I am sorry, I asked a misguided question here: Reference request: smooth affine curves are planar, here is my attempt at a better question.

Let $\mathfrak{p}$ be a prime ideal of $\mathbb{C}[x, y, z]$ of height 2. It defines an integral affine scheme of relative dimension 1 over $\mathrm{Spec}\:\mathbb{C}$ (I found this: https://math.stackexchange.com/a/49285 useful to clarify how dimension works here). This curve may or may not admit a $\mathbb{C}$-locally closed immersion in the projective plane.

My question is, can we determine whether there exists such an immersion algorithmically. The input is an explicit set of generators for $\mathfrak{p}$ (there can be more than 2 generators of course, but still a finite number because the algebra is Noetherian). The output is yes or no (does there exist such an immersion or not). Simpler algorithms are preferred. If you also give an implentation in some CAS that is great.