The Stacks project

33.43 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 33.43.2. Let $X$ be a separated, 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. Since $X$ is irreducible, $U = X \setminus \{ x\} $ is not closed in $X$. In particular, the immersion $U \to X$ is not proper. By Morphisms, Lemma 29.41.7 (here we use $X$ is separated), $U \to \mathop{\mathrm{Spec}}(k)$ is not proper either. $\square$

Lemma 33.43.3. 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 33.38.12 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 33.43.4. 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 33.43.3 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 33.43.5. 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 33.43.3 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 33.43.6. 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 33.43.5. 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 33.43.7. 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 33.43.6. 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. $\square$

Lemma 33.43.8. Let $k$ be a field. Let $X$ be a curve over $k$. Let $x \in X$ be a closed point. We think of $x$ as a (reduced) closed subscheme of $X$ with sheaf of ideals $\mathcal{I}$. The following are equivalent

  1. $\mathcal{O}_{X, x}$ is regular,

  2. $\mathcal{O}_{X, x}$ is normal,

  3. $\mathcal{O}_{X, x}$ is a discrete valuation ring,

  4. $\mathcal{I}$ is an invertible $\mathcal{O}_ X$-module,

  5. $x$ is an effective Cartier divisor on $X$.

If $k$ is perfect or if $\kappa (x)$ is separable over $k$, these are also equivalent to

  1. $X \to \mathop{\mathrm{Spec}}(k)$ is smooth at $x$.

Proof. Since $X$ is a curve, the local ring $\mathcal{O}_{X, x}$ is a Noetherian local domain of dimension $1$ (Lemma 33.20.3). Parts (4) and (5) are equivalent by definition and are equivalent to $\mathcal{I}_ x = \mathfrak m_ x \subset \mathcal{O}_{X, x}$ having one generator (Divisors, Lemma 31.15.2). The equivalence of (1), (2), (3), (4), and (5) therefore follows from Algebra, Lemma 10.119.7. The final statement follows from Lemma 33.25.8 in case $k$ is perfect. If $\kappa (x)/k$ is separable, then the equivalence follows from Algebra, Lemma 10.140.5. $\square$

Remark 33.43.9. Let $k$ be a field. Let $X$ be a regular curve over $k$. By Lemmas 33.43.8 and 33.43.6 there exists a nonsingular projective curve $\overline{X}$ which is a compactification of $X$, i.e., there exists an open immersion $j : X \to \overline{X}$ such that the complement consists of a finite number of closed points. If $k$ is perfect, then $X$ and $\overline{X}$ are smooth over $k$ and $\overline{X}$ is a smooth projective compactification of $X$.

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

Proof. Choose $X \to \overline{X}$ with $\overline{X} \setminus X = \{ x_1, \ldots , x_ r\} $ as in Lemma 33.43.6. Then $\overline{X}$ is a curve as well. If $r = 0$, then $X = \overline{X}$ is H-projective over $k$. Thus we may assume $r \geq 1$ and our goal is to show that $X$ is affine. By Lemma 33.38.2 it suffices to show that $\overline{X} \setminus \{ x_1\} $ is affine. This reduces us to the claim stated in the next paragraph.

Let $X$ be an H-projective curve over $k$. Let $x \in X$ be a closed point such that $\mathcal{O}_{X, x}$ is a discrete valuation ring. Claim: $U = X \setminus \{ x\} $ is affine. By Lemma 33.43.8 the point $x$ defines an effective Cartier divisor of $X$. For $n \geq 1$ denote $nx = x + \ldots + x$ the $n$-fold sum, see Divisors, Definition 31.13.6. Denote $\mathcal{O}_{nx}$ the structure sheaf of $nx$ viewed as a coherent module on $X$. Since every invertible module on the local scheme $nx$ is trivial the first short exact sequence of Divisors, Remark 31.14.11 reads

\[ 0 \to \mathcal{O}_ X \xrightarrow {1} \mathcal{O}_ X(nx) \to \mathcal{O}_{nx} \to 0 \]

in our case. Note that $\dim _ k H^0(X, \mathcal{O}_{nx}) \geq n$. Namely, by Lemma 33.33.3 we have $H^0(X, \mathcal{O}_{nx}) = \mathcal{O}_{X, x}/(\pi ^ n)$ where $\pi $ in $\mathcal{O}_{X, x}$ is a uniformizer and the powers $\pi ^ i$ map to $k$-linearly independent elements in $\mathcal{O}_{X, x}/(\pi ^ n)$ for $i = 0, 1, \ldots , n - 1$. We have $\dim _ k H^1(X, \mathcal{O}_ X) < \infty $ by Cohomology of Schemes, Lemma 30.19.2. If $n > \dim _ k H^1(X, \mathcal{O}_ X)$ we conclude from the long exact cohomology sequence that there exists an $s \in \Gamma (X, \mathcal{O}_ X(nx))$ which is not a section of $\mathcal{O}_ X$. If we take $n$ minimal with this property, then $s$ will map to a generator of the stalk $\left(\mathcal{O}_ X(nx)\right)_ x$ since otherwise it would define a section of $\mathcal{O}_ X((n - 1)x) \subset \mathcal{O}_ X(nx)$. For this $n$ we conclude that $s_0 = 1$ and $s_1 = s$ generate the invertible module $\mathcal{L} = \mathcal{O}_ X(nx)$.

Consider the corresponding morphism $f = \varphi _{\mathcal{L}, (s_0, s_1)} : X \to \mathbf{P}^1_ k$ of Constructions, Section 27.13. Observe that the inverse image of $D_{+}(T_0)$ is $U = X \setminus \{ x\} $ as the section $s_0$ of $\mathcal{L}$ only vanishes at $x$. In particular, $f$ is non-constant, i.e., $\mathop{\mathrm{Im}}(f)$ has more than one point. Hence $f$ must map the generic point $\eta $ of $X$ to the generic point of $\mathbf{P}^1_ k$. Hence if $y \in \mathbf{P}^1_ k$ is a closed point, then $f^{-1}(\{ y\} )$ is a closed set of $X$ not containing $\eta $, hence finite. Finally, $f$ is proper1. By Cohomology of Schemes, Lemma 30.21.22 we conclude that $f$ is finite. Hence $U = f^{-1}(D_{+}(T_0))$ is affine. $\square$

The following lemma combined with Lemma 33.43.2 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 33.43.11. 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 33.43.10. 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] Namely, a H-projective variety is a proper variety by Morphisms, Lemma 29.43.13. A morphism of varieties whose source is a proper variety is a proper morphism by Morphisms, Lemma 29.41.7.
[2] One can avoid using this lemma which relies on the theorem of formal functions. Namely, $X$ 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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