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Adding points never raises h^1, and h^1 stabilizes

Statement

Assume the Axiom of Choice, inherited from the sheaf-cohomology suppliers below. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field), let $D$ be a divisor on $C$ (Divisors on a smooth proper curve) and let $p \in C$ be a closed point with residue degree $d = [κ (p) : k]$ (Degree divisor proper curve). Write $h^{i} (D) = \dim_{k} H^{i} (C, O_{C} (D))$ (The Riemann-Roch dimension l(D)).

The long exact sequence of the short exact sequence $0 \to O_{C} (D) \to O_{C} (D + p) \to i_{p, *} κ (p) \to 0$ contains, after identifying $H^{0} (C, i_{p, *} κ (p))$ with $κ (p)$ , the exact sequence of finite-dimensional $k$ -vector spaces and $k$ -linear maps $0 \to H^{0} (C, O_{C} (D)) \to H^{0} (C, O_{C} (D + p)) \to κ (p) \overset{\partial}{\to} H^{1} (C, O_{C} (D)) \to H^{1} (C, O_{C} (D + p)) \to 0,$ whose middle arrow $\partial$ is the connecting map. Consequently $H^{1} (C, O_{C} (D + p)) ≅ H^{1} (C, O_{C} (D)) / im \partial,$ so that $h^{1} (D + p) \leq h^{1} (D)$ , with equality if and only if $\partial = 0$ , and $h^{1} (D) - h^{1} (D + p) = \dim_{k} im \partial \leq d .$
Consequently for every effective divisor $E \geq 0$ one has $h^{1} (D + E) \leq h^{1} (D)$ : the integer $h^{1}$ is antitone in the divisor (Divisor support positive negative parts).
In particular, for a fixed divisor $D_{0}$ and a fixed effective divisor $A \geq 0$ , the sequence $n \mapsto h^{1} (D_{0} + n A)$ is non-increasing and therefore stabilizes: it is constant for all sufficiently large $n$ . Equivalently, removing points from a divisor can only raise or preserve $h^{1}$ : if $D^{'} \leq D^{''}$ then $h^{1} (D^{''}) \leq h^{1} (D^{'})$ .

The short exact sequence and the cohomology of the skyscraper $i_{p, *} κ (p)$ are supplied by The exact sequence for adding one point to a divisor. Its construction uses the current Cartier-divisor and order interfaces Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors and Cartier and Weil divisors agree on a smooth curve.

Facts & Assumptions

Given: a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ , divisors $D$ , $D_{0}$ , $A$ on $C$ with $A \geq 0$ , and a closed point $p \in C$ .

[F1]

Divisors and degrees. A divisor on $C$ is a finite formal sum $\sum_{x} n_{x} [x]$ over the closed points, $D \leq E$ means that $E - D$ is effective, $\deg_{k}$ is additive and $\deg_{k} (E) \geq 0$ for effective $E$ , and the residue degree satisfies $[κ (p) : k] = \dim_{k} κ (p) \geq 1$ for every closed point $p$ (Divisors on a smooth proper curve, Divisor support positive negative parts, Degree divisor proper curve, Effective divisors have nonnegative degree).

[F2]

Adding one point. The sequence $0 \to O_{C} (D) \to O_{C} (D + p) \to i_{p, *} κ (p) \to 0$ is a short exact sequence of coherent $O_{C}$ -modules, $H^{0} (C, i_{p, *} κ (p)) ≅ κ (p)$ with $k$ -dimension $d = [κ (p) : k]$ , and $H^{q} (C, i_{p, *} κ (p)) = 0$ for every $q \geq 1$ (The exact sequence for adding one point to a divisor).

[F3]

Long exact sequence. A short exact sequence of abelian sheaves on $C$ gives a natural long exact sequence of sheaf cohomology groups $H^{q}$ , with the connecting map $H^{0} \to H^{1}$ (Long exact sequence of sheaf cohomology, Sheaf cohomology as right derived global sections).

[F4]

Finiteness and the notation $h^{i}$ . For every divisor $D^{'}$ the groups $H^{q} (C, O_{C} (D^{'}))$ are finite-dimensional $k$ -vector spaces for all $q \geq 0$ , vanishing for $q \geq 2$ , and $h^{i} (D^{'}) = \dim_{k} H^{i} (C, O_{C} (D^{'}))$ is a nonnegative integer (The Riemann-Roch dimension l(D), Finite-dimensionality of the Riemann-Roch space, Finite-dimensional coherent cohomology over a field, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

[F5]

$k$ -linearity of the long exact sequence. For an $O_{C}$ -module $F$ the cohomology groups are the right derived objects of the global-sections functor of Sheaf cohomology as right derived global sections, and multiplication by a scalar $c \in k$ on $F$ is the $O_{C}$ -module endomorphism given by multiplication by the global function $c$ , so functoriality of the derived objects turns it into the scalar action on $H^{q} (C, F)$ ; a morphism of $O_{C}$ -modules commutes with these endomorphisms and hence induces a $k$ -linear map on cohomology. Consequently the maps and the connecting map in the long exact sequence of [F3], applied to a short exact sequence of $O_{C}$ -modules, are $k$ -linear, and the terms are the finite-dimensional $k$ -vector spaces of the proper finiteness theorem (Finite-dimensional coherent cohomology over a field, Long exact sequence of sheaf cohomology).

[F6]

Linear algebra over $k$ . For a linear map $T : V \to W$ with $V$ finite-dimensional one has $\dim_{k} V = \dim_{k} \ker T + \dim_{k} im T$ (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ), the formula $\tilde{T} (v + \ker T) = T (v)$ defines an isomorphism $V / \ker T \to im T$ (First isomorphism theorem for vector spaces: $V / \ker T$ is isomorphic to $im T$ ), and for $W \leq V$ with $V$ finite-dimensional, $\dim_{k} (V / W) = \dim_{k} V - \dim_{k} W$ (A quotient basis lifts to a basis adapted to $W$ ).

[F7]

The Axiom of Choice is used exactly through the sheaf-cohomology suppliers [F2], [F3] and the finiteness supplier [F4]; no further selection is made below (The Axiom of Choice).

Proof

technique · direct; read the segment of the long exact sequence around the connecting map at one point, compute the dimension drop through the first isomorphism theorem, iterate over the points of an effective divisor, and observe that a non-increasing sequence of nonnegative integers is eventually constant

1.1F2F3F4F5

The exact segment. By [F2] the sequence $0 \to O_{C} (D) \to O_{C} (D + p) \to i_{p, *} κ (p) \to 0$ is a short exact sequence of coherent $O_{C}$ -modules with $H^{0} (C, i_{p, *} κ (p)) ≅ κ (p)$ and $H^{1} (C, i_{p, *} κ (p)) = 0$ . The long exact sequence of [F3] therefore contains the exact segment $H^{0} (C, O_{C} (D)) \to H^{0} (C, O_{C} (D + p)) \to H^{0} (C, i_{p, *} κ (p)) \overset{\partial}{\to} H^{1} (C, O_{C} (D)) \to H^{1} (C, O_{C} (D + p)) \to H^{1} (C, i_{p, *} κ (p))$ ; substituting the identification $H^{0} (C, i_{p, *} κ (p)) ≅ κ (p)$ of [F2] and its vanishing in degree one, and using the injectivity of the first map and the exactness at $H^{0} (C, O_{C} (D + p))$ , gives the exact sequence $0 \to H^{0} (C, O_{C} (D)) \to H^{0} (C, O_{C} (D + p)) \to κ (p) \overset{\partial}{\to} H^{1} (C, O_{C} (D)) \to H^{1} (C, O_{C} (D + p)) \to 0$ , with $\partial$ the connecting map; by [F4] and [F5] all six terms are finite-dimensional $k$ -vector spaces and all maps are $k$ -linear.

2.1F4F5F6step 1.1

The dimension drop. Exactness of the sequence of step 1.1 at $H^{1} (C, O_{C} (D))$ identifies $im \partial$ with the kernel of the map $H^{1} (C, O_{C} (D)) \to H^{1} (C, O_{C} (D + p))$ , and exactness at $H^{1} (C, O_{C} (D + p))$ says that this map is surjective; hence the first isomorphism theorem of [F6] identifies the $k$ -vector spaces $H^{1} (C, O_{C} (D + p))$ and $H^{1} (C, O_{C} (D)) / im \partial$ . Taking dimensions with the quotient formula of [F6] gives $h^{1} (D + p) = \dim_{k} H^{1} (C, O_{C} (D)) - \dim_{k} im \partial = h^{1} (D) - \dim_{k} im \partial$ , so $h^{1} (D + p) \leq h^{1} (D)$ , with equality exactly when $im \partial = 0$ , that is, exactly when the connecting map $\partial$ is zero. Since $\partial$ is $k$ -linear out of the finite-dimensional space $κ (p)$ of dimension $d$ , rank-nullity gives $\dim_{k} im \partial = \dim_{k} κ (p) - \dim_{k} \ker \partial \leq \dim_{k} κ (p) = d$ by [F4], so the drop is at most $d$ .

3.1F1step 2.1

Antitonicity in the divisor. Let $E \geq 0$ be effective and write its finite support with multiplicities as $E = \sum_{i} c_{i} [x_{i}]$ ; consider the finite chain of divisors $D \leq D + [x_{1}] \leq D + 2 [x_{1}] \leq \dots \leq D + E$ that adds one copy of a closed point at a time. Every consecutive pair is of the form $D^{'} \leq D^{'} + q$ for a closed point $q$ , so step 2.1 applied to the divisor $D^{'}$ and the point $q$ gives $h^{1} (D^{'} + q) \leq h^{1} (D^{'})$ ; chaining these inequalities along the finite chain gives $h^{1} (D + E) \leq h^{1} (D)$ .

4.1F1F4step 3.1

Stabilization and the downward reading. Fix a divisor $D_{0}$ and an effective divisor $A \geq 0$ . For every $n \geq 0$ one has $D_{0} + n A \leq D_{0} + (n + 1) A$ , so step 3.1 applied to the divisor $D_{0} + n A$ gives $h^{1} (D_{0} + (n + 1) A) \leq h^{1} (D_{0} + n A)$ : the sequence $n \mapsto h^{1} (D_{0} + n A)$ is non-increasing. Its values are nonnegative integers bounded above by $h^{1} (D_{0})$ by step 3.1; a strict decrease lowers the value by at least one, so the sequence has at most $h^{1} (D_{0})$ strict decreases and is therefore constant for all sufficiently large $n$ , which is the asserted stabilization. Finally, if $D^{'} \leq D^{''}$ are divisors, then $D^{''} = D^{'} + E$ for the effective divisor $E = D^{''} - D^{'}$ and step 3.1 gives $h^{1} (D^{''}) \leq h^{1} (D^{'})$ ; reading $D^{'}$ as obtained from $D^{''}$ by removing the points of $E$ , removing points from a divisor can only raise or preserve $h^{1}$ .

5.1F2F3F4F7step 1.1step 2.1step 3.1step 4.1∎

Conclusion and choice accounting. Step 1.1 gives the exact sequence and the identification of the connecting map, step 2.1 the inequality, the equality criterion and the bound $d$ on the drop, step 3.1 the antitonicity for every effective divisor, and step 4.1 the stabilization and the equivalent downward reading; this proves parts (1), (2) and (3) of the Statement. The Axiom of Choice is used only through the sheaf-cohomology suppliers of [F2] and [F3] and the finiteness supplier of [F4], as recorded in [F7]; the point $q$ added at each stage of step 3.1 is one of the finitely many points of $E$ , so no further selection is made.

Depends on

Used by

Vanishing of H¹ in a fixed ample direction Theorem

Dependency tree · two levels

110 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

William Fulton, Algebraic Curves (Internet Archive copy), Chs. 8 and 6 (standard reference, not scraped)
Michael Artin, MIT 18.721 Introduction to Algebraic Geometry (July 20, 2020 notes), Ch. 8 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025), Chs. 18.5 and 21 (standard reference, not scraped)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)