Macaulay2 » Documentation
Packages » SpecialFanoFourfolds :: SpecialGushelMukaiFourfold
next | previous | forward | backward | up | index | toc

SpecialGushelMukaiFourfold -- the class of all special Gushel-Mukai fourfolds


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.

See also

Functions and methods returning a special Gushel-Mukai fourfold :

Methods that use a special Gushel-Mukai fourfold :

For the programmer

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