The Stacks project

Lemma 32.42.5. Let $X$ be a separated scheme of finite type over $k$. If $\dim (X) \leq 1$, then there exists an open immersion $j : X \to \overline{X}$ with the following properties

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

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

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

Proof. By Lemma 32.42.3 we may assume $X$ is a locally closed subscheme of $\mathbf{P}^ d_ k$ for some $d$. Let $\overline{X} \subset \mathbf{P}^ d_ k$ be the scheme theoretic image of $X \to \mathbf{P}^ d_ k$, see Morphisms, Definition 28.6.2. The description in Morphisms, Lemma 28.7.7 gives properties (1) and (2). Then $\dim (X) = 1 \Rightarrow \dim (\overline{X}) = 1$ for example by looking at generic points, see Lemma 32.20.3. As $\overline{X}$ is Noetherian, it then follows that $\overline{X} \setminus X = \{ x_1, \ldots , x_ n\} $ is a finite set of closed points. $\square$

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