Definition 32.42.1. Let $k$ be a field. A *curve* is a variety of dimension $1$ over $k$.

## 32.42 Curves

In the Stacks project we will use the following as our definition of a curve.

Two standard examples of curves over $k$ are the affine line $\mathbf{A}^1_ k$ and the projective line $\mathbf{P}^1_ k$. The scheme $X = \mathop{\mathrm{Spec}}(k[x, y]/(f))$ is a curve if and only if $f \in k[x, y]$ is irreducible.

Our definition of a curve has the same problems as our definition of a variety, see the discussion following Definition 32.3.1. Moreover, it means that every curve comes with a specified field of definition. For example $X = \mathop{\mathrm{Spec}}(\mathbf{C}[x])$ is a curve over $\mathbf{C}$ but we can also view it as a curve over $\mathbf{R}$. The scheme $\mathop{\mathrm{Spec}}(\mathbf{Z})$ isn't a curve, even though the schemes $\mathop{\mathrm{Spec}}(\mathbf{Z})$ and $\mathbf{A}^1_{\mathbf{F}_ p}$ behave similarly in many respects.

Lemma 32.42.2. Let $X$ be an irreducible scheme of dimension $> 0$ over a field $k$. Let $x \in X$ be a closed point. The open subscheme $X \setminus \{ x\} $ is not proper over $k$.

**Proof.**
Namely, choose a specialization $x' \leadsto x$ with $x' \not= x$ (for example take $x'$ to be the generic point). By Schemes, Lemma 25.20.4 there exists a morphism $a : \mathop{\mathrm{Spec}}(A) \to X$ where $A$ is a valuation ring with fraction field $K$ such that the generic point of $\mathop{\mathrm{Spec}}(A)$ maps to $x'$ and the closed point of $\mathop{\mathrm{Spec}}(A)$ maps to $x$. The morphism $\mathop{\mathrm{Spec}}(K) \to X \setminus \{ x\} $ does not extend to a morphism $b : \mathop{\mathrm{Spec}}(A) \to X \setminus \{ x\} $ since by the uniqueness in Schemes, Lemma 25.22.1 we would have $a = b$ as morphisms into $X$ which is absurd. Hence the valuative criterion (Schemes, Proposition 25.20.6) shows that $X \to \mathop{\mathrm{Spec}}(k)$ is not universally closed, hence not proper.
$\square$

Lemma 32.42.3. Let $X$ be a separated finite type scheme over a field $k$. If $\dim (X) \leq 1$ then $X$ is H-quasi-projective over $k$.

**Proof.**
By Proposition 32.37.12 the scheme $X$ has an ample invertible sheaf $\mathcal{L}$. By Morphisms, Lemma 28.37.3 we see that $X$ is isomorphic to a locally closed subscheme of $\mathbf{P}^ n_ k$ over $\mathop{\mathrm{Spec}}(k)$. This is the definition of being H-quasi-projective over $k$, see Morphisms, Definition 28.38.1.
$\square$

Lemma 32.42.4. Let $X$ be a proper scheme over a field $k$. If $\dim (X) \leq 1$ then $X$ is H-projective over $k$.

**Proof.**
By Lemma 32.42.3 we see that $X$ is a locally closed subscheme of $\mathbf{P}^ n_ k$ for some field $k$. Since $X$ is proper over $k$ it follows that $X$ is a closed subscheme of $\mathbf{P}^ n_ k$ (Morphisms, Lemma 28.39.7).
$\square$

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

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

**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.
$\square$

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 $i = 1, \ldots , n$.

**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.
$\square$

Observe that if an affine scheme $X$ over $k$ is proper over $k$ then $X$ is finite over $k$ (Morphisms, Lemma 28.42.11) and hence has dimension $0$ (Algebra, Lemma 10.52.2 and Proposition 10.59.6). Hence a scheme of dimension $> 0$ over $k$ cannot be both affine and proper over $k$. Thus the possibilities in the following lemma are mutually exclusive.

Lemma 32.42.7. Let $X$ be a curve over $k$. Then either $X$ is an affine scheme or $X$ is H-projective over $k$.

**Proof.**
Choose $X \to \overline{X}$ as in Lemma 32.42.5. By Lemma 32.37.4 we can find a globally generated invertible sheaf $\mathcal{L}$ on $\overline{X}$ and a section $s \in \Gamma (\overline{X}, \mathcal{L})$ such that $X = \overline{X}_ s$. Choose a basis $s = s_0, s_1, \ldots , s_ m$ of the finite dimensional $k$-vector space $\Gamma (\overline{X}, \mathcal{L})$ (Cohomology of Schemes, Lemma 29.19.2). We obtain a corresponding morphism

such that the inverse image of $D_{+}(T_0)$ is $X$, see Constructions, Lemma 26.13.1. In particular, $f$ is non-constant, i.e., $\mathop{\mathrm{Im}}(f)$ has more than one point. A topological argument shows that $f$ maps the generic point $\eta $ of $\overline{X}$ to a nonclosed point of $\mathbf{P}^ n_ k$. Hence if $y \in \mathbf{P}^ n_ k$ is a closed point, then $f^{-1}(\{ y\} )$ is a closed set of $\overline{X}$ not containing $\eta $, hence finite. By Cohomology of Schemes, Lemma 29.21.2^{1} we conclude that $f$ is finite. Hence $X = f^{-1}(D_{+}(T_0))$ is affine.
$\square$

The following lemma combined with Lemma 32.42.2 tells us that given a separated scheme $X$ of dimension $1$ and of finite type over $k$, then $X \setminus Z$ is affine, whenever the closed subset $Z$ meets every irreducible component of $X$.

Lemma 32.42.8. Let $X$ be a separated scheme of finite type over $k$. If $\dim (X) \leq 1$ and no irreducible component of $X$ is proper of dimension $1$, then $X$ is affine.

**Proof.**
Let $X = \bigcup X_ i$ be the decomposition of $X$ into irreducible components. We think of $X_ i$ as an integral scheme (using the reduced induced scheme structure, see Schemes, Definition 25.12.5). In particular $X_ i$ is a singleton (hence affine) or a curve hence affine by Lemma 32.42.7. Then $\coprod X_ i \to X$ is finite surjective and $\coprod X_ i$ is affine. Thus we see that $X$ is affine by Cohomology of Schemes, Lemma 29.13.3.
$\square$

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