The Stacks project

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32.43 Degrees on curves

We start defining the degree of an invertible sheaf and more generally a locally free sheaf on a proper scheme of dimension $1$ over a field. In Section 32.32 we defined the Euler characteristic of a coherent sheaf $\mathcal{F}$ on a proper scheme $X$ over a field $k$ by the formula

\[ \chi (X, \mathcal{F}) = \sum (-1)^ i \dim _ k H^ i(X, \mathcal{F}). \]

Definition 32.43.1. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The degree of $\mathcal{L}$ is defined by

\[ \deg (\mathcal{L}) = \chi (X, \mathcal{L}) - \chi (X, \mathcal{O}_ X) \]

More generally, if $\mathcal{E}$ is a locally free sheaf of rank $n$ we define the degree of $\mathcal{E}$ by

\[ \deg (\mathcal{E}) = \chi (X, \mathcal{E}) - n\chi (X, \mathcal{O}_ X) \]

Observe that this depends on the triple $\mathcal{E}/X/k$. If $X$ is disconnected and $\mathcal{E}$ is finite locally free (but not of constant rank), then one can modify the definition by summing the degrees of the restriction of $\mathcal{E}$ to the connected components of $X$. If $\mathcal{E}$ is just a coherent sheaf, there are several different ways of extending the definition1. In a series of lemmas we show that this definition has all the properties one expects of the degree.

Lemma 32.43.2. Let $k \subset k'$ be an extension of fields. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of constant rank $n$. Then the degree of $\mathcal{E}/X/k$ is equal to the degree of $\mathcal{E}_{k'}/X_{k'}/k'$.

Proof. More precisely, set $X_{k'} = X \times _{\mathop{\mathrm{Spec}}(k)} \mathop{\mathrm{Spec}}(k')$. Let $\mathcal{E}_{k'} = p^*\mathcal{E}$ where $p : X_{k'} \to X$ is the projection. By Cohomology of Schemes, Lemma 29.5.2 we have $H^ i(X_{k'}, \mathcal{E}_{k'}) = H^ i(X, \mathcal{E}) \otimes _ k k'$ and $H^ i(X_{k'}, \mathcal{O}_{X_{k'}}) = H^ i(X, \mathcal{O}_ X) \otimes _ k k'$. Hence we see that the Euler characteristics are unchanged, hence the degree is unchanged. $\square$

Lemma 32.43.3. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $0 \to \mathcal{E}_1 \to \mathcal{E}_2 \to \mathcal{E}_3 \to 0$ be a short exact sequence of locally free $\mathcal{O}_ X$-modules each of finite constant rank. Then

\[ \deg (\mathcal{E}_2) = \deg (\mathcal{E}_1) + \deg (\mathcal{E}_3) \]

Proof. Follows immediately from additivity of Euler characteristics (Lemma 32.32.2) and additivity of ranks. $\square$

Lemma 32.43.4. Let $k$ be a field. Let $f : X' \to X$ be a birational morphism of proper schemes of dimension $\leq 1$ over $k$. Then

\[ \deg (f^*\mathcal{E}) = \deg (\mathcal{E}) \]

for every finite locally free sheaf of constant rank. More generally it suffices if $f$ induces a bijection between irreducible components of dimension $1$ and isomorphisms of local rings at the corresponding generic points.

Proof. The morphism $f$ is proper (Morphisms, Lemma 28.39.7) and has fibres of dimension $\leq 0$. Hence $f$ is finite (Cohomology of Schemes, Lemma 29.21.2). Thus

\[ Rf_*f^*\mathcal{E} = f_*f^*\mathcal{E} = \mathcal{E} \otimes _{\mathcal{O}_ X} f_*\mathcal{O}_ X \]

Since $f$ induces an isomorphism on local rings at generic points of all irreducible components of dimension $1$ we see that the kernel and cokernel

\[ 0 \to \mathcal{K} \to \mathcal{O}_ X \to f_*\mathcal{O}_ X \to \mathcal{Q} \to 0 \]

have supports of dimension $\leq 0$. Note that tensoring this with $\mathcal{E}$ is still an exact sequence as $\mathcal{E}$ is locally free. We obtain

\begin{align*} \chi (X, \mathcal{E}) - \chi (X', f^*\mathcal{E}) & = \chi (X, \mathcal{E}) - \chi (X, f_*f^*\mathcal{E}) \\ & = \chi (X, \mathcal{E}) - \chi (X, \mathcal{E} \otimes f_*\mathcal{O}_ X) \\ & = \chi (X, \mathcal{K} \otimes \mathcal{E}) - \chi (X, \mathcal{Q} \otimes \mathcal{E}) \\ & = n\chi (X, \mathcal{K}) - n\chi (X, \mathcal{Q}) \\ & = n\chi (X, \mathcal{O}_ X) - n\chi (X, f_*\mathcal{O}_ X) \\ & = n\chi (X, \mathcal{O}_ X) - n\chi (X', \mathcal{O}_{X'}) \end{align*}

which proves what we want. The first equality as $f$ is finite, see Cohomology of Schemes, Lemma 29.2.4. The second equality by projection formula, see Cohomology, Lemma 20.45.2. The third by additivity of Euler characteristics, see Lemma 32.32.2. The fourth by Lemma 32.32.3. $\square$

Lemma 32.43.5. Let $k$ be a field. Let $X$ be a proper curve over $k$ with generic point $\xi $. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $n$ and let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Then

\[ \chi (X, \mathcal{E} \otimes \mathcal{F}) = r \deg (\mathcal{E}) + n \chi (X, \mathcal{F}) \]

where $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi $ is the rank of $\mathcal{F}$.

Proof. Let $\mathcal{P}$ be the property of coherent sheaves $\mathcal{F}$ on $X$ expressing that the formula of the lemma holds. We claim that the assumptions (1) and (2) of Cohomology of Schemes, Lemma 29.12.6 hold for $\mathcal{P}$. Namely, (1) holds because the Euler characteristic and the rank $r$ are additive in short exact sequences of coherent sheaves. And (2) holds too: If $Z = X$ then we may take $\mathcal{G} = \mathcal{O}_ X$ and $\mathcal{P}(\mathcal{O}_ X)$ is true by the definition of degree. If $i : Z \to X$ is the inclusion of a closed point we may take $\mathcal{G} = i_*\mathcal{O}_ Z$ and $\mathcal{P}$ holds by Lemma 32.32.3 and the fact that $r = 0$ in this case. $\square$

Let $k$ be a field. Let $X$ be a finite type scheme over $k$ of dimension $\leq 1$. Let $C_ i \subset X$, $i = 1, \ldots , t$ be the irreducible components of dimension $1$. We view $C_ i$ as a scheme by using the induced reduced scheme structure. Let $\xi _ i \in C_ i$ be the generic point. The multiplicity of $C_ i$ in $X$ is defined as the length

\[ m_ i = \text{length}_{\mathcal{O}_{X, \xi _ i}} \mathcal{O}_{X, \xi _ i} \]

This makes sense because $\mathcal{O}_{X, \xi _ i}$ is a zero dimensional Noetherian local ring and hence has finite length over itself (Algebra, Proposition 10.59.6). See Chow Homology, Section 41.9 for additional information. It turns out the degree of a locally free sheaf only depends on the restriction of the irreducible components.

Lemma 32.43.6. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $n$. Then

\[ \deg (\mathcal{E}) = \sum m_ i \deg (\mathcal{E}|_{C_ i}) \]

where $C_ i \subset X$, $i = 1, \ldots , t$ are the irreducible components of dimension $1$ with reduced induced scheme structure and $m_ i$ is the multiplicity of $C_ i$ in $X$.

Proof. Observe that the statement makes sense because $C_ i \to \mathop{\mathrm{Spec}}(k)$ is proper of dimension $1$ (Morphisms, Lemmas 28.39.6 and 28.39.4). Consider the open subscheme $U_ i = X \setminus (\bigcup _{j \not= i} C_ j)$ and let $X_ i \subset X$ be the scheme theoretic closure of $U_ i$. Note that $X_ i \cap U_ i = U_ i$ (scheme theoretically) and that $X_ i \cap U_ j = \emptyset $ (set theoretically) for $i \not= j$; this follows from the description of scheme theoretic closure in Morphisms, Lemma 28.7.7. Thus we may apply Lemma 32.43.4 to the morphism $X' = \bigcup X_ i \to X$. Since it is clear that $C_ i \subset X_ i$ (scheme theoretically) and that the multiplicity of $C_ i$ in $X_ i$ is equal to the multiplicity of $C_ i$ in $X$, we see that we reduce to the case discussed in the following paragraph.

Assume $X$ is irreducible with generic point $\xi $. Let $C = X_{red}$ have multiplicity $m$. We have to show that $\deg (\mathcal{E}) = m \deg (\mathcal{E}|_ C)$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the ideal defining the closed subscheme $C$. Let $e \geq 0$ be minimal such that $\mathcal{I}^{e + 1} = 0$ (Cohomology of Schemes, Lemma 29.10.2). We argue by induction on $e$. If $e = 0$, then $X = C$ and the result is immediate. Otherwise we set $\mathcal{F} = \mathcal{I}^ e$ viewed as a coherent $\mathcal{O}_ C$-module (Cohomology of Schemes, Lemma 29.9.8). Let $X' \subset X$ be the closed subscheme cut out by the coherent ideal $\mathcal{I}^ e$ and let $m'$ be the multiplicity of $C$ in $X'$. Taking stalks at $\xi $ of the short exact sequence

\[ 0 \to \mathcal{F} \to \mathcal{O}_ X \to \mathcal{O}_{X'} \to 0 \]

we find (use Algebra, Lemmas 10.51.3, 10.51.6, and 10.51.5) that

\[ m = \text{length}_{\mathcal{O}_{X, \xi }} \mathcal{O}_{X, \xi } = \dim _{\kappa (\xi )} \mathcal{F}_\xi + \text{length}_{\mathcal{O}_{X', \xi }} \mathcal{O}_{X', \xi } = r + m' \]

where $r$ is the rank of $\mathcal{F}$ as a coherent sheaf on $C$. Tensoring with $\mathcal{E}$ we obtain a short exact sequence

\[ 0 \to \mathcal{E}|_ C \otimes \mathcal{F} \to \mathcal{E} \to \mathcal{E} \otimes \mathcal{O}_{X'} \to 0 \]

By induction we have $\chi (\mathcal{E} \otimes \mathcal{O}_{X'}) = m' \deg (\mathcal{E}|_ C)$. By Lemma 32.43.5 we have $\chi (\mathcal{E}|_ C \otimes \mathcal{F}) = r \deg (\mathcal{E}|_ C) + n \chi (\mathcal{F})$. Putting everything together we obtain the result. $\square$

Lemma 32.43.7. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{E}$, $\mathcal{V}$ be locally free $\mathcal{O}_ X$-modules of constant finite rank. Then

\[ \deg (\mathcal{E} \otimes \mathcal{V}) = \text{rank}(\mathcal{E}) \deg (\mathcal{V}) + \text{rank}(\mathcal{V}) \deg (\mathcal{E}) \]

Proof. By Lemma 32.43.6 and elementary arithmetic, we reduce to the case of a proper curve. This case follows from Lemma 32.43.5. $\square$

Lemma 32.43.8. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$, and let $\mathcal{E}$ be a locally free $\mathcal{O}_ X$-module of rank $n$. Then

\[ \deg (\mathcal{E}) = \deg (\wedge ^ n(\mathcal{E})) = \deg (\det (\mathcal{E})) \]

Proof. By Lemma 32.43.6 and elementary arithmetic, we reduce to the case of a proper curve. Then there exists a modification $f : X' \to X$ such that $f^*\mathcal{E}$ has a filtration whose successive quotients are invertible modules, see Divisors, Lemma 30.35.1. By Lemma 32.43.4 we may work on $X'$. Thus we may assume we have a filtration

\[ 0 = \mathcal{E}_0 \subset \mathcal{E}_1 \subset \mathcal{E}_2 \subset \ldots \subset \mathcal{E}_ n = \mathcal{E} \]

by locally free $\mathcal{O}_ X$-modules with $\mathcal{L}_ i = \mathcal{E}_ i/\mathcal{E}_{i - 1}$ is invertible. By Modules, Lemma 17.23.1 and induction we find $\det (\mathcal{E}) = \mathcal{L}_1 \otimes \ldots \otimes \mathcal{L}_ n$. Thus the equality follows from Lemma 32.43.7 and additivity (Lemma 32.43.3). $\square$

Lemma 32.43.9. Let $k$ be a field, let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $D$ be an effective Cartier divisor on $X$. Then $D$ is finite over $\mathop{\mathrm{Spec}}(k)$ of degree $\deg (D) = \dim _ k \Gamma (D, \mathcal{O}_ D)$. For a locally free sheaf $\mathcal{E}$ of rank $n$ we have

\[ \deg (\mathcal{E}(D)) = n\deg (D) + \deg (\mathcal{E}) \]

where $\mathcal{E}(D) = \mathcal{E} \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(D)$.

Proof. Since $D$ is nowhere dense in $X$ (Divisors, Lemma 30.13.4) we see that $\dim (D) \leq 0$. Hence $D$ is finite over $k$ by Lemma 32.20.2. Since $k$ is a field, the morphism $D \to \mathop{\mathrm{Spec}}(k)$ is finite locally free and hence has a degree (Morphisms, Definition 28.45.1), which is clearly equal to $\dim _ k \Gamma (D, \mathcal{O}_ D)$ as stated in the lemma. By Divisors, Definition 30.14.1 there is a short exact sequence

\[ 0 \to \mathcal{O}_ X \to \mathcal{O}_ X(D) \to i_*i^*\mathcal{O}_ X(D) \to 0 \]

where $i : D \to X$ is the closed immersion. Tensoring with $\mathcal{E}$ we obtain a short exact sequence

\[ 0 \to \mathcal{E} \to \mathcal{E}(D) \to i_*i^*\mathcal{E}(D) \to 0 \]

The equation of the lemma follows from additivity of the Euler characteristic (Lemma 32.32.2) and Lemma 32.32.3. $\square$

Lemma 32.43.10. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced and connected. Let $\kappa = H^0(X, \mathcal{O}_ X)$. Then $\kappa /k$ is a finite extension of fields and $w = [\kappa : k]$ divides

  1. $\deg (\mathcal{E})$ for all locally free $\mathcal{O}_ X$-modules $\mathcal{E}$,

  2. $[\kappa (x) : k]$ for all closed points $x \in X$, and

  3. $\deg (D)$ for all closed subschemes $D \subset X$ of dimension zero.

Proof. See Lemma 32.9.3 for the assertions about $\kappa $. For every quasi-coherent $\mathcal{O}_ X$-module, the $k$-vector spaces $H^ i(X, \mathcal{F})$ are $\kappa $-vector spaces. The divisibilities easily follow from this statement and the definitions. $\square$

Lemma 32.43.11. Let $k$ be a field. Let $f : X \to Y$ be a nonconstant morphism of proper curves over $k$. Let $\mathcal{E}$ be a locally free $\mathcal{O}_ Y$-module. Then

\[ \deg (f^*\mathcal{E}) = \deg (X/Y) \deg (\mathcal{E}) \]

Proof. The degree of $X$ over $Y$ is defined in Morphisms, Definition 28.48.8. Thus $f_*\mathcal{O}_ X$ is a coherent $\mathcal{O}_ Y$-module of rank $\deg (X/Y)$, i.e., $\deg (X/Y) = \dim _{\kappa (\xi )} (f_*\mathcal{O}_ X)_\xi $ where $\xi $ is the generic point of $Y$. Thus we obtain

\begin{align*} \chi (X, f^*\mathcal{E}) & = \chi (Y, f_*f^*\mathcal{E}) \\ & = \chi (Y, \mathcal{E} \otimes f_*\mathcal{O}_ X) \\ & = \deg (X/Y) \deg (\mathcal{E}) + n \chi (Y, f_*\mathcal{O}_ X) \\ & = \deg (X/Y) \deg (\mathcal{E}) + n \chi (X, \mathcal{O}_ X) \end{align*}

as desired. The first equality as $f$ is finite, see Cohomology of Schemes, Lemma 29.2.4. The second equality by projection formula, see Cohomology, Lemma 20.45.2. The third equality by Lemma 32.43.5. $\square$

The following is a trivial but important consequence of the results on degrees above.

Lemma 32.43.12. Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module.

  1. If $\mathcal{L}$ has a nonzero section, then $\deg (\mathcal{L}) \geq 0$.

  2. If $\mathcal{L}$ has a nonzero section $s$ which vanishes at a point, then $\deg (\mathcal{L}) > 0$.

  3. If $\mathcal{L}$ and $\mathcal{L}^{-1}$ have nonzero sections, then $\mathcal{L} \cong \mathcal{O}_ X$.

  4. If $\deg (\mathcal{L}) \leq 0$ and $\mathcal{L}$ has a nonzero section, then $\mathcal{L} \cong \mathcal{O}_ X$.

  5. If $\mathcal{N} \to \mathcal{L}$ is a nonzero map of invertible $\mathcal{O}_ X$-modules, then $\deg (\mathcal{L}) \geq \deg (\mathcal{N})$ and if equality holds then it is an isomorphism.

Proof. Let $s$ be a nonzero section of $\mathcal{L}$. Since $X$ is a curve, we see that $s$ is a regular section. Hence there is an effective Cartier divisor $D \subset X$ and an isomorphism $\mathcal{L} \to \mathcal{O}_ X(D)$ mapping $s$ the canonical section $1$ of $\mathcal{O}_ X(D)$, see Divisors, Lemma 30.14.10. Then $\deg (\mathcal{L}) = \deg (D)$ by Lemma 32.43.9. As $\deg (D) \geq 0$ and $= 0$ if and only if $D = \emptyset $, this proves (1) and (2). In case (3) we see that $\deg (\mathcal{L}) = 0$ and $D = \emptyset $. Similarly for (4). To see (5) apply (1) and (4) to the invertible sheaf

\[ \mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}^{\otimes -1} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{N}, \mathcal{L}) \]

which has degree $\deg (\mathcal{L}) - \deg (\mathcal{N})$ by Lemma 32.43.7. $\square$

Lemma 32.43.13. Let $k$ be a field. Let $X$ be a proper scheme over $k$ which is reduced, connected, and equidimensional of dimension $1$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. If $\deg (\mathcal{L}|_ C) \leq 0$ for all irreducible components $C$ of $X$, then either $H^0(X, \mathcal{L}) = 0$ or $\mathcal{L} \cong \mathcal{O}_ X$.

Proof. Let $s \in H^0(X, \mathcal{L})$ be nonzero. Since $X$ is reduced there exists an irreducible component $C$ of $X$ with $s|_ C \not= 0$. But if $s|_ C$ is nonzero, then $s$ is nonwhere vanishing on $C$ by Lemma 32.43.12. This in turn implies $s$ is nowhere vanishing on every irreducible component of $X$ meeting $C$. Since $X$ is connected, we conclude that $s$ vanishes nowhere and the lemma follows. $\square$

Lemma 32.43.14. Let $k$ be a field. Let $X$ be a proper curve over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Then $\mathcal{L}$ is ample if and only if $\deg (\mathcal{L}) > 0$.

Proof. If $\mathcal{L}$ is ample, then there exists an $n > 0$ and a section $s \in H^0(X, \mathcal{L}^{\otimes n})$ with $X_ s$ affine. Since $X$ isn't affine (otherwise by Morphisms, Lemma 28.42.11 $X$ would be finite), we see that $s$ vanishes at some point. Hence $\deg (\mathcal{L}^{\otimes n}) > 0$ by Lemma 32.43.12. By Lemma 32.43.7 we conclude that $\deg (\mathcal{L}) = 1/n\deg (\mathcal{L}^{\otimes n}) > 0$.

Assume $\deg (\mathcal{L}) > 0$. Then

\[ \dim _ k H^0(X, \mathcal{L}^{\otimes n}) \geq \chi (X, \mathcal{L}^ n) = n\deg (\mathcal{L}) + \chi (X, \mathcal{O}_ X) \]

grows linearly with $n$. Hence for any finite collection of closed points $x_1, \ldots , x_ t$ of $X$, we can find an $n$ such that $\dim _ k H^0(X, \mathcal{L}^{\otimes n}) > \sum \dim _ k \kappa (x_ i)$. (Recall that by Hilbert Nullstellensatz, the extension fields $k \subset \kappa (x_ i)$ are finite, see for example Morphisms, Lemma 28.19.3). Hence we can find a nonzero $s \in H^0(X, \mathcal{L}^{\otimes n})$ vanishing in $x_1, \ldots , x_ t$. In particular, if we choose $x_1, \ldots , x_ t$ such that $X \setminus \{ x_1, \ldots , x_ t\} $ is affine, then $X_ s$ is affine too (for example by Properties, Lemma 27.26.4 although if we choose our finite set such that $\mathcal{L}|_{X \setminus \{ x_1, \ldots , x_ t\} }$ is trivial, then it is immediate). The conclusion is that we can find an $n > 0$ and a nonzero section $s \in H^0(X, \mathcal{L}^{\otimes n})$ such that $X_ s$ is affine.

We will show that for every quasi-coherent sheaf of ideals $\mathcal{I}$ there exists an $m > 0$ such that $H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes m})$ is zero. This will finish the proof by Cohomology of Schemes, Lemma 29.17.1. To see this we consider the maps

\[ \mathcal{I} \xrightarrow {s} \mathcal{I} \otimes \mathcal{L}^{\otimes n} \xrightarrow {s} \mathcal{I} \otimes \mathcal{L}^{\otimes 2n} \xrightarrow {s} \ldots \]

Since $\mathcal{I}$ is torsion free, these maps are injective and isomorphisms over $X_ s$, hence the cokernels have vanishing $H^1$ (by Cohomology of Schemes, Lemma 29.9.10 for example). We conclude that the maps of vector spaces

\[ H^1(X, \mathcal{I}) \to H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes n}) \to H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes 2n}) \to \ldots \]

are surjective. On the other hand, the dimension of $H^1(X, \mathcal{I})$ is finite, and every element maps to zero eventually by Cohomology of Schemes, Lemma 29.17.4. Thus for some $e > 0$ we see that $H^1(X, \mathcal{I} \otimes \mathcal{L}^{\otimes en})$ is zero. This finishes the proof. $\square$

Lemma 32.43.15. Let $k$ be a field. Let $X$ be a proper scheme of dimension $\leq 1$ over $k$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $C_ i \subset X$, $i = 1, \ldots , t$ be the irreducible components of dimension $1$. The following are equivalent:

  1. $\mathcal{L}$ is ample, and

  2. $\deg (\mathcal{L}|_{C_ i}) > 0$ for $i = 1, \ldots , t$.

Proof. Let $x_1, \ldots , x_ r \in X$ be the isolated closed points. Think of $x_ i = \mathop{\mathrm{Spec}}(\kappa (x_ i))$ as a scheme. Consider the morphism of schemes

\[ f : C_1 \amalg \ldots \amalg C_ t \amalg x_1 \amalg \ldots \amalg x_ r \longrightarrow X \]

This is a finite surjective morphism of schemes proper over $k$ (details omitted). Thus $\mathcal{L}$ is ample if and only if $f^*\mathcal{L}$ is ample (Cohomology of Schemes, Lemma 29.17.2). Thus we conclude by Lemma 32.43.14. $\square$

Lemma 32.43.16. 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, 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 32.20.3). 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 30.15.2). The equivalence of (1), (2), (3), (4), and (5) therefore follows from Algebra, Lemma 10.118.7. The final statement follows from Lemma 32.25.8. $\square$

Lemma 32.43.17. Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Then there exist

  1. an invertible $\mathcal{O}_ X$-module $\mathcal{L}$ with $\dim _ k H^0(X, \mathcal{L}) = 1$ and $H^1(X, \mathcal{L}) = 0$, and

  2. an invertible $\mathcal{O}_ X$-module $\mathcal{N}$ with $\dim _ k H^0(X, \mathcal{N}) = 0$ and $H^1(X, \mathcal{N}) = 0$.

Proof. Choose a closed immersion $i : X \to \mathbf{P}^ n_ k$ (Lemma 32.42.4). Setting $\mathcal{L} = i^*\mathcal{O}_{\mathbf{P}^ n}(d)$ for $d \gg 0$ we see that there exists an invertible sheaf $\mathcal{L}$ with $H^0(X, \mathcal{L}) \not= 0$ and $H^1(X, \mathcal{L}) = 0$ (see Cohomology of Schemes, Lemma 29.17.1 for vanishing and the references therein for nonvanishing). We will finish the proof of (1) by descending induction on $t = \dim _ k H^0(X, \mathcal{L})$. The base case $t = 1$ is trivial. Assume $t > 1$.

Let $U \subset X$ be the nonempty open subset of nonsingular points studied in Lemma 32.25.8. Let $s \in H^0(X, \mathcal{L})$ be nonzero. There exists a closed point $x \in U$ such that $s$ does not vanish in $x$. Let $\mathcal{I}$ be the ideal sheaf of $i : x \to X$ as in Lemma 32.43.16. Look at the short exact sequence

\[ 0 \to \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L} \to \mathcal{L} \to i_*i^*\mathcal{L} \to 0 \]

Observe that $H^0(X, i_*i^*\mathcal{L}) = H^0(x, i^*\mathcal{L})$ has dimension $1$ as $x$ is a $k$-rational point ($k$ is algebraically closed). Since $s$ does not vanish at $x$ we conclude that

\[ H^0(X, \mathcal{L}) \longrightarrow H^0(X, i_*i^*\mathcal{L}) \]

is surjective. Hence $\dim _ k H^0(X, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}) = t - 1$. Finally, the long exact sequence of cohomology also shows that $H^1(X, \mathcal{I} \otimes _{\mathcal{O}_ X} \mathcal{L}) = 0$ thereby finishing the proof of the induction step.

To get an invertible sheaf as in (2) take an invertible sheaf $\mathcal{L}$ as in (1) and do the argument in the previous paragraph one more time. $\square$

Lemma 32.43.18. Let $k$ be an algebraically closed field. Let $X$ be a proper curve over $k$. Set $g = \dim _ k H^1(X, \mathcal{O}_ X)$. For every invertible $\mathcal{O}_ X$-module $\mathcal{L}$ with $\deg (\mathcal{L}) \geq 2g - 1$ we have $H^1(X, \mathcal{L}) = 0$.

Proof. Let $\mathcal{N}$ be the invertible module we found in Lemma 32.43.17 part (2). The degree of $\mathcal{N}$ is $\chi (X, \mathcal{N}) - \chi (X, \mathcal{O}_ X) = 0 - (1 - g) = g - 1$. Hence the degree of $\mathcal{L} \otimes \mathcal{N}^{\otimes - 1}$ is $\deg (\mathcal{L}) - (g - 1) \geq g$. Hence $\chi (X, \mathcal{L} \otimes \mathcal{N}^{\otimes -1}) \geq g + 1 - g = 1$. Thus there is a nonzero global section $s$ whose zero scheme is an effective Cartier divisor $D$ of degree $\deg (\mathcal{L}) - (g - 1)$. This gives a short exact sequence

\[ 0 \to \mathcal{N} \xrightarrow {s} \mathcal{L} \to i_*(\mathcal{L}|_ D) \to 0 \]

where $i : D \to X$ is the inclusion morphism. We conclude that $H^0(X, \mathcal{L})$ maps isomorphically to $H^0(D, \mathcal{L}|_ D)$ which has dimension $\deg (\mathcal{L}) - (g - 1)$. The result follows from the definition of degree. $\square$

[1] If $X$ is a proper curve and $\mathcal{F}$ is a coherent sheaf on $X$, then one often defines the degree as $\chi (X, \mathcal{F}) - r\chi (X, \mathcal{O}_ X)$ where $r = \dim _{\kappa (\xi )} \mathcal{F}_\xi $ is the rank of $\mathcal{F}$ at the generic point $\xi $ of $X$.

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