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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The empty divisor, its Euler characteristic and the genus boundary cases

Example

Assume the Axiom of Choice inherited from the current cohomology, dimension and Riemann-Roch suppliers.

Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ of genus $g = g (C)$ (Genus via the Euler characteristic), and let $D = 0$ be the empty divisor. Then $l (0) = h^{0} (C, O_{C}) = 1, i (0) = h^{1} (C, O_{C}) = g,$ so the Euler characteristic of the structure sheaf is $χ (C, O_{C}) = h^{0} (C, O_{C}) - h^{1} (C, O_{C}) = 1 - g,$ and Riemann-Roch on the empty divisor reads $1 - g = 0 + 1 - g,$ the normalization that makes $χ (C, O_{C}) = 1 - g$ the definition of the genus. The complete linear system $∣ 0 ∣$ consists of the single divisor $0$ in every genus, so $\dim_{k} ∣ 0 ∣ = 0 = l (0) - 1$ . The two boundary genera: for $g = 0$ one has $χ = 1$ with $i (0) = 0$ , so the empty divisor is nonspecial — for $P_{k}^{1}$ this reads $h^{0} (O) = 1$ , $h^{1} (O) = 0$ — while for $g = 1$ one has $χ = 0$ with $l (0) = 1 = i (0)$ , the first and simplest case of a divisor of degree zero that is special, where the Riemann inequality for $D = 0$ is the strict bound $1 \geq 0$ with gap $i (0) = 1$ . In genus zero the equality case $l (D) = \deg_{k} (D) + 1 - g$ holds at $D = 0$ , and it fails at $D = 0$ as soon as $g \geq 1$ .

Scaffold repair, recorded for the owner. The frozen scaffold cited the examples-page item ex-cohomology-o-d-projective-line-all-d for the projective-line values $h^{0} (O) = 1$ , $h^{1} (O) = 0$ . Examples-page items are leaves and cannot carry a load; the citation is replaced by the published A-page corollaries Global sections of projective twists and Top cohomology of projective twists together with Divisors on the projective line are classified by degree, which give the same values at $d = 0$ and $g (P_{k}^{1}) = 0$ . Every promised claim is preserved.

The current Riemann-Roch as l minus i supplies the Riemann-Roch identity, The dimension of a complete linear system supplies the dimension formula for $∣ 0 ∣$ , and Complete linear system identifies the linear system. The current structure-sheaf and cohomology suppliers give the empty divisor and genus boundary values.

Facts & Assumptions

Given: the Axiom of Choice inherited from the current cohomology, dimension and Riemann-Roch suppliers; a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ of genus $g$ , and the empty divisor $D = 0$ on $C$ .

[F1]

The structure sheaf: the canonical map $k \to H^{0} (C, O_{C})$ is an isomorphism, so $h^{0} (C, O_{C}) = l (0) = 1$ ; in particular the zero divisor has a one-dimensional space of sections, spanned by the constant function $1$ (Functions on a proper curve, The Riemann-Roch dimension l(D)).

[F2]

The index of speciality and the genus: $i (0) = h^{1} (C, O_{C}) = g (C)$ , the genus, which is the dimension of $H^{1}$ of the structure sheaf (The index of speciality i(D), Genus via the Euler characteristic, The Riemann-Roch dimension l(D)).

[F3]

Euler characteristic: $χ (C, O_{C}) = h^{0} (C, O_{C}) - h^{1} (C, O_{C}) = 1 - g$ , the Euler characteristic of the coherent sheaf $O_{C}$ (Euler characteristic of a coherent sheaf, Genus via the Euler characteristic).

[F4]

Riemann-Roch: for every divisor $D$ one has $l (D) - i (D) = \deg_{k} (D) + 1 - g$ , with $i (D) \geq 0$ and equality $l (D) = \deg_{k} (D) + 1 - g$ exactly when $i (D) = 0$ ; a divisor is nonspecial exactly when $i (D) = 0$ , and special exactly when $i (D) \geq 1$ (Riemann-Roch as l minus i, Special and nonspecial divisors).

[F5]

The complete linear system of the empty divisor: $∣ 0 ∣ = {0}$ is a single point, and $\dim_{k} ∣ 0 ∣ = 0 = l (0) - 1$ for every genus; generally $∣ D ∣$ is nonempty exactly when $l (D) \geq 1$ (The dimension of a complete linear system, Complete linear system, Divisors on a smooth proper curve).

[F6]

The projective line: $P_{k}^{1}$ is a smooth proper geometrically integral curve of genus $0$ ; for $d = 0$ its twists satisfy $H^{0} (P_{k}^{1}, O) ≅ k [x_{0}, x_{1}]_{0} ≅ k$ and $H^{1} (P_{k}^{1}, O) = 0$ (Divisors on the projective line are classified by degree, Global sections of projective twists, Top cohomology of projective twists).

[F7]

The Axiom of Choice is available and is inherited only through the cohomology, degree and Riemann-Roch suppliers recorded above; the computation below evaluates the fixed divisor $D = 0$ and selects nothing (The Axiom of Choice).

Proof

technique · evaluate $h^0$ and $h^1$ at the empty divisor, read off the Euler characteristic and Riemann-Roch identity, identify $|0|$, and check the genus-zero and genus-one boundary cases

1.1F1F2

The dimensions at the empty divisor. By [F1] the space $L (0)$ is one-dimensional, spanned by the constants, so $l (0) = h^{0} (C, O_{C}) = 1$ ; by [F2] the index of speciality is $i (0) = h^{1} (C, O_{C}) = g$ .

2.1F3F4step 1.1

Euler characteristic and Riemann-Roch at the empty divisor. By [F3] the Euler characteristic is $χ (C, O_{C}) = h^{0} - h^{1} = 1 - g$ , and the Riemann-Roch identity [F4] at $D = 0$ , with $\deg_{k} (0) = 0$ because the empty divisor has empty support, reads $l (0) - i (0) = 1 - g = 0 + 1 - g$ , that is, $1 - g = 0 + 1 - g$ : the genus is exactly the normalization constant that makes this identity hold.

2.2F5step 1.1

The complete linear system of the empty divisor. By [F5] the complete linear system $∣ 0 ∣$ consists of the single divisor $0$ , so $\dim_{k} ∣ 0 ∣ = 0 = l (0) - 1$ by step 1.1, in every genus.

3.1F4F6step 1.1step 2.1

The genus boundary cases. If $g = 0$ , then step 1.1 gives $i (0) = 0$ , so by [F4] the empty divisor is nonspecial, $χ = 1$ by step 2.1, and the identity is an equality; the projective line realizes this with $h^{0} (O) = 1$ , $h^{1} (O) = 0$ by [F6]. If $g = 1$ , then step 1.1 gives $l (0) = 1 = i (0)$ , so the empty divisor is special by [F4], $χ = 0$ by step 2.1, and the Riemann inequality for $D = 0$ reads $l (0) = 1 \geq 0 = \deg_{k} (0) + 1 - g$ ; it is strict with gap exactly $i (0) = 1$ . In genus zero the equality case $l (D) = \deg_{k} (D) + 1 - g$ holds at $D = 0$ , and for every $g \geq 1$ it fails at $D = 0$ by step 1.1 and [F4].

4.1F2F3F5F7step 1.1step 2.1step 2.2step 3.1∎

Conclusion and choice accounting. The empty divisor has $l (0) = 1$ , $i (0) = g$ and Euler characteristic $1 - g$ , Riemann-Roch at $D = 0$ is the identity $1 - g = 0 + 1 - g$ , and $∣ 0 ∣ = {0}$ with $\dim_{k} ∣ 0 ∣ = 0$ ; the empty divisor is nonspecial exactly in genus zero and is the simplest special divisor in genus one. The Axiom of Choice is inherited only through the suppliers of [F7]; nothing is selected.

Depends on

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