F0, R0, and T1 should all be matrices over some common polynomial ring, and F0 and T1 should have one row. R0 should be the first syzygy matrix of F0 and T1 should have the same number of columns as F0.
F is a list of length two with F_0=F0 and F_1 the first order perturbations corresponding to T1. R is a list of length two with R_0=R0 and R_1 such that F_0*R_1+F_1*R_0==0. If SanityCheck is set to true, as is the default, then the algorithm will check that this equation is satisfied, and terminate with an error if this is not the case.
The parameters used in the perturbations may be specified by the option DefParam.
For example, consider the cone over the rational normal curve of degree four, see [Pi74]:
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=cotangentCohomology1(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) o6 = ({| -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 |, | t_1x_1+t_2x_0 t_4x_0 ------------------------------------------------------------------------ -t_1x_3-t_2x_2+t_4x_1 t_3x_2 -t_1x_4-t_2x_3+t_3x_3 t_3x_4-t_4x_3 |}, ------------------------------------------------------------------------ {{2} | x_3 x_2 0 x_4 x_3 0 0 0 |, {2} | t_4 0 0 {2} | -x_2 -x_1 x_4 0 0 0 x_4 x_3 | {2} | -t_2 t_1 0 {2} | x_1 x_0 -x_3 0 0 x_4 0 0 | {2} | 0 0 0 {2} | 0 0 -x_3 -x_2 -x_1 0 -x_3 -x_2 | {2} | 0 0 -t_4 {2} | 0 0 x_2 x_1 x_0 -x_3 0 0 | {2} | 0 0 0 {2} | 0 0 0 0 0 x_2 x_1 x_0 | {2} | 0 0 t_1 ------------------------------------------------------------------------ 0 0 0 0 0 |}) 0 -t_3 0 0 0 | -t_3 0 0 0 -t_3 | -t_2 t_1 0 -t_4 -t_3 | 0 0 -t_4 -t_3 0 | 0 0 t_2-t_3 0 0 | o6 : Sequence |