D.5.2.2 mod_versal

Procedure from library deform.lib (see deform_lib).

Usage:

mod_versal(Mo,I[,d,any]); I=ideal, M=module, d=int, any =list

Compute:

miniversal deformation of coker(Mo) over Qo=Po/Io, Po=basering;

Create:

Ringsr (exported):
’my’Px = extending the basering by new variables (deformation parameters), the new variables precede the old ones,
the ordering is the product between "my_ord"
and "ord(Po)"
’my’Qx = Px/Io extending Qo (returns as basering),
’my’Ox = Px/(Io+Js) ring of the versal deformation of coker(Ms),
’my’So = embedding-ring of the versal base space. (default ’my’="")

Matrices (in Qx, exported):
Js = giving the versal base space (obstructions),
Ms = giving the versal family of Mo,
Ls = giving the lifting of syzygies Lo=syz(Mo),

If d is defined (!=0), it computes up to degree d.
If ’any’ is defined and any[1] is no string, interactive version.
Otherwise ’any’ gives predefined strings:"my","param","order","out" ("my" prefix-string, "param" is a letter (e.g. "A") for the name of first parameter or (e.g. "A(") for index parameter variables, "ord" ordering string for ring extension), "out" name of output-file).

Note:

printlevel < 0 no output at all,
printlevel >=0,1,2,.. informs you, what is going on,
this proc uses ’execute’.

Example:

LIB "deform.lib";
int p = printlevel;
printlevel = 1;
ring  Ro = 0,(x,y),wp(3,4);
ideal Io = x4+y3;
matrix Mo[2][2] = x2,y,-y2,x2;
mod_versal(Mo,Io);
→ // vdim (Ext^2) = 4
→ // vdim (Ext^1) = 4
→ // ready: Ext1 and Ext2
→ // Ext1 is quasi-homogeneous represented: 3,6,1,4
→ // infinitesimal extension
→ x2-Ax-B,   y+Cx+D,
→ -y2+Cxy+Dy,x2+Ax+B
→ // start deg = 2
→ // start deg = 3
→ // start deg = 4
→ // start deg = 5
→ // finished in degree 
→ 5
→ // quasi-homogeneous weights of miniversal base
→ 3,6,1,4
→ // Result belongs to qring Qx
→ // Equations of total space of miniversal deformation are in Js
→ -2AB+A3+3CD2-BC3-3AC2D+A2C3,
→ -B2+A2B+D3-3BC2D+ABC3
→ // Matrix of the deformed module is Ms and lifted syzygies are Ls.
→ // Make Qx the basering and list objects defined in Qx by typing:
→    listvar(ring);setring Qx; show(Qx);listvar(ideal);listvar(matrix);
→ // NOTE: rings Qx, Ox, So are still alive!
→ // (use: 'kill_rings();' to remove them)
printlevel = p;
if(system("with","Namespaces")) {
if(nameof(Current) == "Ring" ) {
kill Top::Px,Top::Qx,Top::So;
} else {
kill Ring::Px,Ring::So;
}
}
kill Px,Qx,So;