Each element of the sequence (f,r,g,c) is a list of matrices in increasing powers of t. Their sums satisfy the deformation equation transpose ( (sum f)*(sum r))+(sum c)*sum(g)==0 up to powers of t equal to the length of f.
Each element of the output sequence (F,R,G,C) is a list of matrices in increasing powers of t. The first three matrices of the sequence are gotten from (f,r,g) by appending one matrix to each list in the sequence, and furthermore the columns of C_0 are multiples of those of c_0. The other matrices are chosen to satisfy the deformation equation transpose ((sum F)*(sum R))+(sum C)*(sum G)==0 up to powers of t equal to the length of F, provided that there is such a solution.
For example, consider the cone over the rational normal curve of degree four:
i1 : S=QQ[x_0..x_4]; |
i2 : I=minors(2,matrix {{x_0,x_1,x_2,x_3},{x_1,x_2,x_3,x_4}}); o2 : Ideal of S |
i3 : F0=gens I o3 = | -x_1^2+x_0x_2 -x_1x_2+x_0x_3 -x_2^2+x_1x_3 -x_1x_3+x_0x_4 ------------------------------------------------------------------------ -x_2x_3+x_1x_4 -x_3^2+x_2x_4 | 1 6 o3 : Matrix S <--- S |
i4 : T1=cotanComplexOne(F0); 6 4 o4 : Matrix S <--- S |
i5 : R0=gens ker F0; 6 8 o5 : Matrix S <--- S |
i6 : (f,r)=firstOrderDeformations(F0,R0,T1); |
We now lift the first order deformations to second order:
i7 : T2=cotanComplexTwo(F0); 8 3 o7 : Matrix S <--- S |
i8 : c={substitute(T2,ring f_0)}; |
i9 : g={}; |
i10 : (F,R,G,C)=liftDeformation(f,r,g,c); Calculating residual terms Lifting Family and Equations Lifting Relations and Coefficients Doing Sanity Check |
i11 : sum F -- equations for family o11 = | t_1x_1+t_2x_0-x_1^2+x_0x_2 t_4x_0-x_1x_2+x_0x_3 ----------------------------------------------------------------------- -t_1t_4-t_1x_3-t_2x_2+t_4x_1-x_2^2+x_1x_3 ----------------------------------------------------------------------- t_2t_3-t_3^2+t_3x_2-x_1x_3+x_0x_4 ----------------------------------------------------------------------- t_3t_4-t_1x_4-t_2x_3+t_3x_3-x_2x_3+x_1x_4 t_3x_4-t_4x_3-x_3^2+x_2x_4 | 1 6 o11 : Matrix (S[t , t , t , t ]) <--- (S[t , t , t , t ]) 1 2 3 4 1 2 3 4 |
i12 : sum G -- base equations o12 = | t_2t_3-t_3^2 | | t_1t_3 | | t_3t_4 | 3 1 o12 : Matrix (S[t , t , t , t ]) <--- (S[t , t , t , t ]) 1 2 3 4 1 2 3 4 |