A.4.5 Deformations

• The libraries sing.lib, respextively deform.lib, contain procedures to compute total and base space of the miniversal (= semiuniversal) deformation of an isolated complete intersection singularity, respectively of an arbitrary isolated singularity.
• The procedure deform in sing.lib returns a matrix whose columns represent all 1st order deformations. More precisely, if is the ideal generated by , then any infinitesimal deformation of over is given by , where , and where is a -linear combination of the .
• The procedure versal in deform.lib computes a formal miniversal deformation up to a certain order which can be prescribed by the user. For a complete intersection the 1st order part is already miniversal.
• The procedure versal extends the basering to a new ring with additional deformation parameters which contains the equations for the miniversal base space and the miniversal total space.
• There are default names for the objects created, but the user may also choose their own names.
• If the user sets printlevel=2; before running versal, some intermediate results are shown. This is useful since versal is already complicated and might run for some time on more complicated examples. (type help versal;)

We compute for the same examples as in the section T1 and T2 the miniversal deformations:

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 def L=versal(j); // using default names ==> // smooth base space ==> // ready: T_1 and T_2 ==> ==> ==> // 'versal' returned a list, say L, of four rings. In L[1] are stored: ==> // as matrix Fs: Equations of total space of the miniversal deformation\ , ==> // as matrix Js: Equations of miniversal base space, ==> // as matrix Rs: syzygies of Fs mod Js. ==> // To access these data, type ==> def Px=L[1]; setring Px; print(Fs); print(Js); print(Rs); ==> ==> // L[2] = L[1]/Fo extending Qo=Po/Fo, ==> // L[3] = the embedding ring of the versal base space, ==> // L[4] = L[1]/Js extending L[3]/Js. ==> def Px=L[1]; 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 ==> // objects belonging to this ring: ==> // Rs [0] matrix 2 x 1 ==> // Fs [0] matrix 1 x 2 ==> // Js [0] matrix 1 x 0 listvar(matrix); ==> // Rs [0] matrix 2 x 1 ==> // Fs [0] matrix 1 x 2 ==> // Js [0] matrix 1 x 0 // ___ 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+F+Ez=0, y2+z2+x3+D+Cx+Bx2+Ay=0 } // --> { (A,B,C,D,E,F) } //---------------------------------------------------- // general case (cone over rational normal curve of degree 4): kill L; 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; // Call parameters of the miniversal base A(1),A(2),...: def L=versal(i,0,"","A("); ==> // ready: T_1 and T_2 ==> // start computation in degree 2. ==> // ** J is no standard basis ==> ==> ==> // 'versal' returned a list, say L, of four rings. In L[1] are stored: ==> // as matrix Fs: Equations of total space of the miniversal deformation\ , ==> // as matrix Js: Equations of miniversal base space, ==> // as matrix Rs: syzygies of Fs mod Js. ==> // To access these data, type ==> def Px=L[1]; setring Px; print(Fs); print(Js); print(Rs); ==> ==> // L[2] = L[1]/Fo extending Qo=Po/Fo, ==> // L[3] = the embedding ring of the versal base space, ==> // L[4] = L[1]/Js extending L[3]/Js. ==> "// used time:",timer-time,"sec"; // time of last command ==> // used time: 0 sec def Def_rPx=L[1]; setring Def_rPx; Fs; ==> Fs[1,1]=-u^2+z*v+A(2)*u+A(4)*v ==> Fs[1,2]=-z*u+y*v-A(1)*u+A(4)*u ==> Fs[1,3]=-y*u+x*v+A(3)*u+A(4)*z ==> Fs[1,4]=z^2-y*u+A(1)*z+A(2)*y ==> Fs[1,5]=y*z-x*u+A(2)*x-A(3)*z ==> Fs[1,6]=-y^2+x*z+A(1)*x+A(3)*y Js; ==> Js[1,1]=A(2)*A(4) ==> Js[1,2]=A(1)*A(4)-A(4)^2 ==> Js[1,3]=-A(3)*A(4) // 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) | // -u^2+x*v+A(2)*u+A(4)*v=0, -z*u+y*v-A(1)*u+A(3)*u=0, // -y*u+x*v+A(3)*u+A(4)*z=0, z^2-y*u+A(1)*z+A(2)*y=0, // y*z-x*u+A(2)*x-A(3)*z=0, -y^2+x*z+A(1)*x+A(3)*y=0 } // --> { A(1..4) | // A(2)*A(4) = -A(3)*A(4) = -A(1)*A(4)+A(4)^2 = 0 } //----------------------------------------------------