F0, T1, and T2 should all be matrices over some common polynomial ring, and F0 should have one row. If T1 and T2 are omitted, then bases of the first and second cotangent cohomology modules for F0 are used.
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. Furthermore, F_0=F0, R_0=gens ker F0, the columns of C_0 are multiples of those of T2 and F_1 consists of first order perturbations corresponding to T1. Thus, if T1 and T2 are tangent and obstruction spaces for some deformation functor, then F and G represent a versal family and equations for a versal base space.
Several options are available to control the termination of the calculation. The calculation will terminate at the very latest after reaching order equal to the option HighestOrder. If any single lifting step takes longer than TimeLimit the algorithm will terminate earlier. If PolynomialCheck is set to true, then the algorithm will check if the present solution lifts to infinite order and terminate if this is the case. If SanityCheck is set to true, then the algorithm will check that the present solution really does solve the deformation equation, and terminate if this is not the case. Finally, Verbosity may be used to control the verbosity of the output.
The option SmartLift is also available, which controls whether the algorithm spends extra time trying to find liftings which introduce no new obstructions at the next highest order. The option CorrectionMatrix may be used to control which liftings are considered.
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 : (F,R,G,C)=versalDeformation(F0); Calculating first order deformations and obstruction space Calculating first order relations Starting lifting Order 2 Order 3 Solution is polynomial |
Equations for a versal base space are
i5 : sum G o5 = | t_2t_3-t_3^2 | | t_1t_3 | | t_3t_4 | 3 1 o5 : Matrix (S[t , t , t , t ]) <--- (S[t , t , t , t ]) 1 2 3 4 1 2 3 4 |
The versal family is given by
i6 : sum F o6 = | 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 o6 : Matrix (S[t , t , t , t ]) <--- (S[t , t , t , t ]) 1 2 3 4 1 2 3 4 |
We may also consider the example of the cone over the del Pezzo surface of degree six:
i7 : S=QQ[x1,x2,x3,x4,x5,x6,z]; |
i8 : I=ideal {x1*x4-z^2,x2*x5-z^2,x3*x6-z^2,x1*x3-z*x2,x2*x4-z*x3,x3*x5-z*x4,x4*x6-z*x5,x5*x1-z*x6,x6*x2-z*x1}; o8 : Ideal of S |
i9 : F0=gens I; 1 9 o9 : Matrix S <--- S |
i10 : (F,R,G,C)=versalDeformation(F0); Calculating first order deformations and obstruction space Calculating first order relations Starting lifting Order 2 Order 3 Solution is polynomial |
Equations for a versal base space are
i11 : sum G o11 = | t_2t_3 | | t_1t_3-t_3^2 | 2 1 o11 : Matrix (S[t , t , t ]) <--- (S[t , t , t ]) 1 2 3 1 2 3 |
The versal family is given by
i12 : sum F o12 = | -t_1t_2+t_1z+t_2z+x1x4-z2 t_2z+t_3z+x2x5-z2 t_1z+x3x6-z2 ----------------------------------------------------------------------- t_1x2+x1x3-x2z t_2x3+x2x4-x3z t_3x4+x3x5-x4z t_1x5-t_3x5+x4x6-x5z ----------------------------------------------------------------------- t_2x6+t_3x6+x1x5-x6z x2x6-x1z | 1 9 o12 : Matrix (S[t , t , t ]) <--- (S[t , t , t ]) 1 2 3 1 2 3 |
We may also compute local multigraded Hilbert schemes. Here, we consider the Borel fixed ideal for the multigraded Hilbert scheme of the diagonal in a product of three projective planes:
i13 : S=QQ[x1,x2,x3,y1,y2,y3,z1,z2,z3,Degrees=> {{1,0,0},{1,0,0},{1,0,0},{0,1,0},{0,1,0},{0,1,0},{0,0,1},{0,0,1},{0,0,1}}]; |
i14 : I=ideal {y1*z2, x1*z2, y2*z1, y1*z1, x2*z1, x1*z1, x1*y2, x2*y1, x1*y1, x2*y2*z2}; o14 : Ideal of S |
i15 : (F,R,G,C)=versalDeformation(gens I,normalModule({0,0,0},gens I), CT^2({0,0,0},gens I)); Calculating first order relations Starting lifting Order 2 Order 3 Order 4 Order 5 Order 6 Solution is polynomial |
Local equations for the multigraded Hilbert scheme are
i16 : sum G o16 = | t_2t_3t_4-t_2t_4t_7-t_1t_3t_8+t_1t_7t_8+t_1t_3t_13-t_2t_3t_13-t_1t_7t | t_1t_3t_4-t_2t_3t_4-t_1t_7t_8+t_2t_7t_8-t_1t_3t_13+t_2t_3t_13+t_1t_7t | t_1t_3t_16-t_2t_7t_16-t_1t_14t_16+t_2t_14t_16-t_3t_15t_16+t_7t_15t_16 | t_1t_3t_18-t_2t_7t_18-t_1t_14t_18+t_2t_14t_18-t_3t_15t_18+t_7t_15t_18 | t_2t_4t_17-t_1t_8t_17+t_1t_13t_17-t_2t_13t_17-t_4t_15t_17+t_8t_15t_17 | t_2t_4t_18-t_1t_8t_18+t_1t_13t_18-t_2t_13t_18-t_4t_15t_18+t_8t_15t_18 | t_3t_4t_17-t_7t_8t_17-t_3t_13t_17+t_7t_13t_17-t_4t_14t_17+t_8t_14t_17 | t_3t_4t_16-t_7t_8t_16-t_3t_13t_16+t_7t_13t_16-t_4t_14t_16+t_8t_14t_16 ----------------------------------------------------------------------- _13+t_2t_7t_13-t_3t_4t_15+t_4t_7t_15+t_3t_8t_15-t_7t_8t_15 | _13-t_2t_7t_13-t_1t_4t_14+t_2t_4t_14+t_1t_8t_14-t_2t_8t_14 | | | | | | | 8 1 o16 : Matrix (S[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) <--- (S[t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t , t ]) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 |
At this point, the multigraded HIlbert scheme has 7 irreducible components:
i17 : # primaryDecomposition ideal sum G o17 = 7 |