4.6 Deformations

We give three examples, the first being a hypersurface, the second a complete intersection, the third no complete intersection and compute in each of the cases the miniversal deformation:

LIB "deform.lib";
ring R=32003,(x,y,z),ds;
//----------------------------------------------------
// hypersurface case (from series T[p,q,r]):
int p,q,r = 3,3,4;
poly f = x^p+y^q+z^r+xyz;
print(deform(f));
==> z3,z2,yz,xz,z,y,x,1
// the miniversal deformation of f=0 is the projection from the
// miniversal total space to the miniversal base space:
// { (A,B,C,D,E,F,G,H,x,y,z) | x3+y3+xyz+z4+A+Bx+Cxz+Dy+Eyz+Fz+Gz2+Hz3 =0 }
//  --> { (A,B,C,D,E,F,G,H) }
//----------------------------------------------------
// complete intersection case (from series P[k,l]):
int k,l =3,2;
ideal j=xy,x^k+y^l+z2;
print(deform(j));
==> 0,0, 0,0,z,1,
==> y,x2,x,1,0,0 
versal(j);                  // using default names
==> // smooth base space
==> // ready: T1 and T2
==> 
==> // 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;
show(Px);                   // show is a procedure from inout.lib
==> // ring: (32003),(A,B,C,D,E,F,x,y,z),(ds(6),ds(3),C);
==> // minpoly = 0
listvar(ideal);
// ___ Equations of miniversal base space ___:
Js;
==> 
// ___ Equations of miniversal total space ___:
Fs;
==> Fs[1,1]=xy+Ez+F
==> Fs[1,2]=y2+z2+x3+Ay+Bx2+Cx+D
// the miniversal deformation of V(j) is the projection from the
// miniversal total space to the miniversal base space:
// { (A,B,C,D,E,F,x,y,z) | xy+A+Bz=0, y2+z2+x3+C+Dx+Ex2+Fy=0 }
//  --> { (A,B,C,D,E,F) }
//----------------------------------------------------
// general case (cone over rational normal curve of degree 4):
ring r1=0,(x,y,z,u,v),ds;
matrix m[2][4]=x,y,z,u,y,z,u,v;
ideal i=minor(m,2);                 // 2x2 minors of matrix m
int time=timer;
// Def_r is the name of the miniversal base space with
// parameters A(1),...,A(4)
versal(i,0,"Def_r","A(");
==> // ready: T1 and T2
==> 
==> // Result belongs to ring Def_rPx.
==> // Equations of total space of miniversal deformation are 
==> // given by Fs, equations of miniversal base space by Js.
==> // Make Def_rPx the basering and list objects defined in Def_rPx by t
==> yping:
==>    setring Def_rPx; show(Def_rPx);
==>    listvar(matrix);
==> // NOTE: rings Def_rQx, Def_rPx, Def_rSo are alive!
==> // (use 'kill_rings("Def_r");' to remove)
"// used time:",timer-time,"sec";   // time for miniversal
==> // used time: 0 sec
// the miniversal deformation of V(i) is the projection from the
// miniversal total space to the miniversal base space:
// { (A(1..4),x,y,z,u,v) |
//         -y^2+x*z+A(2)*x-A(3)*y=0, -y*z+x*u-A(1)*x-A(3)*z=0,
//         -y*u+x*v-A(3)*u-A(4)*z=0, -z^2+y*u-A(1)*y-A(2)*z=0,
//         -z*u+y*v-A(2)*u-A(4)*u=0, -u^2+z*v+A(1)*u-A(4)*v=0 }
//  --> { A(1..4) |
//         -A(1)*A(4) = A(3)*A(4) = -A(2)*A(4)-A(4)^2 = 0 }
//----------------------------------------------------