-- Parameterized curves
-- the coordinate ring of P^4
Target::=QQ[x[0..4]];
-- the coordinate ring of P^1
Domain::=QQ[s,t];
-- the ring where to perform elimination
Tot::=QQ[s,t,x[0..4]];
-- we make ready to work in the ring Tot
Use Tot;
-- the polynomials defining the parametrization of the curve, i.e. the map P^1 --> P^4
F0:=s^4;
F1:=s^3t;
F2:=s^2t^2;
F3:=st^3;
F4:=t^4;
-- the ideal we need to perfom elimination
I:=Ideal(x[0]-F0,x[1]-F1,x[2]-F2,x[3]-F3,x[4]-F4);
-- eliminating s and t we determine the kernel and hence the ideal of the curve
K:=Elim(s..t,I);
-- we make ready to work in the coordinate ring of P^4
Use Target;
-- we bring the generators of K in the Target ring
L:=BringIn(Gens(K));
-- we define the ideal of the curve
IC:=Ideal(L);
-- the Hilbert function of the curve
Hilbert(Target/IC);
-- Es 1a try to drop one of the Fi at the time and find the equations of the curve in P^3; using the Hilbert function determine the dimension of the spaces of degree 1 and 2 forms containing them.
-- Es 2a determine the equation of some rational plane curves of degrees 3 and 4; are these curves smooth? how can we answer using the Hilbert function?