D.5.2.1 versal

Procedure from library deform.lib (see deform_lib).

Usage:

versal(Fo[,d,any]); Fo=ideal, d=int, any=list

Compute:

miniversal deformation of Fo up to degree d (default d=100),

Create:

Rings (exported):
’my’Px = extending the basering Po by new variables given by "A,B,.." (deformation parameters), returns as basering; the new variables precede the old ones, the ordering is the product between "ls" and "ord(Po)"
’my’Qx = Px/Fo extending Qo=Po/Fo,
’my’So = the embedding-ring of the versal base space,
’my’Ox = Px/Js extending So/Js. (default my="")

Matrices (in Px, exported):
Js = giving the versal base space (obstructions),
Fs = giving the versal family of Fo,
Rs = giving the lifting of Ro=syz(Fo).

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, "order" 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     = 0;
ring r1        = 0,(x,y,z,u,v),ds;
matrix m[2][4] = x,y,z,u,y,z,u,v;
ideal Fo       = minor(m,2);
// cone over rational normal curve of degree 4
versal(Fo);
→ // ready: T_1 and T_2
→ // start computation in degree 2.
→ 
→ // Result belongs to ring Px.
→ // Equations of total space of miniversal deformation are 
→ // given by Fs, equations of miniversal base space by Js.
→ // Make Px the basering and list objects defined in Px by typing:
→    setring Px; show(Px);
→    listvar(matrix);
→ // NOTE: rings Qx, Px, So are alive!
→ // (use 'kill_rings("");' to remove)
setring Px;
// ___ Equations of miniversal base space ___:
Js;"";
→ Js[1,1]=BD
→ Js[1,2]=-AD+D2
→ Js[1,3]=-CD
→ 
// ___ Equations of miniversal total space ___:
Fs;"";
→ Fs[1,1]=-u2+zv+Bu+Dv
→ Fs[1,2]=-zu+yv-Au+Du
→ Fs[1,3]=-yu+xv+Cu+Dz
→ Fs[1,4]=z2-yu+Az+By
→ Fs[1,5]=yz-xu+Bx-Cz
→ Fs[1,6]=-y2+xz+Ax+Cy
→