By curve I mean a scheme of pure dimension $1$. Let $(\Gamma(C_{red},\mathcal{L}))$ be a complete linear system on $C_{red}$ which gives a projective embedding $C_{red} \hookrightarrow \mathbb{P}^n$ for some integer $n$. Assume that there exists a closed immersion of $C$ into $\mathbb{P}^n$. Under what condition does there exist a complete linear system on $C$ which embeds it into $\mathbb{P}^n$ for the same integer $n$?
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