--Tony's exercise: let F be a Pgeneric* quartic form in k[x,y,z,t], is it possible to write F=LC+MNQ where deg L=deg M=dg N=1, deg C=3 e deg Q=2?
--this is equivalent to the following: consider the varieties of reducible forms
--V degree 3 forms splitting as (3,1)
--W degree 3 forms splitting as (2,1,1)
--is dim J(V,W)=34?
Use R::=Q[x,y,z,t];
--to compute the dimension of the join using Terracini's Lemma we need generic forms
L:=x;
M:=y;
N:=z;
C:=Randomized(DensePoly(3));
Q:=Randomized(DensePoly(2));
--tangent space to V in the point LC
I:=Ideal(L,C);
--tangent space to W in the point MNQ
J:=Ideal(M*N,M*Q,N*Q);
--then we need to compute the linear span of the tangent spaces
Hilbert(R/(I+J),4);
--hence the dimension of the join is 34-Hilbert(R/(I+J),4)