Description
F0X and F0Y are matrices over a common ring and respectively encode equations a subscheme/quotient module and its ambient scheme/module.
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.
If Projective is set to true then only degree zero deformations are considered.
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 computes the versal deformation of this pair:
i1 : R=QQ[x,y];
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i2 : F0Y=basis(3,R);
1 4
o2 : Matrix R <-- R
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i3 : F0X=basis(2,R);
1 3
o3 : Matrix R <-- R
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i4 : D=nestedVersalDeformation(F0X,F0Y,Verbose=>4);
Calculating first order relations
Calculating standard expressions for obstructions
Eliminating unnecessary deformation parameters
Starting lifting
Order 2
Lifting family for X
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Lifting family for Y
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Calculating tangent cone for new obstructions
Calculating joint residual terms
Calculating Joint Obstruction Equations
Changing Families
Changing Relations
Lifting Submodule Relation
Correcting Coefficients
Doing Sanity Check
Checking polynomial lifting
Order 3
Lifting family for X
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Lifting family for Y
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Calculating tangent cone for new obstructions
Calculating joint residual terms
Calculating Joint Obstruction Equations
Changing Families
Changing Relations
Lifting Submodule Relation
Correcting Coefficients
Doing Sanity Check
Checking polynomial lifting
Solution is polynomial
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i5 : transpose (families D)_0
o5 = {-2} | s_2s_3-s_3^2-s_1s_4+s_1s_5+ys_1+xs_2+x2 |
{-2} | s_3s_4-s_1s_6+ys_3+xs_4+xy |
{-2} | -s_4^2+s_4s_5+s_2s_6-s_3s_6+ys_5+xs_6+y2 |
3 1
o5 : Matrix (R[s ..s , t ..t ]) <-- (R[s ..s , t ..t ])
1 6 1 10 1 6 1 10
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i6 : transpose (families D)_1
o6 = {-3} | -t_3t_4t_7+t_4t_5t_7+t_1t_6t_7-t_1t_7^2+t_2t_4t_8-t_4^2t_8-t_1t
{-3} | -t_4t_6t_7+t_1t_7t_8+t_2t_4t_10-t_4^2t_10-t_1t_5t_10+yt_4t_6+xt
{-3} | t_4t_6t_8-t_1t_8^2-t_4t_6t_9+t_1t_8t_9-t_3t_4t_10+t_4t_5t_10+t_
{-3} | t_6^2t_7-t_6t_7^2-t_5t_6t_8-t_3t_7t_8+t_5t_7t_8+t_2t_8^2-t_4t_8
------------------------------------------------------------------------
_5t_8-t_2t_4t_9+t_4^2t_9+t_1t_5t_9+yt_3t_4+xt_3t_5-yt_4t_5-xt_5^2-yt_1t_
_5t_6-yt_4t_7-yt_1t_8-xt_2t_8+xt_4t_8-xt_1t_10+y2t_4+xyt_5+x2t_6+x2y
1t_6t_10-t_1t_7t_10-xt_6^2+xt_6t_7+xt_3t_8+yt_4t_8-xt_5t_8-yt_4t_9-yt_1t
^2+t_5t_6t_9-t_2t_8t_9+t_4t_8t_9+t_3t_5t_10-t_5^2t_10-t_2t_6t_10+t_4t_6t
------------------------------------------------------------------------
6-xt_2t_6+xt_4t_6+xt_2t_7-2xt_4t_7-xt_1t_8+xt_1t_9+y2t_1+xyt_2+x2t_3+x3
_10-xt_4t_10+xyt_7+x2t_8+xy2
_10+t_2t_7t_10-t_4t_7t_10-yt_6^2+yt_6t_7-yt_7^2+yt_3t_8-2yt_5t_8-xt_6t_8
------------------------------------------------------------------------
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-xt_7t_8+yt_5t_9+xt_6t_9+yt_2t_10+xt_3t_10-yt_4t_10+xyt_9+x2t_10+y3 |
4 1
o6 : Matrix (R[s ..s , t ..t ]) <-- (R[s ..s , t ..t ])
1 6 1 10 1 6 1 10
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i7 : obstructions D
o7 = | 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
| 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_4t_7-t_1t_8
| s_2s_4+s_1s_6-s_6t_4-s_4t_5-s_2t_6+t_5t_6-t_2t_8+t_4t_8-t_1t_10
| s_3s_5+s_1s_6-s_3t_7-s_1t_8+t_4t_8-t_4t_9-t_1t_10
| s_4^2+s_3s_6-t_6^2-s_4t_7+t_6t_7-s_2t_8+t_3t_8-t_5t_8-t_4t_10
| s_4^2-s_4s_5+s_5^2-s_2s_6+2s_3s_6-t_6^2+t_6t_7-t_7^2+t_3t_8-2t_5t_8-s_
| s_4s_6+s_5s_6-t_6t_8-t_7t_8-s_4t_9+t_6t_9-s_2t_10+t_3t_10
------------------------------------------------------------------------
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2t_6+t_4t_6+t_2t_7-2t_4t_7-t_1t_8+t_1t_9 |
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3t_9+t_5t_9-s_1t_10+t_2t_10-t_4t_10 |
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12 1
o7 : Matrix (R[s ..s , t ..t ]) <-- (R[s ..s , t ..t ])
1 6 1 10 1 6 1 10
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Setting IncludeTrivial to false will limit possible first order deformations of the subscheme/quotient module to those coming from CT^1 F0X, resulting in a subfamily:
i8 : D=nestedVersalDeformation(F0X,F0Y,Verbose=>4,IncludeTrivial=>false);
Calculating first order relations
Calculating standard expressions for obstructions
Eliminating unnecessary deformation parameters
Starting lifting
Order 2
Lifting family for X
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Lifting family for Y
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Calculating tangent cone for new obstructions
Calculating joint residual terms
Calculating Joint Obstruction Equations
Changing Families
Changing Relations
Lifting Submodule Relation
Correcting Coefficients
Doing Sanity Check
Checking polynomial lifting
Order 3
Lifting family for X
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Lifting family for Y
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Calculating tangent cone for new obstructions
Calculating joint residual terms
Calculating Joint Obstruction Equations
Changing Families
Changing Relations
Lifting Submodule Relation
Correcting Coefficients
Doing Sanity Check
Checking polynomial lifting
Order 4
Lifting family for X
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Lifting family for Y
Calculating tangent cone for obstructions
Calculating residual terms
Lifting Family
Calculating Obstruction Equations
Lifting Relations and Coefficients
Calculating tangent cone for new obstructions
Calculating joint residual terms
Calculating Joint Obstruction Equations
Changing Families
Changing Relations
Lifting Submodule Relation
Correcting Coefficients
Doing Sanity Check
Checking polynomial lifting
Solution is polynomial
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i9 : transpose (families D)_0
o9 = {-2} | -s_1s_3+ys_1+xs_2+x2 |
{-2} | -s_1s_4+xs_3+xy |
{-2} | -s_3^2+s_2s_4+xs_4+y2 |
3 1
o9 : Matrix (R[s ..s , t ..t ]) <-- (R[s ..s , t ..t ])
1 4 1 10 1 4 1 10
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i10 : transpose (families D)_1
o10 = {-3} | -t_3t_4t_7+t_4t_5t_7+t_1t_6t_7-t_1t_7^2+t_2t_4t_8-t_4^2t_8-t_
{-3} | -t_4t_6t_7+t_1t_7t_8+t_2t_4t_10-t_4^2t_10-t_1t_5t_10+yt_4t_6+
{-3} | t_4t_6t_8-t_1t_8^2-t_4t_6t_9+t_1t_8t_9-t_3t_4t_10+t_4t_5t_10+
{-3} | -s_4t_4t_8+s_1t_8^2-t_4t_8^2+s_4t_4t_9-s_1t_8t_9+2t_4t_8t_9-t
-----------------------------------------------------------------------
1t_5t_8-t_2t_4t_9+t_4^2t_9+t_1t_5t_9+yt_3t_4+xt_3t_5-yt_4t_5-xt_5^2
xt_5t_6-yt_4t_7-yt_1t_8-xt_2t_8+xt_4t_8-xt_1t_10+y2t_4+xyt_5+x2t_6+
t_1t_6t_10-t_1t_7t_10-xt_6^2+xt_6t_7+xt_3t_8+yt_4t_8-xt_5t_8-yt_4t_
_4t_9^2+s_4t_1t_10-s_1t_6t_10+t_4t_6t_10-t_4t_7t_10-t_1t_9t_10-yt_6
-----------------------------------------------------------------------
-yt_1t_6-xt_2t_6+xt_4t_6+xt_2t_7-2xt_4t_7-xt_1t_8+xt_1t_9+y2t_1+xyt_2+
x2y
9-yt_1t_10-xt_4t_10+xyt_7+x2t_8+xy2
^2+yt_6t_7-yt_7^2+yt_3t_8-2yt_5t_8-xt_6t_8-xt_7t_8+yt_5t_9+xt_6t_9+yt_
-----------------------------------------------------------------------
x2t_3+x3 |
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2t_10+xt_3t_10-yt_4t_10+xyt_9+x2t_10+y3 |
4 1
o10 : Matrix (R[s ..s , t ..t ]) <-- (R[s ..s , t ..t ])
1 4 1 10 1 4 1 10
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i11 : obstructions D
o11 = | 0
| 0
| 0
| 0
| s_1s_2-s_1t_3+t_3t_4-t_4t_5-t_1t_6
| s_2^2+2s_1s_3-s_4t_1-s_3t_2-s_2t_3+t_3t_5-t_5^2-t_2t_6+t_4t_6+t_2t_7-
| s_1s_3-s_1t_6+t_4t_6-t_4t_7-t_1t_8
| s_2s_3+s_1s_4-s_4t_4-s_3t_5-s_2t_6+t_5t_6-t_2t_8+t_4t_8-t_1t_10
| s_1s_4-s_1t_8+t_4t_8-t_4t_9-t_1t_10
| s_3^2-t_6^2-s_3t_7+t_6t_7-s_2t_8+t_3t_8-t_5t_8-t_4t_10
| s_3^2-s_2s_4-t_6^2+t_6t_7-t_7^2+t_3t_8-2t_5t_8+t_5t_9-s_1t_10+t_2t_10
| s_3s_4-t_6t_8-t_7t_8-s_3t_9+t_6t_9-s_2t_10+t_3t_10
-----------------------------------------------------------------------
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2t_4t_7-t_1t_8+t_1t_9 |
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-t_4t_10 |
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12 1
o11 : Matrix (R[s ..s , t ..t ]) <-- (R[s ..s , t ..t ])
1 4 1 10 1 4 1 10
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