Lemma 53.9.1. Let $Z \subset \mathbf{P}^2_ k$ be a closed subscheme which is equidimensional of dimension $1$ and has no embedded points (equivalently $Z$ is Cohen-Macaulay). Then the ideal $I(Z) \subset k[T_0, T_1, T_2]$ corresponding to $Z$ is principal.

**Proof.**
This is a special case of Divisors, Lemma 31.31.3 (see also Varieties, Lemma 33.34.4). The parenthetical statement follows from the fact that a $1$ dimensional Noetherian scheme is Cohen-Macaulay if and only if it has no embedded points, see Divisors, Lemma 31.4.4.
$\square$

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