The Stacks project

33.42 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 33.3.1. 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 33.42.2. 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 26.20.4 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 26.22.1 we would have $a = b$ as morphisms into $X$ which is absurd. Hence the valuative criterion (Schemes, Proposition 26.20.6) shows that $X \setminus \{ x\} \to \mathop{\mathrm{Spec}}(k)$ is not universally closed, hence not proper. $\square$

Lemma 33.42.3. 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 33.37.12 the scheme $X$ has an ample invertible sheaf $\mathcal{L}$. By Morphisms, Lemma 29.39.3 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 29.40.1. $\square$

Lemma 33.42.4. 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 33.42.3 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 29.41.7). $\square$

Lemma 33.42.5. 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 33.42.3 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 29.6.2. The description in Morphisms, Lemma 29.7.7 gives properties (1) and (2). Then $\dim (X) = 1 \Rightarrow \dim (\overline{X}) = 1$ for example by looking at generic points, see Lemma 33.20.3. 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 33.42.6. 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 33.42.5. Consider the normalization $X'$ of $\overline{X}$ in $X$. By Lemma 33.27.3 the morphism $X' \to \overline{X}$ is finite. By Morphisms, Lemma 29.44.16 $X' \to \overline{X}$ is projective. By Morphisms, Lemma 29.43.16 we see that $X' \to \overline{X}$ is H-projective. By Morphisms, Lemma 29.43.7 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 29.53.16. Then $X \to X'$ and $\{ x'_1, \ldots , x'_ m\} $ is as desired. $\square$

Lemma 33.42.7. Let $X$ be a separated scheme of finite type over $k$ with $\dim (X) \leq 1$. Then there exists a commutative diagram

\[ \xymatrix{ \overline{Y}_1 \amalg \ldots \amalg \overline{Y}_ n \ar[rd] & Y_1 \amalg \ldots \amalg Y_ n \ar[r]_-\nu \ar[d] \ar[l]^ j & X_{k'} \ar[r] \ar[d] & X \ar[d]^ f \\ & \mathop{\mathrm{Spec}}(k'_1) \amalg \ldots \amalg \mathop{\mathrm{Spec}}(k'_ n) \ar[r] & \mathop{\mathrm{Spec}}(k') \ar[r] & \mathop{\mathrm{Spec}}(k) } \]

of schemes with the following properties:

  1. $k'/k$ is a finite purely inseparable extension of fields,

  2. $\nu $ is the normalization of $X_{k'}$,

  3. $j$ is an open immersion with dense image,

  4. $k'_ i/k'$ is a finite separable extension for $i = 1, \ldots , n$,

  5. $\overline{Y}_ i$ is smooth, projective, geometrically irreducible dimension $\leq 1$ over $k'_ i$.

Proof. As we may replace $X$ by its reduction, we may and do assume $X$ is reduced. Choose $X \to \overline{X}$ as in Lemma 33.42.6. If we can show the lemma for $\overline{X}$, then the lemma follows for $X$ (details omitted). Thus we may and do assume $X$ is projective.

Choose $k'/k$ finite purely inseparable such that the normalization of $X_{k'}$ is geometrically normal over $k'$, see Lemma 33.27.4. Denote $Y = (X_{k'})^\nu $ the normalization; for properties of the normalization, see Section 33.27. Then $Y$ is geometrically regular as normal and regular are the same in dimension $\leq 1$, see Properties, Lemma 28.12.6. Hence $Y$ is smooth over $k'$ by Lemma 33.12.6. Let $Y = Y_1 \amalg \ldots \amalg Y_ n$ be the decomposition of $Y$ into irreducible components. Set $k'_ i = \Gamma (Y_ i, \mathcal{O}_{Y_ i})$. These are finite separable extensions of $k'$ by Lemma 33.9.3. The proof is finished by Lemma 33.9.4. $\square$

Observe that if an affine scheme $X$ over $k$ is proper over $k$ then $X$ is finite over $k$ (Morphisms, Lemma 29.44.11) and hence has dimension $0$ (Algebra, Lemma 10.53.2 and Proposition 10.60.7). 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 33.42.8. 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 33.42.5. By Lemma 33.37.4 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 30.19.2). We obtain a corresponding morphism

\[ f : \overline{X} \longrightarrow \mathbf{P}^ m_ k \]

such that the inverse image of $D_{+}(T_0)$ is $X$, see Constructions, Lemma 27.13.1. 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 30.21.21 we conclude that $f$ is finite. Hence $X = f^{-1}(D_{+}(T_0))$ is affine. $\square$

The following lemma combined with Lemma 33.42.2 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 33.42.9. 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 26.12.5). In particular $X_ i$ is a singleton (hence affine) or a curve hence affine by Lemma 33.42.8. 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 30.13.3. $\square$

[1] One can avoid using this lemma which relies on the theorem of formal functions. Namely, $\overline{X}$ is projective hence it suffices to show a proper morphism $f : X \to Y$ with finite fibres between quasi-projective schemes over $k$ is finite. To do this, one chooses an affine open of $X$ containing the fibre of $f$ over a point $y$ using that any finite set of points of a quasi-projective scheme over $k$ is contained in an affine. Shrinking $Y$ to a small affine neighbourhood of $y$ one reduces to the case of a proper morphism between affines. Such a morphism is finite by Morphisms, Lemma 29.44.7.

Comments (4)

Comment #1425 by Erik Visse on

In the paragraph following Lemma 32.28.4 there is a reference to Lemma 28.44.7. In section 28.44 there is another lemma (28.44.10) whose proof is basically Lemma 28.44.7 and some easy facts that states exactly the (first) result in the above mentioned paragraph. Maybe the reference could be changed to 28.44.10 for convenience.

Comment #4588 by Fred Vu on

In Lemma 32.42.2 (tag 0A24), the last sentence of the proof should read instead of .

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