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Riemann-Roch as l minus i

Statement

Assume the Axiom of Choice, inherited from the Riemann-Roch, genus and finiteness suppliers below. Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) with genus $g = g (C) = h^{1} (C, O_{C})$ (Genus via the Euler characteristic), and let $D$ be a divisor on $C$ (Divisors on a smooth proper curve). Then $l (D) - i (D) = h^{0} (C, O_{C} (D)) - h^{1} (C, O_{C} (D)) = \deg_{k} (D) + 1 - g,$ where $l (D) = h^{0} (D)$ is the dimension of the Riemann-Roch space, $i (D) = h^{1} (D)$ the index of speciality (The Riemann-Roch dimension l(D), The index of speciality i(D)). Moreover $i (D) \geq 0$ , so $l (D) \geq \deg_{k} (D) + 1 - g,$ with equality if and only if $i (D) = 0$ ; a divisor is called nonspecial exactly when $i (D) = 0$ , a terminology fixed in def-nonspecial-divisor, which follows on this page and consumes the present theorem. For the zero divisor the identity reads $1 - g = 0 + 1 - g$ with $i (0) = g$ . The index of speciality remains an unknown defect: no duality identifies it with the space of sections of a complementary divisor, and no threshold statement about $2 g - 2$ is made.

The attachment of $O_{C} (D)$ and the identification of $L (D)$ with $H^{0} (C, O_{C} (D))$ use the current interfaces Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors and Cartier and Weil divisors agree on a smooth curve, inherited through Riemann-Roch for curves: the Euler-characteristic form.

Facts & Assumptions

Given: a field $k$ , a smooth proper geometrically integral curve $C$ over $k$ with genus $g = g (C) = h^{1} (C, O_{C})$ , and a divisor $D$ on $C$ .

[F1]

The curve $C$ is proper, of finite type and of chain dimension one over the field $k$ , and a divisor on $C$ is a finite formal integral combination of closed points with $k$ -degree $\deg_{k} (D)$ (Curves over a field, Divisors on a smooth proper curve).

[F2]

Notation: $l (D) = h^{0} (D) = \dim_{k} H^{0} (C, O_{C} (D))$ and $h^{i} (D) = \dim_{k} H^{i} (C, O_{C} (D))$ for every $i \geq 0$ ; both are nonnegative integers, and for the zero divisor $l (0) = \dim_{k} H^{0} (C, O_{C}) = 1$ (The Riemann-Roch dimension l(D)).

[F3]

The index of speciality: $i (D) = h^{1} (C, O_{C} (D)) = \dim_{k} H^{1} (C, O_{C} (D))$ is a nonnegative integer, and $i (0) = h^{1} (C, O_{C}) = g (C)$ is the genus; it depends only on the linear equivalence class of $D$ (The index of speciality i(D), Genus via the Euler characteristic).

[F4]

Riemann-Roch in Euler-characteristic form: $h^{0} (D) - h^{1} (D) = χ (C, O_{C} (D)) = \deg_{k} (D) + 1 - g$ ; no Serre duality is used, and $h^{1} (D)$ is left as an unknown nonnegative integer (Riemann-Roch for curves: the Euler-characteristic form).

[F5]

The current interfaces Invertible sheaf of cartier divisor, Rational sections of line bundles are Cartier divisors and Cartier and Weil divisors agree on a smooth curve supply the attachment of $O_{C} (D)$ and the identification of $L (D)$ with its global sections; this use is inherited from [F4].

[F6]

The Axiom of Choice is used exactly through the Riemann-Roch supplier [F4], the genus definition [F3] and the finiteness supplier [F2]; no further selection is made below (The Axiom of Choice).

Proof

technique · direct; substitute the definitions $l(D)=h^0(D)$ and $i(D)=h^1(D)$ into the Euler-characteristic Riemann-Roch identity, and read off the inequality and its equality case from $i(D)\ge0$

1.1F1F2F3F4

Set-up. By [F1] the curve $C$ is proper over $k$ and $D$ is a divisor on $C$ with $k$ -degree $\deg_{k} (D)$ . By [F2] $l (D) = h^{0} (D)$ and by [F3] $i (D) = h^{1} (C, O_{C} (D))$ , a nonnegative integer; by [F4] applied to the divisor $D$ and the genus $g = g (C)$ of [F3], $h^{0} (D) - h^{1} (D) = \deg_{k} (D) + 1 - g$ .

2.1F2F3step 1.1

The defect identity. Substituting the definitions of $l$ and $i$ from step 1.1 into the Riemann-Roch identity gives $l (D) - i (D) = h^{0} (D) - h^{1} (D) = \deg_{k} (D) + 1 - g$ , the first displayed identity; all three expressions are integers, the left side because it is a difference of dimensions.

3.1F2F3step 2.1

The inequality and the equality case. Solving the identity of step 2.1 for $l (D)$ gives $l (D) = \deg_{k} (D) + 1 - g + i (D)$ ; since $i (D) \geq 0$ by [F3], this gives $l (D) \geq \deg_{k} (D) + 1 - g$ . If $l (D) = \deg_{k} (D) + 1 - g$ , then subtracting gives $i (D) = 0$ ; conversely if $i (D) = 0$ , the same identity gives $l (D) = \deg_{k} (D) + 1 - g$ . Hence equality holds if and only if $i (D) = 0$ , and $i (D) \geq 0$ is the nonnegativity of the dimension $h^{1} (D)$ of [F2].

3.2F2F3step 2.1

The zero divisor. By [F2] $l (0) = 1$ , and by [F3] $i (0) = g (C) = g$ ; the identity of step 2.1 at $D = 0$ therefore reads $1 - g = 0 + 1 - g$ , so the zero divisor is nonspecial exactly when $g = 0$ , and it is special with defect $g$ when $g > 0$ .

4.1F2F3F4F5F6step 2.1step 3.1step 3.2∎

Conclusion and choice accounting. Steps 2.1, 3.1 and 3.2 give the defect identity, the inequality with its equality case, and the zero-divisor reading, all for an arbitrary divisor $D$ on $C$ ; the index of speciality appears only as the dimension $i (D)$ of [F3], with no duality identification and no threshold statement. The Axiom of Choice is used only through the suppliers recorded in [F6], namely the Riemann-Roch theorem [F4], the genus definition [F3] and the finiteness supplier [F2]; the flagged dictionary [F5] records the inherited obligation on $O_{C} (D)$ .

Depends on

Used by

h¹ of a line bundle equals the dimension of the space of dual sections Corollary
H¹ of a line bundle vanishes above degree 2g - 2 Corollary
Riemann-Roch in exact form for divisors of degree above 2g - 2 Corollary
The dimension of a complete linear system Corollary
A negative right-hand side does not contradict Riemann-Roch Counterexample
The Riemann inequality is not an equality for special divisors Counterexample
Special and nonspecial divisors Definition
A principal divisor of degree zero on the projective line Example
A smooth conic with a rational point is a projective line Example
A sufficiently positive divisor is nonspecial and Riemann-Roch counts its sections Example
Riemann-Roch on the projective line for every degree Example
The empty divisor, its Euler characteristic and the genus boundary cases Example
Why the sharp degree thresholds wait for the duality pair Remark
The full Riemann-Roch theorem for divisors on a smooth proper curve Theorem

Dependency tree · two levels

63 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), Chs. 8 and 6 (standard reference, not scraped)
Michael Artin, MIT 18.721 Introduction to Algebraic Geometry (July 20, 2020 notes), Ch. 8 (standard reference, not scraped)
The Stacks Project, Algebraic Curves (tag 0BRV) (standard reference, not scraped)