8. A Gushel-Mukai fourfold is said to be special if it contains a surface whose cohomology class does not come from the Grassmannian G(1,4). The special Gushel-Mukai fourfolds are parametrized by a countable union of (not necessarily irreducible) hypersurfaces in the corresponding moduli space, labelled by the integers d ≥10 with d = 0, 2, 4 (mod 8); the number d is called the discriminant of the fourfold. For precise definition and results, we refer mainly to the paper Special prime Fano fourfolds of degree 10 and index 2, by O. Debarre, A. Iliev, and L. Manivel.
An object of the class SpecialGushelMukaiFourfold is basically a couple (S,X), where X is (the ideal of) a Gushel-Mukai fourfold and S is (the ideal of) a surface contained in X. The main constructor for the objects of the class is the method specialGushelMukaiFourfold, and the discriminant d can be calculated by the method discriminant(SpecialGushelMukaiFourfold).
The object SpecialGushelMukaiFourfold is a type, with ancestor classes MutableHashTable < HashTable < Thing.