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Riemann-Roch in exact form for divisors of degree above 2g - 2

Statement

Assume the Axiom of Choice as inherited from the duality suppliers. Let $C$ be a smooth proper geometrically integral curve over a field $k$ of genus $g$ and let $D$ be a divisor with $\deg_{k} (D) > 2 g - 2$ . Then $l (D) = \deg_{k} (D) + 1 - g,$ that is, $D$ is nonspecial in the sense of Special and nonspecial divisors, and $h^{1} (C, O_{C} (D)) = 0$ .

Facts & Assumptions

Given: A field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ of genus $g$ ; a divisor $D$ on $C$ with $\deg_{k} (D) > 2 g - 2$ .

[F1]

For an invertible $O_{C}$ -module $L$ with $\deg (L) > 2 g - 2$ one has $H^{1} (C, L) = 0$ and $h^{0} (C, L) = \deg (L) + 1 - g$ . (H^1 of a line bundle vanishes above degree 2g - 2)

[F2]

For a divisor $D$ one has $l (D) = \dim_{k} L (D) = \dim_{k} H^{0} (C, O_{C} (D)) = h^{0} (D)$ , the space $L (D)$ consisting of the rational functions whose divisor plus $D$ is effective, and $i (D) = h^{1} (C, O_{C} (D)) = \dim_{k} H^{1} (C, O_{C} (D))$ . (The Riemann-Roch dimension l(D), The space L(D), The index of speciality i(D))

[F3]

A divisor $D$ is nonspecial when $i (D) = 0$ , that is when $H^{1} (C, O_{C} (D)) = 0$ ; by Riemann-Roch as $l$ minus $i$ this holds exactly when $l (D) = \deg_{k} (D) + 1 - g$ . (Special and nonspecial divisors, Riemann-Roch as l minus i)

[F4]

Full Riemann-Roch: for every divisor $D$ one has $l (D) - l (K_{C} - D) = \deg_{k} (D) + 1 - g$ , equivalently $h^{0} (C, O_{C} (D)) - h^{1} (C, O_{C} (D)) = \deg_{k} (D) + 1 - g$ . (The full Riemann-Roch theorem for divisors on a smooth proper curve, Riemann-Roch as l minus i)

[F5]

Divisor degree is $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ and is additive (Degree divisor proper curve). Every invertible sheaf is $O_{C} (E)$ with $E$ unique modulo principal divisors (Cartier and Weil divisors agree on a smooth curve). Principal divisors have degree zero (Principal divisors on a normal proper curve have degree zero), so defining $\deg (O_{C} (E)) : = \deg_{k} (E)$ is independent of the representative. In particular $\deg (O_{C} (D)) = \deg_{k} (D)$ .

[F6]

The Axiom of Choice: every family of nonempty sets has a choice function. (The Axiom of Choice)

Proof

Proof technique: direct; apply the high-degree vanishing corollary to $O_{C} (D)$ and translate its output into the divisor language.

1.1F1F5given

(Set-up.) The invertible sheaf $L = O_{C} (D)$ has degree $\deg (L) = \deg_{k} (D)$ by [F5], so the hypothesis gives $\deg (L) > 2 g - 2$ , and [F1] applies to $L$ .

2.1F1step 1.1

By [F1] one has $H^{1} (C, L) = 0$ and $h^{0} (C, L) = \deg (L) + 1 - g = \deg_{k} (D) + 1 - g$ .

3.1F2F3step 2.1

By [F2] the vanishing of step 2.1 reads $i (D) = h^{1} (C, O_{C} (D)) = 0$ , so $D$ is nonspecial in the sense of [F3]; and $l (D) = h^{0} (C, O_{C} (D)) = h^{0} (C, L) = \deg_{k} (D) + 1 - g$ , using step 2.1.

4.1F4step 3.1

(Consistency with full Riemann-Roch.) Substituting $l (D) = \deg_{k} (D) + 1 - g$ from step 3.1 into the full Riemann-Roch identity of [F4] gives $l (K_{C} - D) = 0$ , equivalently $h^{0} (C, O_{C} (K_{C} - D)) = 0$ ; the equality $l (D) = \deg_{k} (D) + 1 - g$ is exactly the characterization of nonspeciality recorded in [F3], so the two formulations agree.

5.1F6step 2.1step 3.1step 4.1∎

Steps 2.1, 3.1 and 4.1 establish $l (D) = \deg_{k} (D) + 1 - g$ , the vanishing $H^{1} (C, O_{C} (D)) = 0$ , and the nonspeciality of $D$ ; the Axiom of Choice [F6] is used exactly through the duality suppliers cited above.

Depends on

Used by

A genus-one curve with a rational point embeds as a plane cubic Corollary
Degree 2g does not force very ampleness Counterexample
Degree 2g-1 does not force base-point-freeness Counterexample
A degree-n line bundle on a genus-one curve has an n-dimensional space of sections for n > 0 Example
Line bundles of degree at least 2g are base-point-free Theorem
Line bundles of degree at least 2g+1 are very ample Theorem

Dependency tree · two levels

69 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), Ch. 8 (standard reference, not scraped)
Ravi Vakil, The Rising Sea (version of October 21, 2025) (standard reference, not scraped)
MIT 18.725 Algebraic Geometry (Fall 2015) course notes, Lectures 24-25 (standard reference, not scraped)