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Transversals and Tangents: Interpretation of Result
Up to now s and t were treated as variables. Now we have to consider them as parameters:
  ring S = (0,s,t), (a,b,c,d,e,f,g,h,k,l), lp;
  def F=imap(R,F);
One of the components is contained in E2:
  def G=F[1];
  G[1..7];
==> g f e d c b a
One component is all of S:
  std(F[6]);
==> _[1]=1
The other 6 components contain the desired data. As an example, let us consider one of these:
  std(F[3]);
==>
    _[1]=(s-1)*k2-2*kl-l2
    _[2]=(s-1)*h+(2t-2)*k+(t-1)*l
    _[3]=fl-gk
    _[4]=(s-1)*fk-2*gk-gl
    _[5]=(s-1)*f2-2*fg-g2
    _[6]=el-g2
    _[7]=ek-fg
    _[8]=d+f+g
    _[9]=c
    _[10]=2*b+e
    _[11]=a
    
  • Generators 2,8,9,10,11 are linear and linearly independent.
  • Generator 1 factorizes into 2 linear polynomials over the field extension of the rationals by the square root of s.
  • Modulo each of the two factors of generator 1, generator 3 decomposes into a linear factor and an excess factor in E3.
A direct computation then shows that these seven linear generators are linearly independent and that modulo those F[3] is generated by one quadratic polynomial.

The last thing which is left to check is that there are no allowed parameters s and t for which all 10 coordinates a,...,l vanish simultaneously.

KL, 06/03 http://www.singular.uni-kl.de