Singular Manual: Deformations

| ### 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: (ZZ/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]=y2+z2+x3+Cy+Dx2+Ex+F
    ==> Fs[1,2]=xy+Az+B
      // 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.
    ==>
    ==>
    ==> // '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 }
      //----------------------------------------------------

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