LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)audited 2026-10-02

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Sufficiently positive divisors in a fixed direction are nonspecial

Statement

Assume the Axiom of Choice as inherited from the vanishing theorem. Let $k$ be a field, 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 and let $A$ be an effective divisor on $C$ with $O_{C} (A) ≅ φ^{*} O_{P_{k}^{1}} (1)$ — for instance $A = (f)_{\infty}$ for a nonconstant $f \in k (C)^{\times}$ , as produced by Finite morphisms from a curve to the projective line (Divisors on a smooth proper curve). Let $D_{0}$ be a divisor on $C$ . Then there is an integer $n_{0}$ , depending on $D_{0}$ and on the fixed morphism $φ$ (through $A$ ), such that every divisor $D$ with $D \geq D_{0} + n_{0} A$ is nonspecial: $H^{1} (C, O_{C} (D)) = 0.$ Explicitly, every divisor of the form $D_{0} + n A + E$ with $n \geq n_{0}$ and $E$ effective is nonspecial.

Fixed direction only. This is only a statement about the fixed ample direction $A$ : it is not claimed that every divisor of degree greater than $2 g - 2$ is nonspecial, which is a different statement requiring the duality pair that follows this page. No threshold in terms of $\deg_{k} (D)$ alone and no Serre duality is used or asserted here.

The example uses the finite-map construction Finite morphisms from a curve to the projective line and the fixed-direction vanishing theorem Vanishing of H^1 in a fixed ample direction, as stated in [F1].

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 $O_{C} (A) ≅ φ^{*} O_{P_{k}^{1}} (1)$ , and a divisor $D_{0}$ on $C$ .

[F1]

Vanishing theorem: for the fixed curve, morphism $φ$ and effective divisor $A$ , and for the given $D_{0}$ , 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 $D \geq D_{0} + n_{0} A$ (Vanishing of H^1 in a fixed ample direction).

[F2]

Nonspecial divisors: $i (D) = h^{1} (C, O_{C} (D)) = \dim_{k} H^{1} (C, O_{C} (D))$ is the index of speciality, and $D$ is nonspecial exactly when $i (D) = 0$ , that is, exactly when $H^{1} (C, O_{C} (D)) = 0$ ; $D$ is special exactly when $i (D) \geq 1$ (Special and nonspecial divisors, The index of speciality i(D), The Riemann-Roch dimension l(D)).

[F3]

Divisors: a divisor on $C$ is a finite formal integral combination of closed points, effective when all coefficients are nonnegative, and $D \geq D^{'}$ means that $D - D^{'}$ is effective (Divisors on a smooth proper curve).

[F4]

The genus $g = g (C) = h^{1} (C, O_{C})$ is a nonnegative integer; the threshold $2 g - 2$ and the duality pair are not part of this lemma and no statement about them is made here (Genus via the Euler characteristic).

[F5]

The Axiom of Choice is available and is inherited only through the vanishing theorem [F1]; the argument below transforms the vanishing statement into the definition of nonspeciality and selects nothing beyond the integer $n_{0}$ supplied by [F1] (The Axiom of Choice).

Proof

technique · apply the fixed-direction vanishing theorem to $D_0$, then translate $h^1(D)=0$ into nonspeciality

1.1F1

The fixed-direction threshold. By [F1] applied to the given divisor $D_{0}$ and the fixed morphism $φ$ and divisor $A$ , there is an integer $n_{0}$ with $H^{1} (C, O_{C} (D_{0} + n A + E)) = 0$ for every $n \geq n_{0}$ and every effective $E$ , equivalently $h^{1} (D) = 0$ for every divisor $D$ with $D \geq D_{0} + n_{0} A$ ; this integer depends only on $D_{0}$ and on $φ$ through the fixed sheaf $O_{C} (A) ≅ φ^{*} O (1)$ , not on $D$ or $E$ .

2.1F2step 1.1

Nonspeciality above the threshold. Let $D$ be a divisor with $D \geq D_{0} + n_{0} A$ . By step 1.1, $h^{1} (D) = 0$ , and by [F2] the index of speciality vanishes exactly for the nonspecial divisors, so $D$ is nonspecial, that is, $H^{1} (C, O_{C} (D)) = 0$ . The same applies to every $D = D_{0} + n A + E$ with $n \geq n_{0}$ and $E$ effective, since such a divisor dominates $D_{0} + n_{0} A$ and is of the form covered by [F1].

3.1F1F2F3F4F5step 1.1step 2.1∎

Conclusion, the fixed-direction restriction and choice accounting. Steps 1.1 and 2.1 show that every divisor $D \geq D_{0} + n_{0} A$ , in particular every $D_{0} + n A + E$ with $n \geq n_{0}$ and $E$ effective, is nonspecial with $H^{1} (C, O_{C} (D)) = 0$ . Nothing is asserted about divisors merely of large degree: the lemma provides no universal bound in terms of $\deg_{k} (D)$ and $2 g - 2$ , and no Serre duality enters, as [F4] records. The integer $n_{0}$ is the one supplied by [F1] for $D_{0}$ and the fixed morphism; it is not chosen, and the Axiom of Choice is inherited only through [F1], as recorded in [F5].

Depends on

Used by

Dependency tree · two levels

77 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), Ch. 18.5 and Ch. 21 (standard reference, not scraped)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)