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SpecialCubicFourfold -- the class of all special cubic fourfolds


A cubic fourfold is a smooth cubic hypersurface in $\mathbb{P}^5$. A cubic fourfold $X\subset \mathbb{P}^5$ is special of discriminant $d>6$ if it contains an algebraic surface $S$, and the discriminant of the saturated lattice spanned by $h^2$ and $[S]$ in $H^{2,2}(X,\mathbb{Z}):=H^4(X,\mathbb{Z})\cap H^2(\Omega_X^2)$ is $d$, where $h$ denotes the class of a hyperplane section of $X$. The set $\mathcal{C}_d$ of special cubic fourfolds of discriminant $d$ is either empty or an irreducible divisor inside the moduli space of cubic fourfolds $\mathcal{C}$. Moreover, $\mathcal{C}_d\neq \emptyset$ if and only if $d>6$ and $d=$0 or 2 (mod 6). For the general theory, see the papers Special cubic fourfolds and Some rational cubic fourfolds, by B. Hassett.

An object of the class SpecialCubicFourfold is basically represented by a couple (S,X), where $X$ is a cubic fourfold and $S$ is a surface contained in $X$. The surface $S$ is required to be smooth or with at most a finite number $n$ of non-normal nodes. This number $n$ (if known) can be specified manually using the option NumNodes. The main constructor for the objects of the class is the function specialCubicFourfold.

See also

Functions and methods returning a special cubic fourfold :

Methods that use a special cubic fourfold :

For the programmer

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