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The canonical divisor has degree 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 $g$ with canonical divisor $K_{C}$ . Then $\deg_{k} (K_{C}) = 2 g - 2,$ independently of the choice of the nonzero rational differential defining $K_{C}$ .

Facts & Assumptions

Given: A field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ of genus $g$ ; a canonical divisor $K_{C} = div (ω)$ for a nonzero rational differential $ω$ .

[F1]

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

[F2]

The canonical bundle has exactly $g$ independent sections: $h^{0} (C, ω_{C}) = g$ , and equivalently $l (K_{C}) = g$ for any canonical divisor $K_{C}$ . (The canonical bundle has exactly g independent sections)

[F3]

For the zero divisor $O_{C} (0) = O_{C}$ one has $H^{0} (C, O_{C}) ≅ k$ canonically, hence $l (0) = 1$ because $l (D) = \dim_{k} H^{0} (C, O_{C} (D))$ for every divisor $D$ . (Functions on a proper curve, The Riemann-Roch dimension l(D))

[F4]

The canonical sheaf is $ω_{C} = Ω_{C / k}^{1}$ ; a canonical divisor is the divisor $K_{C} = div (ω)$ of any nonzero rational differential, and the divisors of the nonzero rational differentials form a single linear equivalence class, so $ω_{C} ≅ O_{C} (K_{C})$ and any two canonical divisors differ by the divisor of a nonzero rational function. (Canonical bundle and canonical divisors, Divisors of rational differentials form one linear equivalence class)

[F5]

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

[F6]

For a divisor $D$ the Riemann-Roch space is $L (D) = {f \in k (C)^{\times} : div (f) + D \geq 0} \cup {0}$ , a $k$ -subspace of $k (C)$ with $l (D) = \dim_{k} L (D) = \dim_{k} H^{0} (C, O_{C} (D))$ . (The space L(D), The Riemann-Roch dimension l(D))

[F7]

On $C$ a divisor is a finite formal sum of closed points with $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ , an additive integer-valued function of divisors; in particular $K_{C} - K_{C} = 0$ and $\deg_{k}$ is defined on the divisor $K_{C}$ . (Degree divisor proper curve)

[F8]

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

Proof

Proof technique: direct; evaluate full Riemann-Roch at the canonical divisor and use $l (K_{C}) = g$ , $l (0) = 1$ .

1.1F4F5F7given

(Set-up.) By [F7] a divisor on $C$ is a finite sum of closed points with an additive degree, so $K_{C} - K_{C} = 0$ and $\deg_{k} (K_{C})$ is defined; by [F4] $K_{C}$ is the divisor of a nonzero rational differential and $ω_{C} ≅ O_{C} (K_{C})$ , and by [F5] the genus is $g = h^{1} (C, O_{C})$ .

1.2F2F3F6given

(Two evaluations.) By [F2] one has $l (K_{C}) = g$ , and by [F3] and [F6] one has $l (0) = \dim_{k} H^{0} (C, O_{C}) = 1$ .

2.1F1step 1.1

(Riemann-Roch at the canonical divisor.) Applying [F1] to the divisor $D = K_{C}$ gives $l (K_{C}) - l (K_{C} - K_{C}) = \deg_{k} (K_{C}) + 1 - g$ .

3.1step 1.2step 2.1algebra

Since $K_{C} - K_{C} = 0$ by step 1.1, substituting the two evaluations of step 1.2 into the identity of step 2.1 gives $g - 1 = \deg_{k} (K_{C}) + 1 - g$ , hence $\deg_{k} (K_{C}) = 2 g - 2$ .

4.1F1F2F4step 1.2step 3.1

(Independence of the differential.) Let $ω^{'}$ be another nonzero rational differential with canonical divisor $K_{C}^{'} = div (ω^{'})$ ; by [F4] $K_{C}^{'}$ is again a canonical divisor and $K_{C}^{'} - K_{C}^{'} = 0$ , and by [F2] $l (K_{C}^{'}) = g$ ; applying [F1] to $D = K_{C}^{'}$ and substituting $l (K_{C}^{'}) = g$ together with $l (0) = 1$ from step 1.2 gives $g - 1 = \deg_{k} (K_{C}^{'}) + 1 - g$ , hence $\deg_{k} (K_{C}^{'}) = 2 g - 2$ exactly as in step 3.1.

5.1F8step 3.1step 4.1∎

Steps 3.1 and 4.1 prove $\deg_{k} (K_{C}) = 2 g - 2$ for every choice of nonzero rational differential, so the degree is independent of that choice; the Axiom of Choice [F8] is used exactly through the duality suppliers cited above.

Depends on

Used by

H¹ of a line bundle vanishes above degree 2g - 2 Corollary
The genus of a smooth plane curve in terms of its degree 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
Serre duality on the projective line, twist by twist Example
The full Riemann-Roch theorem on the projective line, in every degree Example
The canonical bundle of a genus-one curve is trivial Theorem
The canonical map: base-point-freeness and the hyperelliptic exception Theorem
The Riemann-Hurwitz formula with the different Theorem

Dependency tree · two levels

64 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)