NestedDeformation -- The class of all nested deformations

Description

A NestedDeformation is a type of MutableHashTable encoding the deformation of a pair consisting of a scheme $Y$ and subscheme $X$ or of a module $M_Y$ and quotient module $M_X$. A NestedDeformation is not typically created by hand, but rather with the methods setupNestedDeformation, nestedHilbertScheme, nestedVersalDeformation, versalDeformation, or liftDeformation.

A NestedDeformation has keys "FX" , "RX" , "CX" , "FY" , "RY" , "CY" , "CXY" , "G" , "H" , "Xperturb" , "Yperturb" , "T2basis" , "O1" . The values of "FX" , "RX" , "G" , "CX" , and "FY" , "RY" , "G" , "CY" , are lists of matrices and respectively encode deformations of $X$ and $Y$ or $M_X$ and $M_Y$ as described in the documentation node for versalDeformation. The values of "CXY" , "H" , are also lists of matrices. Taking sums of all lists involved, one has the additional deformation equation $$ (\mathrm{transpose}\ \mathrm{flatten}\ (FX\cdot H-FY))+CXY\cdot G=0 $$ modulo an appropriate power of the maximal ideal demonstrating the deformation of $X$ (respectively $M_X$) as a subscheme (respectively quotient) of the deformation of $Y$ (respectively $M_Y$).

The values of "Xperturb" , "Yperturb" , "T2basis" , and "O1" are for internal use and may be ignored by the user.

The user may extract the obstruction equations and families of a NestedDeformation via the methods obstructions and families.

See nestedHilbertScheme and nestedVersalDeformation for several examples.

Methods that use an object of class NestedDeformation:

families(NestedDeformation) -- see families -- outputs the perturbed families of a NestedDeformation
isPolynomial(NestedDeformation) -- see isPolynomial -- checks if a deformation lifts to arbitrary order
liftDeformation(NestedDeformation) -- lift a nested deformation to the next order
obstructions(NestedDeformation) -- see obstructions -- outputs the obstruction equations of a NestedDeformation
sanityCheck(ZZ,NestedDeformation) -- see sanityCheck -- checks if deformation data satisfies the deformation equation at a certain order
versalDeformation(NestedDeformation) -- continues calculation of a versal deformation

For the programmer

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

The source of this document is in VersalDeformations.m2:941:0.