|
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];
One component is all of S:
std(F[6]);
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.
|