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SpecialFanoFourfolds :: SpecialCubicFourfold

SpecialCubicFourfold -- the class of all special cubic fourfolds

Description

A cubic fourfold is a smooth cubic hypersurface in 5. A cubic fourfold X⊂ℙ5 is special of discriminant d>6 if it contains an algebraic surface S, and the discriminant of the saturated lattice spanned by h2 and [S] in H2,2(X,ℤ):=H4(X,ℤ)∩H2X2) is d, where h denotes the class of a hyperplane section of X. The set Cd of special cubic fourfolds of discriminant d is either empty or an irreducible divisor inside the moduli space of cubic fourfolds C. Moreover, Cd≠Ø 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 a couple (S,X), where X is (the principal ideal of) a cubic fourfold and S is (the ideal of) 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 method specialCubicFourfold, and the discriminant d can be calculated by the method discriminant(SpecialCubicFourfold).

Functions and methods returning an object of class SpecialCubicFourfold :

Methods that use an object of class SpecialCubicFourfold :

For the programmer

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