D.5.8.17 T_12

Procedure from library sing.lib (see sing_lib).

Usage:

T_12(i[,any]); i = ideal

Return:

T_12(i): list of 2 modules:
* standard basis of T_1-module =T_1(i), 1st order deformations
* standard basis of T_2-module =T_2(i), obstructions of R=P/i
If a second argument is present (of any type) return a list of 9 modules, matrices, integers:
[1]= standard basis of T_1-module
[2]= standard basis of T_2-module
[3]= vdim of T_1
[4]= vdim of T_2
[5]= matrix, whose cols present infinitesimal deformations
[6]= matrix, whose cols are generators of relations of i(=syz(i))
[7]= matrix, presenting Hom_P(syz/kos,R), lifted to P
[8]= presentation of T_1-module, no std basis
[9]= presentation of T_2-module, no std basis

Display:

k-dimension of T_1 and T_2 if printlevel >= 0 (default)

Note:

Use proc miniversal from deform.lib to get miniversal deformation of i, the list contains all objects used by proc miniversal

Example:

LIB "sing.lib";
int p      = printlevel;
printlevel = 1;
ring r     = 200,(x,y,z,u,v),(c,ws(4,3,2,3,4));
ideal i    = xz-y2,yz2-xu,xv-yzu,yu-z3,z2u-yv,zv-u2;
//a cyclic quotient singularity
list L     = T_12(i,1);
→ // dim T_1 = 5
→ // dim T_2 = 3
print(L[5]);             //matrix of infin. deformations
→ 0,  0,  0,  0,  0,  
→ yz, y,  z2, 0,  0,  
→ -z3,-z2,-zu,yz, yu, 
→ -z2,-z, -u, 0,  0,  
→ zu, u,  v,  -z2,-zu,
→ 0,  0,  0,  u,  v   
printlevel = p;