// The baby calculation of 3.2

RR<a,b,c,d,x,y,z,t> := PolynomialRing(Rationals(),8);

L := [a*x+b*y, a*z+b*t+c*x+d*y, c*z+d*t];
Dimension(Scheme(Spec(RR), L));
I := Ideal(L);
X := PrimaryDecomposition(I);
#X; X[2]; X[1];
(a*d-b*c)*(d*x-b*z) in I;
(x*t-y*z)*(d*x-b*z) in I;
(a*d-b*c)*(a*t-c*y) in I;
(x*t-y*z)*(a*t-c*y) in I;
(d*y-b*t+a*z-c*x) in I;
(a*d-b*c)*(d*y-b*t+a*z-c*x) in I;
(x*t-y*z)*(d*y-b*t+a*z-c*x) in I;
((a*d-b*c)*(x*t-y*z) - (a*z-c*x)^2) in I;
(a*d-b*c)*((a*d-b*c)*(x*t-y*z) - (a*z-c*x)^2) in I;
(x*t-y*z)*((a*d-b*c)*(x*t-y*z) - (a*z-c*x)^2) in I;