liftDeformation(NestedDeformation) -- lift a nested deformation to the next order

Function: liftDeformation
Usage:

E = liftDeformation D
Inputs:
- D, an instance of the type NestedDeformation
Optional inputs:
- DegreeBound => ..., default value 0, determines the degree limit used to compute the tangent cone of obstruction equations
- SanityCheck => ..., default value true, checks if lifting solves deformation equation
- Verbose => ..., default value 0, control the verbosity of output
Outputs:
- E, an instance of the type NestedDeformation

Description

Lifts a NestedDeformation D to the next order. If SanityCheck is set to true, as is the default, then the algorithm will check that the lifted solution really does solve the deformation equation, and terminate with an error if this is not the case.

For example, consider embedded deformations of the fat points in the plane defined by the square and cube of the homogeneous maximal ideal. The following lifts a first order NestedDeformation for this pair to second order:

i1 : R=QQ[x,y];

i2 : F0Y=basis(3,R);

             1      4
o2 : Matrix R  <-- R

i3 : F0X=basis(2,R);

             1      3
o3 : Matrix R  <-- R

i4 : T1Y=normalMatrix(F0Y);

             4      12
o4 : Matrix R  <-- R

i5 : T1X=normalMatrix(F0X);

             3      6
o5 : Matrix R  <-- R

i6 : T2X=CT^2(F0X);

             2      1
o6 : Matrix R  <-- R

i7 : T2Y=CT^2(F0Y);

             3      3
o7 : Matrix R  <-- R

i8 : T2XY=nestedObstruction(F0X,F0Y);

             4      8
o8 : Matrix R  <-- R

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

i10 : E=liftDeformation D;

i11 : obstructions E

o11 = | 0                                                                    
      | 0                                                                    
      | 0                                                                    
      | 0                                                                    
      | s_1s_2+s_1s_3-s_5t_1-s_3t_2-s_1t_3+t_3t_4-t_4t_5-t_1t_6+t_2t_7-t_4t_7
      | s_2^2-s_2s_3+s_3^2+2s_1s_4-s_1s_5-s_6t_1-s_4t_2-s_2t_3+t_3t_5-t_5^2-t
      | s_3^2+s_1s_4-s_5t_4-s_3t_5-s_1t_6+t_4t_6+t_5t_7-t_7^2-t_4t_8-t_1t_9+t
      | s_2s_4+s_1s_6-s_6t_4-s_4t_5-s_2t_6+t_5t_6-t_2t_9+t_4t_9-t_1t_12      
      | s_3s_5+s_1s_6-s_5t_7-s_3t_8+t_7t_8-s_1t_9+t_4t_9-t_4t_11-t_1t_12     
      | s_4^2+s_3s_6-t_6^2-s_6t_7-s_4t_8+t_6t_8-s_2t_9+t_3t_9-t_5t_9+t_7t_9-t
      | s_4^2-s_4s_5+s_5^2-s_2s_6+2s_3s_6-t_6^2+t_6t_8-t_8^2+t_3t_9-2t_5t_9+t
      | s_4s_6+s_5s_6-t_6t_9-t_8t_9-s_6t_10+t_9t_10-s_4t_11+t_6t_11-s_2t_12+t
      -----------------------------------------------------------------------
                                                                            |
                                                                            |
                                                                            |
                                                                            |
      +t_1t_10                                                              |
      _2t_6+t_4t_6+t_5t_7-t_7^2+t_2t_8-2t_4t_8-t_1t_9+t_4t_10+t_1t_11       |
      _4t_10                                                                |
                                                                            |
                                                                            |
      _4t_12                                                                |
      _7t_9-s_5t_10+t_8t_10-s_3t_11+t_5t_11-t_7t_11-s_1t_12+t_2t_12-t_4t_12 |
      _3t_12-t_7t_12                                                        |

                                 12                         1
o11 : Matrix (R[s ..s , t ..t  ])   <-- (R[s ..s , t ..t  ])
                 1   6   1   12             1   6   1   12

Ways to use this method:

liftDeformation(NestedDeformation) -- lift a nested deformation to the next order

The source of this document is in VersalDeformations.m2:1470:0.