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Vanishing of H^1 in a fixed ample direction

Statement

Assume the Axiom of Choice as inherited from Serre vanishing and the ample-powers theorem. It also supplies the Dependent Choice premise of the curve Cartier-to-Weil result through AC implies DC implies countable choice. Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field). Let $φ : C \to P_{k}^{1}$ be a finite $k$ -morphism (such a morphism exists by Finite morphisms from a curve to the projective line) and let $A$ be an effective divisor on $C$ with $O_{C} (A) ≅ φ^{*} O_{P_{k}^{1}} (1)$ (Divisors on a smooth proper curve). Write $L = O_{C} (A)$ (The Riemann-Roch dimension l(D)).

Then for every divisor $D_{0}$ on $C$ there is an integer $n_{0}$ such that $H^{1} (C, O_{C} (D_{0} + n A + E)) = 0 for every n \geq n_{0} and every effective divisor E,$ equivalently $h^{1} (D) = 0$ for every divisor $D$ with $D \geq D_{0} + n_{0} A$ . The bound $n_{0}$ may depend on $D_{0}$ and on the fixed morphism $φ$ , but not on $E$ or on the degree of $E$ . No Serre duality and no Riemann-Roch threshold $2 g - 2$ are used.

The divisor-to-sheaf interface used below first identifies the Weil divisors $D_{0}$ and $A$ as Cartier divisors, constructs their sheaves, and applies the Cartier addition/tensor isomorphism; the actual interfaces and their uses are recorded in [F6]. The finite morphism and pole-divisor realization in [F1] are the stated interface of Finite morphisms from a curve to the projective line. The proof uses the closed H-very ample witness from [F4] and the established affine-base implication in [F9] for Serre vanishing, and uses Finite-dimensionality of the Riemann-Roch space to establish coherence of $O_{C} (D_{0})$ as recorded in [F10].

Facts & Assumptions

Given: a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ , a finite $k$ -morphism $φ : C \to P_{k}^{1}$ , an effective divisor $A$ on $C$ with $O_{C} (A) ≅ φ^{*} O (1)$ , the invertible sheaf $L = O_{C} (A)$ , and a divisor $D_{0}$ .

[F1]

The morphism and its twist: every smooth proper geometrically integral curve over $k$ admits a finite locally free $k$ -morphism to $P_{k}^{1}$ , and for a nonconstant $f \in k (C)^{\times}$ with pole divisor $A = (f)_{\infty}$ the sheaf $O_{C} (A)$ is isomorphic to $φ_{f}^{*} O_{P_{k}^{1}} (1)$ (Finite morphisms from a curve to the projective line, Finite morphisms of schemes).

[F2]

Ampleness of the twisting sheaf on $P_{k}^{1}$ : the identity of $P_{k}^{1}$ over $Spec k$ is a closed immersion pulling $O (1)$ back to $O (1)$ , so $O (1)$ is H-very ample relative to $Spec k$ ; since the base is affine, Relative very ampleness implies relative ampleness makes $O (1)$ ample in the absolute sense of Absolute ampleness by affine section opens (Relative very ampleness in the finite projective-space convention, Twists of a quasi-coherent sheaf, Invertible sheaves).

[F3]

Ampleness pulls back along finite morphisms: the pullback $g^{*} L$ of an ample invertible sheaf along a finite morphism $g$ is ample (Finite pullback preserves absolute ampleness).

[F4]

Ample powers are closed H-very ample over a proper finite-type base: for a proper finite-type morphism $X \to S$ with $S$ Noetherian and $L$ an ample invertible $O_{X}$ -module, there is $d_{0} \geq 1$ such that $L^{\otimes d}$ is closed H-very ample relative to $S$ for every $d \geq d_{0}$ , witnessed by a closed immersion into a relative projective space with $O (1)$ pulling back to $L^{\otimes d}$ (High powers of an ample line bundle embed a proper scheme, Locally Noetherian and Noetherian schemes, Proper morphisms, Locally finite type and finite type morphisms, Relative very ampleness in the finite projective-space convention).

[F5]

Serre vanishing: for a Noetherian commutative ring $A$ , a scheme $X$ projective over $A$ in the finite-dimensional H-projective convention, an ample invertible $O_{X}$ -module $L$ and a coherent $O_{X}$ -module $F$ , there is $m_{0}$ with $H^{q} (X, F \otimes L^{\otimes m}) = 0$ for every $q > 0$ and every $m \geq m_{0}$ (Serre vanishing for coherent sheaves and ample twists, Projective morphisms before Proj, Coherent module sheaves, Sheaf cohomology as right derived global sections); a field is Noetherian and its spectrum is Noetherian (A field has only the zero ideal and itself, hence is Noetherian, The spectrum of a Noetherian ring is a Noetherian topological space).

[F6]

Divisor and tensor dictionary. The divisors $D_{0}$ and $A$ on the smooth curve are Weil divisors. The curve Cartier-to-Weil result Cartier and Weil divisors agree on a smooth curve identifies them as Cartier divisors; Invertible sheaf of cartier divisor constructs their associated sheaves, and The sheaf of a Cartier divisor is invertible proves those sheaves invertible. For every integer $n \geq 0$ , repeated application of Addition of Cartier divisors is tensor product of their sheaves gives $O_{C} (D_{0} + n A) ≅ O_{C} (D_{0}) \otimes O_{C} (A)^{\otimes n} ≅ O_{C} (D_{0}) \otimes L^{\otimes n}$ . The rational-section identification with $L (D_{0} + n A)$ is the one in The space L(D), using Rational sections of line bundles are Cartier divisors. This is the actual interface used in step 4.1.

[F7]

Monotonicity of $h^{1}$ : for every divisor $D$ and every effective divisor $E \geq 0$ one has $h^{1} (D + E) \leq h^{1} (D)$ , where $h^{1} (D) = \dim_{k} H^{1} (C, O_{C} (D))$ ; equivalently $h^{1}$ is antitone in the divisor (Adding points never raises h^1, and h^1 stabilizes, The Riemann-Roch dimension l(D)).

[F8]

The Axiom of Choice is inherited from the ample-powers theorem [F4] and Serre vanishing [F5]. In ZF, AC implies DC by AC implies DC implies countable choice, so the stated assumption supplies the Dependent Choice premise of the curve Cartier-to-Weil result in [F6] (The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain). Apart from the integers supplied by the cited existence theorems, the proof makes no further selection.

[F9]

A closed H-very ample invertible sheaf on a scheme over the affine base $Spec k$ is ample in the absolute sense by Relative very ampleness implies relative ampleness; the implication applies to the closed immersion and pulled-back $O (1)$ witness supplied in step 2.1. This is the route establishing ampleness of $L^{\otimes d}$ for Serre vanishing, rather than an assumed closure of ampleness under tensor powers.

[F10]

The actual lemma Finite-dimensionality of the Riemann-Roch space applies to the given smooth proper geometrically integral curve and divisor $D_{0}$ ; it proves that $O_{C} (D_{0})$ is coherent. This supplies the coherent-sheaf hypothesis of [F5] in step 3.1.

Proof

technique · pull the ample twisting sheaf on $\mathbb P^1_k$ back along the finite morphism, use that an ample power of the pullback embeds $C$ projectively, apply Serre vanishing to the twist of $\mathcal O_C(D_0)$, and remove the extra effective summand by monotonicity of $h^1$

1.1F1F2F3

Ampleness of $L$ . By [F2] the twisting sheaf $O (1)$ is ample on $P_{k}^{1}$ , and by the hypothesis $φ$ is a finite $k$ -morphism with $O_{C} (A) ≅ φ^{*} O (1)$ ; hence $L = O_{C} (A)$ is isomorphic to the pullback of an ample invertible sheaf along a finite morphism, and [F3] makes $L$ ample.

2.1F1F4F5step 1.1

An ample power embeds $C$ projectively. The curve $C$ is proper and of finite type over the field $k$ by [F1], so the structure morphism $C \to Spec k$ is proper and of finite type, and $Spec k$ is Noetherian by [F5]; with $L$ ample by step 1.1, [F4] provides an integer $d \geq 1$ such that $L^{\otimes d}$ is closed H-very ample relative to $Spec k$ , witnessed by a closed immersion $C ↪ P_{k}^{N}$ with $O (1)$ pulling back to $L^{\otimes d}$ . In particular $C$ is projective over the Noetherian ring $k$ in the H-projective convention of [F5].

3.1F5F9F10step 2.1

Serre vanishing for the twist by a power of $L$ . The field $k$ and $Spec k$ are Noetherian by [F5]. The sheaf $O_{C} (D_{0})$ is coherent by [F10], and the closed H-very ample witness for $L^{\otimes d}$ from step 2.1 makes $L^{\otimes d}$ ample by [F9]. Applying [F5] to the projective $k$ -scheme $C$ , this ample invertible sheaf and the coherent sheaf $O_{C} (D_{0})$ gives an integer $m_{1}$ such that $H^{1} (C, O_{C} (D_{0}) \otimes_{O_{C}} L^{\otimes d m}) = 0 for every m \geq m_{1} .$ Set $m_{0} : = \max (m_{1}, 0)$ . Since the vanishing holds for every $m \geq m_{1}$ , it holds for every $m \geq m_{0}$ , and now all exponents used in the divisor translation are nonnegative.

4.1F6step 3.1

Translation to divisors. By [F6] one has $O_{C} (D_{0}) \otimes L^{\otimes d m} ≅ O_{C} (D_{0} + d m A)$ for every $m \geq 0$ , with $L = O_{C} (A)$ ; since $m_{0} \geq 0$ , step 3.1 gives $H^{1} (C, O_{C} (D_{0} + d m A)) = 0$ for every $m \geq m_{0}$ .

5.1F1F7step 4.1

The stated form. Put $n_{0} : = d m_{0}$ , which is nonnegative since $d \geq 1$ and $m_{0} \geq 0$ . By step 4.1 one has $H^{1} (C, O_{C} (D_{0} + n_{0} A)) = 0$ , and this vanishing spreads to all $n \geq n_{0}$ and all effective $E$ : write $n = n_{0} + r$ with $r \geq 0$ , so $D_{0} + n A + E = D_{0} + n_{0} A + (r A + E)$ , where $r A + E$ is effective because $A$ and $E$ are effective; by [F7] applied to the divisor $D_{0} + n_{0} A$ and the effective divisor $r A + E$ one gets $h^{1} (D_{0} + n A + E) \leq h^{1} (D_{0} + n_{0} A) = 0$ . Thus $H^{1} (C, O_{C} (D_{0} + n A + E)) = 0$ for every $n \geq n_{0}$ and every effective $E$ .

6.1F7step 5.1

The equivalent form. If $D$ is a divisor with $D \geq D_{0} + n_{0} A$ , then $E : = D - (D_{0} + n_{0} A)$ has nonnegative coefficients, that is, $E$ is effective, and $D = D_{0} + n_{0} A + E$ ; step 5.1 at $n = n_{0}$ gives $h^{1} (D) = 0$ . Conversely, if $h^{1} (D) = 0$ for every $D \geq D_{0} + n_{0} A$ , then for every $n \geq n_{0}$ and every effective $E$ the divisor $D_{0} + n A + E$ satisfies $D_{0} + n A + E \geq D_{0} + n_{0} A$ , so $h^{1} (D_{0} + n A + E) = 0$ ; the two formulations are therefore equivalent.

7.1F1F4F5F6F7F8F9F10step 1.1step 2.1step 3.1step 4.1step 5.1step 6.1∎

Conclusion and choice accounting. Step 1.1 makes $L = O_{C} (A)$ ample as the pullback of the ample twisting sheaf of $P_{k}^{1}$ along the finite morphism $φ$ ; step 2.1 exhibits $C$ as projective over the Noetherian ring $k$ through a closed H-very ample witness; step 3.1 applies Serre vanishing to the coherent twist by the ample sheaf $L^{\otimes d}$ and replaces its bound by $m_{0} \geq 0$ ; step 4.1 translates that vanishing to the divisors $D_{0} + d m A$ ; and steps 5.1 and 6.1 spread it to all $n \geq n_{0}$ and all effective summands $E$ , proving both formulations with $n_{0} = d m_{0}$ . The integers $d$ and $m_{0}$ depend only on the morphism $φ$ (through $L$ and the closed immersion witness) and on $D_{0}$ , not on $E$ or on $\deg_{k} E$ . No Serre duality and no threshold $2 g - 2$ is used anywhere; the Choice premise used by the Cartier-to-Weil supplier is explicitly obtained from the stated Axiom of Choice by [F8].

Depends on

Used by

Dependency tree · two levels

165 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)
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)