setupNestedDeformation -- create a first order NestedDeformation from tangent directions and obstruction spaces

Usage:

D = setupNestedDeformation(L)
Inputs:
- L, a list
Optional inputs:
- DefParamX => ..., default value "s"
- DefParamY => ..., default value "t"
- SanityCheck => ..., default value true
- Verbose => ..., default value 0
Outputs:
- D, an instance of the type NestedDeformation

Description

L is a list of the form L={F0X,T1X,T2X,F0Y,T1Y,T2Y,T2XY} or of the form L={F0X,T2X,F0Y,T2Y,T1XY,T2XY} consisting of matrices over a common ring. In the first case, F0X,T1X,T2X and F0Y,T1Y,T2Y satisfy the requirements for the input of versalDeformation(Matrix,Matrix,Matrix) and respectively encode equations, first order deformations, and obstructions for deformations of a subscheme/quotient module and its ambient scheme/module. The matrix T2XY encodes a nested obstruction space for the pair as output by nestedObstruction.

In the second case, the input requirements are similar except that T1XY is a matrix encoding first order deformations of the pair, for example as output by nestedTangent.

If SanityCheck is set to true, as is the default, then the algorithm will check that the relevant deformation equations are satisfied, and terminate with an error if this is not the case.

The parameters used in the perturbations may be specified by the options DefParamX and DefParamY.

For example, consider the cone over the rational normal curve of degree four and a fat point at the origin cut out by the square of the homogeneous maximal ideal. The following creates a NestedDeformation encoding first order deformations of this pair:

i1 : R=QQ[x_0..x_4];

i2 : F0Y=gens minors(2,matrix {{x_0,x_1,x_2,x_3},{x_1,x_2,x_3,x_4}});

             1      6
o2 : Matrix R  <-- R

i3 : F0X=basis(2,R);

             1      15
o3 : Matrix R  <-- R

i4 : T1Y=CT^1(F0Y);

             6      4
o4 : Matrix R  <-- R

i5 : T1X=normalMatrix(F0X);

             15      75
o5 : Matrix R   <-- R

i6 : T2X=CT^2(F0X);

             40      185
o6 : Matrix R   <-- R

i7 : T2Y=CT^2(F0Y);

             8      3
o7 : Matrix R  <-- R

i8 : T2XY=nestedObstruction(F0X,F0Y);

             6
o8 : Matrix R  <-- 0

i9 : D=setupNestedDeformation({F0X,T1X,T2X,F0Y,T1Y,T2Y,T2XY})

o9 = D

o9 : NestedDeformation

i10 : transpose (families D)_0

o10 = {-2} | x_4s_1+x_3s_2+x_2s_3+x_1s_4+x_0s_5+x_0^2                       
      {-2} | x_4s_6+x_3s_7+x_2s_8+x_1s_9+x_0s_10+x_0x_1                     
      {-2} | x_4s_26+x_3s_27+x_2s_28+x_1s_29+x_0s_30+x_1t_1+x_0t_2+x_0x_2   
      {-2} | x_4s_31+x_3s_32+x_2s_33+x_1s_34+x_0s_35+x_0t_4+x_0x_3          
      {-2} | x_4s_46+x_3s_47+x_2s_48+x_1s_49+x_0s_50-x_3t_1-x_2t_2+x_2t_3+x_
      {-2} | x_4s_26+x_3s_27+x_2s_28+x_1s_29+x_0s_30+x_1^2                  
      {-2} | x_4s_31+x_3s_32+x_2s_33+x_1s_34+x_0s_35+x_1x_2                 
      {-2} | x_4s_46+x_3s_47+x_2s_48+x_1s_49+x_0s_50-x_3t_1-x_2t_2+x_1t_4+x_
      {-2} | x_4s_51+x_3s_52+x_2s_53+x_1s_54+x_0s_55-x_4t_1-x_3t_2+x_3t_3+x_
      {-2} | x_4s_46+x_3s_47+x_2s_48+x_1s_49+x_0s_50+x_2^2                  
      {-2} | x_4s_51+x_3s_52+x_2s_53+x_1s_54+x_0s_55+x_2x_3                 
      {-2} | x_4s_61+x_3s_62+x_2s_63+x_1s_64+x_0s_65+x_4t_3-x_3t_4+x_2x_4   
      {-2} | x_4s_61+x_3s_62+x_2s_63+x_1s_64+x_0s_65+x_3^2                  
      {-2} | x_4s_66+x_3s_67+x_2s_68+x_1s_69+x_0s_70+x_3x_4                 
      {-2} | x_4s_71+x_3s_72+x_2s_73+x_1s_74+x_0s_75+x_4^2                  
      -----------------------------------------------------------------------
                  |
                  |
                  |
                  |
      1t_4+x_0x_4 |
                  |
                  |
      1x_3        |
      1x_4        |
                  |
                  |
                  |
                  |
                  |
                  |

                                                               15                                                       1
o10 : Matrix (R[s ..s  , s  ..s  , s  ..s  , s  ..s  , t ..t ])   <-- (R[s ..s  , s  ..s  , s  ..s  , s  ..s  , t ..t ])
                 1   10   26   35   46   55   61   75   1   4             1   10   26   35   46   55   61   75   1   4

i11 : transpose (families D)_1

o11 = {-2} | x_1t_1+x_0t_2-x_1^2+x_0x_2          |
      {-2} | x_0t_4-x_1x_2+x_0x_3                |
      {-2} | -x_3t_1-x_2t_2+x_1t_4-x_2^2+x_1x_3  |
      {-2} | x_2t_3-x_1x_3+x_0x_4                |
      {-2} | -x_4t_1-x_3t_2+x_3t_3-x_2x_3+x_1x_4 |
      {-2} | x_4t_3-x_3t_4-x_3^2+x_2x_4          |

                                                               6                                                       1
o11 : Matrix (R[s ..s  , s  ..s  , s  ..s  , s  ..s  , t ..t ])  <-- (R[s ..s  , s  ..s  , s  ..s  , s  ..s  , t ..t ])
                 1   10   26   35   46   55   61   75   1   4            1   10   26   35   46   55   61   75   1   4

Ways to use setupNestedDeformation:

setupNestedDeformation(List)

For the programmer

The object setupNestedDeformation is a method function with options.

The source of this document is in VersalDeformations.m2:1542:0.