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The full Riemann-Roch theorem for divisors on a smooth proper curve

Statement

Assume the Axiom of Choice as inherited from the coherent-cohomology suppliers. Let $C$ be a smooth proper geometrically integral curve over a field $k$ with genus $g = g (C)$ and canonical divisor $K_{C}$ , the divisor of a nonzero rational differential on $C$ . Then for every divisor $D$ on $C$ , $l (D) - l (K_{C} - D) = \deg_{k} (D) + 1 - g;$ equivalently $h^{0} (C, O_{C} (D)) - h^{1} (C, O_{C} (D))$ is the Euler characteristic of $O_{C} (D)$ and $i (D) = h^{1} (C, O_{C} (D)) = l (K_{C} - D)$ . Both sides of the identity are unchanged if $K_{C}$ is replaced by a linearly equivalent canonical divisor.

Facts & Assumptions

Given: A field $k$ ; a smooth proper geometrically integral curve $C$ over $k$ of genus $g$ ; a canonical divisor $K_{C}$ (the divisor of a nonzero rational differential); an arbitrary divisor $D$ on $C$ .

[F1]

Riemann-Roch as $l$ minus $i$ : for every divisor $D$ on $C$ one has $l (D) - i (D) = h^{0} (C, O_{C} (D)) - h^{1} (C, O_{C} (D)) = \deg_{k} (D) + 1 - g$ , and $i (D) \geq 0$ ; for $D = 0$ this reads $1 - g = 0 + 1 - g$ with $i (0) = g$ . No duality is used there, the index of speciality being left as an unknown defect. (Riemann-Roch as l minus i)

[F2]

With $K_{C}$ a canonical divisor, the index of speciality is realized by dual sections: $h^{1} (C, O_{C} (D)) = h^{0} (C, O_{C} (K_{C} - D)) = l (K_{C} - D)$ for every divisor $D$ . (h^1 of a line bundle equals the dimension of the space of dual sections)

[F3]

For a divisor $D$ one has $l (D) = \dim_{k} L (D) = \dim_{k} H^{0} (C, O_{C} (D)) = h^{0} (D)$ and $i (D) = h^{1} (C, O_{C} (D)) = \dim_{k} H^{1} (C, O_{C} (D)) \geq 0$ ; in particular $l$ and $i$ are nonnegative integers and $i (0) = g$ . (The Riemann-Roch dimension l(D), The index of speciality i(D))

[F4]

The canonical sheaf is $ω_{C} = Ω_{C / k}^{1}$ , and for a nonzero rational differential $ω$ with divisor $K_{C} = div (ω)$ one has $ω_{C} ≅ O_{C} (K_{C})$ ; the divisors of the nonzero rational differentials form a single linear equivalence class, 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 Riemann-Roch space is $L (D) = {f \in k (C)^{\times} : div (f) + D \geq 0} \cup {0}$ , equivalently $L (D) = H^{0} (C, O_{C} (D))$ as a $k$ -subspace of $k (C)$ . (The space L(D))

[F6]

On the smooth curve $C$ divisors are finite sums of closed points with degree $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ , Weil and Cartier divisors agree, and each divisor has an associated invertible sheaf $O_{C} (D)$ with local equations $f_{i}^{- 1} O_{U_{i}}$ , so that $D \mapsto O_{C} (D)$ descends to an isomorphism of divisor classes with the Picard group. (Divisors on a smooth proper curve, Degree divisor proper curve, Invertible sheaf of cartier divisor, Cartier and Weil divisors agree on a smooth curve)

[F7]

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

Proof

technique · direct; combine Riemann-Roch as $l$ minus $i$ with the duality identification of $i(D)$, then check invariance under the choice of canonical divisor

1.1F3F4F5F6given

(Set-up.) Let $D$ be a divisor on $C$ ; by [F6] $D$ is a finite sum of closed points with a degree $\deg_{k} (D)$ and has an associated invertible sheaf $O_{C} (D)$ well defined modulo linear equivalence, and by [F3] and [F5] the integers $l (D) = \dim_{k} H^{0} (C, O_{C} (D))$ and $i (D) = \dim_{k} H^{1} (C, O_{C} (D))$ are defined; by [F4] the divisor $K_{C}$ of a nonzero rational differential is a canonical divisor with $ω_{C} ≅ O_{C} (K_{C})$ , well defined modulo linear equivalence.

2.1F1F3step 1.1given

(Riemann-Roch, no duality.) By [F1] applied to the divisor $D$ one has $l (D) - i (D) = h^{0} (C, O_{C} (D)) - h^{1} (C, O_{C} (D)) = \deg_{k} (D) + 1 - g$ , so the Euler characteristic of $O_{C} (D)$ equals $\deg_{k} (D) + 1 - g$ and $i (D) \geq 0$ by [F3].

2.2F2step 1.1given

(Duality term.) By [F2] applied to the same $D$ and the canonical divisor $K_{C}$ one has $i (D) = h^{1} (C, O_{C} (D)) = h^{0} (C, O_{C} (K_{C} - D)) = l (K_{C} - D)$ .

3.1F1F2step 2.1step 2.2

(Full identity.) Substituting the identification of step 2.2 into the identity of step 2.1 gives $l (D) - l (K_{C} - D) = \deg_{k} (D) + 1 - g$ for every divisor $D$ , that is, both the displayed Riemann-Roch identity and the equivalent formulation $h^{0} (C, O_{C} (D)) - h^{1} (C, O_{C} (D)) = \deg_{k} (D) + 1 - g$ with $i (D) = l (K_{C} - D)$ .

4.1F3F4F5step 3.1

(Invariance in the canonical divisor.) Let $K_{C}^{'}$ be another canonical divisor; by [F4] there is $f \in k (C)^{\times}$ with $K_{C}^{'} = K_{C} + div (f)$ . By [F5], $u \in L (K_{C}^{'} - D)$ exactly when $u f \in L (K_{C} - D)$ , so multiplication by $f$ is a $k$ -linear bijection $L (K_{C}^{'} - D) \to L (K_{C} - D)$ with inverse multiplication by $f^{- 1}$ . Hence $l (K_{C}^{'} - D) = l (K_{C} - D)$ by [F3].

5.1F7step 3.1step 4.1∎

Since the right-hand side $\deg_{k} (D) + 1 - g$ of the identity of step 3.1 does not involve $K_{C}$ at all, step 4.1 shows that both sides are unchanged under replacing $K_{C}$ by a linearly equivalent canonical divisor; this completes the proof of the full Riemann-Roch theorem, and the Axiom of Choice [F7] is used exactly through the coherent-cohomology suppliers cited above.

Depends on

Used by

Riemann-Roch in exact form for divisors of degree above 2g - 2 Corollary
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
Hyperelliptic curves and hyperelliptic maps Definition
A degree-n line bundle on a genus-one curve has an n-dimensional space of sections for n > 0 Example
The full Riemann-Roch theorem on the projective line, in every degree Example
The canonical map: base-point-freeness and the hyperelliptic exception Theorem

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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)