The general type of Gushel-Mukai fourfold (called *ordinary*) can be realized as the intersection of a smooth del Pezzo fivefold $\mathbb{G}(1,4)\cap\mathbb{P}^8\subset \mathbb{P}^8$ with a quadric hypersurface in $\mathbb{P}^8$. A Gushel-Mukai fourfold is said to be *special* if it contains a surface whose cohomology class *does not come* from the Grassmannian $\mathbb{G}(1,4)$. 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 $d \geq 10$ 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 represented by a couple (S,X), where $X$ is a Gushel-Mukai fourfold and $S$ is a surface contained in $X$. The main constructor for the objects of the class is the function specialGushelMukaiFourfold.

- discriminant(SpecialGushelMukaiFourfold) -- discriminant of a special Gushel-Mukai fourfold

- specialGushelMukaiFourfold -- make a special Gushel-Mukai fourfold

- associatedK3surface(SpecialGushelMukaiFourfold) -- K3 surface associated to a rational Gushel-Mukai fourfold
- detectCongruence(SpecialGushelMukaiFourfold) -- see detectCongruence(SpecialGushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold
- detectCongruence(SpecialGushelMukaiFourfold,ZZ) -- detect and return a congruence of (2e-1)-secant curves of degree e inside a del Pezzo fivefold
- discriminant(SpecialGushelMukaiFourfold) -- discriminant of a special Gushel-Mukai fourfold
- fromOrdinaryToGushel(SpecialGushelMukaiFourfold) -- see fromOrdinaryToGushel -- try to deform to a fourfold of Gushel type
- isAdmissibleGM(SpecialGushelMukaiFourfold) -- see isAdmissibleGM -- whether an integer is admissible (in the sense of the theory of GM fourfolds)
- map(SpecialGushelMukaiFourfold) -- associated quadratic map
- mirrorFourfold(SpecialGushelMukaiFourfold) -- see mirrorFourfold -- associated fourfold to a rational cubic or GM fourfold
- parameterCount(SpecialGushelMukaiFourfold) -- count of parameters in the moduli space of GM fourfolds
- toGrass(SpecialGushelMukaiFourfold) -- see toGrass -- Gushel morphism from a GM fourfold to GG(1,4)
- unirationalParametrization(SpecialGushelMukaiFourfold) -- see unirationalParametrization -- unirational parametrization

The object SpecialGushelMukaiFourfold is a type, with ancestor classes HodgeSpecialFourfold < EmbeddedProjectiveVariety < MultiprojectiveVariety < MutableHashTable < HashTable < Thing.