Let $C$ be a curve in $\mathbb{P}^2$. I am wondering if we can find a number $n$ and two embeddings $$ \varphi_i:\mathbb{P}^2\hookrightarrow \mathbb{P}^n, i\in\{1,2\} $$ such that $\varphi_{i}^{-1}(\varphi_1(\mathbb{P}^2)\cap\varphi_2(\mathbb{P}^2))\simeq C$ for $i\in\{1,2\}$.
Maybe this is not always true and the answer depends on $C$. What are the conditions on $C$ for this to be true?
For instance, if $C$ is a line it holds, because we just have to consider two different planes in $\mathbb{P}^3$.
Assume $\deg(C) = d$. Let $f \colon \mathbb{P}^2 \to \mathbb{P}^N$ be the Veronese embedding of degree $d$, so that $C = f(\mathbb{P}^2) \cap \mathbb{P}^{N-1}$. Consider the rank-2 quadric $$ Q = \mathbb{P}^N \cup_{\mathbb{P}^{N-1}} \mathbb{P}^N \subset \mathbb{P}^{N+1}, $$ i.e., the union of two hyperplanes, and let $$ \varphi_i \colon \mathbb{P}^2 \stackrel{f}\to \mathbb{P}^N \hookrightarrow \mathbb{P}^{N+1} $$ be the embedding into these hyperplanes. These have the required properties.