Lemma 32.42.6. Let $X$ be a separated scheme of finite type over $k$. If $X$ is reduced and $\dim (X) \leq 1$, then there exists an open immersion $j : X \to \overline{X}$ such that

$\overline{X}$ is H-projective over $k$, i.e., $\overline{X}$ is a closed subscheme of $\mathbf{P}^ d_ k$ for some $d$,

$j(X) \subset \overline{X}$ is dense and scheme theoretically dense,

$\overline{X} \setminus X = \{ x_1, \ldots , x_ n\} $ for some closed points $x_ i \in \overline{X}$,

the local rings $\mathcal{O}_{\overline{X}, x_ i}$ are discrete valuation rings for $i = 1, \ldots , n$.

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