CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-10-02

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The canonical bundle has exactly g independent sections

Statement

Assume the Axiom of Choice as inherited from the duality and coherent-cohomology suppliers. Let $C$ be a smooth proper geometrically integral curve over a field $k$ of genus $g$ with canonical bundle $ω_{C}$ . Then $h^{0} (C, ω_{C}) = g and h^{1} (C, O_{C}) = g;$ equivalently $l (K_{C}) = g$ for any canonical divisor $K_{C}$ .

Facts & Assumptions

Given: A field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ of genus $g$ ; its canonical bundle $ω_{C} = Ω_{C / k}^{1}$ and a canonical divisor $K_{C}$ .

[F1]

Serre duality for line bundles on $C$ : for every invertible $O_{C}$ -module $L$ the trace pairing $H^{1} (C, L) \times H^{0} (C, ω_{C} \otimes L^{- 1}) \to k$ is perfect, so $h^{1} (C, L) = h^{0} (C, ω_{C} \otimes L^{- 1})$ ; in particular $h^{1} (C, O_{C}) = h^{0} (C, ω_{C})$ and $h^{1} (C, ω_{C}) = h^{0} (C, O_{C})$ . (Serre duality for line bundles on a smooth proper curve, and the residue realization)

[F2]

For a smooth curve $C$ the canonical sheaf $ω_{C} = Ω_{C / k}^{1}$ is invertible, and for a nonzero rational differential $ω$ with divisor $K_{C} = div (ω)$ one has $ω_{C} ≅ O_{C} (K_{C})$ ; any two canonical divisors are linearly equivalent. (Canonical bundle and canonical divisors)

[F3]

The genus of $C$ is $g = g (C) = h^{1} (C, O_{C}) = \dim_{k} H^{1} (C, O_{C})$ , so $g = 1 - χ (O_{C})$ . (Genus via the Euler characteristic)

[F4]

For a proper curve over $k$ that is geometrically connected and geometrically reduced, the canonical map $k \to H^{0} (C, O_{C})$ is an isomorphism; a smooth proper geometrically integral curve is such a curve. (Functions on a proper curve)

[F5]

For a divisor $D$ on $C$ one has $l (D) = \dim_{k} L (D) = \dim_{k} H^{0} (C, O_{C} (D)) = h^{0} (D)$ , and $h^{i} (D) = \dim_{k} H^{i} (C, O_{C} (D))$ , with $H^{0}$ and $H^{1}$ the derived sheaf cohomology of $O_{C} (D)$ . (The Riemann-Roch dimension l(D), Sheaf cohomology as right derived global sections)

[F6]

The index of speciality of $D$ is $i (D) = h^{1} (C, O_{C} (D)) = \dim_{k} H^{1} (C, O_{C} (D))$ . (The index of speciality i(D))

[F8]

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

Proof

Proof technique: direct; dualize $O_{C}$ and $ω_{C}$ and compare with the definition of the genus.

1.1F2F3F4F5given

(Set-up.) By [F3] the genus is $g = h^{1} (C, O_{C})$ , and by [F4] there is a canonical isomorphism $k \tilde{\to} H^{0} (C, O_{C})$ , so that $h^{0} (O_{C}) = 1$ and $l (0) = 1$ in the notation of [F5]; by [F2] the canonical sheaf $Ω_{C / k}^{1}$ is invertible and $ω_{C} ≅ O_{C} (K_{C})$ for the divisor $K_{C}$ of any nonzero rational differential, with $K_{C}$ well defined modulo linear equivalence.

1.2F1given

(Duality.) The duality statement [F1] applied to the invertible sheaves $L = O_{C}$ and $L = ω_{C}$ gives $h^{1} (C, O_{C}) = h^{0} (C, ω_{C} \otimes O_{C}) = h^{0} (C, ω_{C})$ and $h^{1} (C, ω_{C}) = h^{0} (C, ω_{C} \otimes ω_{C}^{- 1}) = h^{0} (C, O_{C})$ , the tensor simplifications being the canonical ones for invertible sheaves.

2.1F3F6step 1.1step 1.2

Combining step 1.2 with step 1.1 gives $h^{0} (C, ω_{C}) = h^{1} (C, O_{C}) = g$ , which is the first displayed identity and, by [F6], also the identity $i (0) = g$ .

3.1F2F5F8step 2.1∎

For any canonical divisor $K_{C}$ one has $ω_{C} ≅ O_{C} (K_{C})$ by [F2], so $l (K_{C}) = h^{0} (C, O_{C} (K_{C})) = h^{0} (C, ω_{C}) = g$ by [F5] and step 2.1; together with step 2.1 this proves both displayed identities and the equivalent formulation, and the Axiom of Choice [F8] is used exactly through the duality and coherent-cohomology suppliers cited above.

Depends on

Used by

The canonical divisor has degree 2g - 2 Corollary
Degree 2g does not force very ampleness Counterexample
Degree 2g-1 does not force base-point-freeness Counterexample
The canonical map of a hyperelliptic curve is not an embedding Counterexample
Hyperelliptic curves and hyperelliptic maps Definition
Adjunction on a smooth plane cubic: the canonical bundle is trivial Example
Adjunction on a smooth plane quartic: the canonical bundle is the hyperplane bundle Example
The canonical bundle of a genus-one curve is trivial Theorem
The canonical map: base-point-freeness and the hyperelliptic exception Theorem

Dependency tree · two levels

102 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)