What are the closed subschemes of $\mathbb P^2_R$ for a ring $R$? In particular, what are the closed subschemes of $\mathbb P^2_k$ for a field $k$?

Question

I am making myself acquainted with the Hilbert Functor $\operatorname{Hilb}_{X/S}$, which is a special case of the $\operatorname{Quot}$ functor. Since $\operatorname{Hilb}_{X/S}$ classifies the isomorphism classes of closed subschemes of $X$ that are isomorphic to $S$, I am interested in the following question:

What are the closed subschemes of $\mathbb P^2_R$ for a ring $R$? More specifically, what are the closed subschemes of $\mathbb P^2_k$ for a field $k$?

I am just asking in order to make myself a little more acquainted with the topic.

Answer 1

Question: "What are the closed subschemes of P2R for a ring R? More specifically, what are the closed subschemes of P2k for a field k?"

Answer: For any closed subschemes $Y \subseteq \mathbb{P}^n_A$ there is a homogeneous ideal $I \subseteq A[x_0,..,x_n]$ with $Y \cong V(I)$. This is Hartshorne, Corr.II.5.16.