The Stacks project

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

  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 $x_ i \in \overline{X}$,

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

Proof. Let $j : X \to \overline{X}$ be as in Lemma 32.42.5. Consider the normalization $X'$ of $\overline{X}$ in $X$. By Lemma 32.27.2 the morphism $X' \to \overline{X}$ is finite. By Morphisms, Lemma 28.42.16 $X' \to \overline{X}$ is projective. By Morphisms, Lemma 28.41.16 we see that $X' \to \overline{X}$ is H-projective. By Morphisms, Lemma 28.41.7 we see that $X' \to \mathop{\mathrm{Spec}}(k)$ is H-projective. Let $\{ x'_1, \ldots , x'_ m\} \subset X'$ be the inverse image of $\{ x_1, \ldots , x_ n\} = \overline{X} \setminus X$. Then $\dim (\mathcal{O}_{X', x'_ i}) = 1$ for all $1 \leq i \leq m$. Hence the local rings $\mathcal{O}_{X', x'}$ are discrete valuation rings by Morphisms, Lemma 28.51.16. Then $X \to X'$ and $\{ x'_1, \ldots , x'_ m\} $ is as desired. $\square$


Comments (2)

Comment #2249 by Paco on

Isn't there a typo in the second sentence of the proof? Shouldn't X' be the normalization of Xbar in X?

There are also:

  • 3 comment(s) on Section 32.42: Curves

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