Pick $p,q,r$ complex numbers (I am most interested in the case when they are positive integers). Define the function

$P_i = px^i + qy^i + rz^i$

where $x,y,z$ are coordinates. I have a few related questions:

Has the algebra generated by $P_1, P_2, P_3, P_4, ...$ been considered before in the literature? For example, is it known how many $P_i$ are needed to generate the rest (this is a ring of invariants under a finite group for a finitely generated ring so there is always a bound for fixed $p,q,r$).

Has the Cohen-Macaulay property been considered for such algebras?

I am also interested in the related question where we impose the relation that $P_1 = 0$, i.e., $px+qy+rz=0$.

More generally, are there known criteria (say, on singularities) for a projective curve/surface that guarantee that it is arithmetically Cohen-Macaulay? (Probably there is nothing for surfaces, but I might hope it is possible for curves). Cohomological criteria aren't so useful in this situation, I think.

(This comes up in a research project related to subspace arrangements -- when $p,q,r$ are positive integers, these algebras are invariant rings under a symmetric group of their coordinate rings -- look at the locus in $C^{p+q+r}$ where there is a group of $p$ equal coordinates, $q$ equal coordinates, and $r$ equal coordinates).

**EDIT:** I know that the algebra can fail to be Cohen-Macaulay in some cases, for example when $(p,q,r)=(3,2,1)$, so I am interested in determining which parameters give which behavior.