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Riemann-Roch on the projective line for every degree

Example

Assume the Axiom of Choice inherited from the current projective-line cohomology and divisor/Picard suppliers.

Let $k$ be a field, let $P_{k}^{1}$ have coordinate $t$ with point at infinity $\infty$ , and let $D = d [\infty]$ for an integer $d$ . Since $O_{P_{k}^{1}} (d)$ is isomorphic to $O_{P_{k}^{1}} (d [\infty])$ — the class of the point at infinity generates $Pic (P_{k}^{1}) = Z$ with $O (1)$ corresponding to $1$ (The Picard group of the projective line) — the explicit cohomology of the twists gives $l (D) = h^{0} (O (d)) = \max (d + 1, 0), i (D) = h^{1} (O (d)) = \max (- d - 1, 0),$ the values being read off from Global sections of projective twists and Top cohomology of projective twists. Hence $l (D) - i (D) = d + 1 = \deg_{k} (D) + 1 - g$ with $g = 0$ for every integer $d$ : on the projective line Riemann-Roch is an identity between explicit numbers, including the negative-degree range where $l (D) = 0$ and $i (D) = - d - 1$ compensates. The divisors of degree at least $- 1$ are exactly the nonspecial ones here, and the complete linear system $∣ D ∣$ of Complete linear system is nonempty exactly for $d \geq 0$ .

Scaffold repair, recorded for the owner. The frozen scaffold statement cited the examples-page item ex-cohomology-o-d-projective-line-all-d for the cohomology of the twists. That item is homed on the examples page cohomology-of-quasi-coherent-sheaves-on-affine-and-projective-schemes-examples, and an examples-page item may not depend on another examples-page item, so the citation is replaced here by the published A-page corollaries Global sections of projective twists and Top cohomology of projective twists, which contain the same values for $n = 1$ ; every promised claim is preserved. The example's negative-degree compensation clause is also corrected to $l (D) = 0$ and $i (D) = - d - 1$ with $l - i = d + 1$ , so that the displayed identity is true at every $d$ .

The current Divisors on the projective line are classified by degree and The Picard group of the projective line give the divisor-to-twist identification used in the calculation. The dimension notation is supplied by The Riemann-Roch dimension l(D), The index of speciality i(D) and Special and nonspecial divisors, and the complete linear system by Complete linear system.

Facts & Assumptions

Given: the Axiom of Choice inherited from the current projective-line cohomology and divisor/Picard suppliers; a field $k$ , the projective line $P_{k}^{1}$ with coordinate $t$ and point at infinity $\infty$ , and the divisor $D = d [\infty]$ for an integer $d$ .

[F1]

Projective-line data: $P_{k}^{1}$ is a smooth proper geometrically integral curve over $k$ of genus $0$ ; the coordinate section $x_{0}$ of $O (1)$ vanishes exactly at infinity with multiplicity one, so $div (x_{0}) = [\infty]$ and $O (1) ≅ O (\infty)$ with $\deg_{k} O (1) = 1$ ; every divisor $D^{'}$ is linearly equivalent to $\deg_{k} (D^{'}) [\infty]$ , and the degree homomorphism on divisor classes is an isomorphism, so linearly equivalent divisors have equal degree and $[κ (\infty) : k] = 1$ (Divisors on the projective line are classified by degree, The Picard group of the projective line).

[F2]

Global sections: for $n = 1$ and $A = k$ one has $H^{0} (P_{k}^{1}, O (d)) ≅ k [x_{0}, x_{1}]_{d}$ for $d \geq 0$ and $H^{0} (P_{k}^{1}, O (d)) = 0$ for $d < 0$ ; the monomials $x_{0}^{a} x_{1}^{d - a}$ for $0 \leq a \leq d$ form a $k$ -basis of $k [x_{0}, x_{1}]_{d}$ , so $h^{0} (O (d)) = d + 1$ for $d \geq 0$ and $h^{0} (O (d)) = 0$ for $d < 0$ (Global sections of projective twists).

[F3]

Top cohomology: for $n = 1$ the group $H^{1} (P_{k}^{1}, O (d))$ vanishes for $d > - 2$ and for $d \leq - 2$ it is free of rank $(\binom{- d - 1}{1}) = - d - 1$ over $k \neq 0$ ; so $h^{1} (O (d)) = 0$ for $d \geq - 1$ and $h^{1} (O (d)) = - d - 1$ for $d \leq - 2$ (Top cohomology of projective twists).

[F4]

Degree: for a divisor $D = \sum_{x} n_{x} [x]$ the $k$ -degree is $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ , sum over the finite support, and it is a group homomorphism on the divisor group (Degree divisor proper curve, Divisors on a smooth proper curve).

[F5]

Riemann-Roch dimensions: $l (D) = \dim_{k} L (D) = h^{0} (D) = \dim_{k} H^{0} (C, O_{C} (D))$ and $i (D) = h^{1} (D) = \dim_{k} H^{1} (C, O_{C} (D))$ , so $l (D) = h^{0} (O (D))$ and $i (D) = h^{1} (O (D))$ for the attached sheaf (The Riemann-Roch dimension l(D), The index of speciality i(D)).

[F6]

Riemann-Roch: $l (D) - i (D) = \deg_{k} (D) + 1 - g$ with $i (D) \geq 0$ , and $l (D) = \deg_{k} (D) + 1 - g$ exactly for the nonspecial divisors (Riemann-Roch as l minus i, Special and nonspecial divisors).

[F7]

Nonspeciality: $D$ is nonspecial exactly when $i (D) = 0$ , and special exactly when $i (D) \geq 1$ ; equivalently nonspeciality is the equality case $l (D) = \deg_{k} (D) + 1 - g$ (Special and nonspecial divisors).

[F8]

The complete linear system $∣ D ∣ = {D^{'} effective : D^{'} is linearly equivalent to D}$ is in bijection with the set of $k$ -lines in $L (D)$ and is empty exactly when $L (D) = 0$ (Complete linear system).

[F9]

The current Divisors on the projective line are classified by degree and The Picard group of the projective line identify the divisor class and attached twist; The Riemann-Roch dimension l(D), The index of speciality i(D), Special and nonspecial divisors and Complete linear system supply the dimension and linear-system interpretations.

[F10]

The Axiom of Choice is available and is inherited only through the suppliers named above; the computations below evaluate explicit formulas and select nothing (The Axiom of Choice).

Verification

technique · direct computation of $l(D)$ and $i(D)$ from the explicit cohomology of the twists, followed by a case check on the sign of $d$ for the Riemann-Roch identity, the nonspecialty threshold and the nonemptiness of the complete linear system

1.1F1F4F5F9

Degree and attached sheaf. By [F1] the residue degree of the point at infinity is $[κ (\infty) : k] = 1$ and the degree homomorphism is defined on linear-equivalence classes, so for $D = d [\infty]$ the degree is $\deg_{k} (D) = d \cdot [κ (\infty) : k] = d$ by [F4]. By [F1] the class of $\infty$ generates $Pic (P_{k}^{1}) ≅ Z$ with $O (1) \mapsto 1$ , so by the current divisor/Picard route [F9] one has $O (D) = O (d [\infty]) ≅ O (1)^{\otimes d} ≅ O (d)$ ; hence $l (D) = l (0 + d [\infty]) = h^{0} (O (d))$ and $i (D) = h^{1} (O (d))$ by [F5], and $g = g (P_{k}^{1}) = 0$ by [F1].

2.1F2F3step 1.1

The explicit values. By [F2] applied with $n = 1$ and $A = k$ , the space $H^{0} (P_{k}^{1}, O (d))$ has dimension $d + 1$ for $d \geq 0$ and vanishes for $d < 0$ , because $k [x_{0}, x_{1}]_{d}$ has dimension $d + 1$ as the space of homogeneous polynomials of degree $d$ in two variables; hence $h^{0} (O (d)) = \max (d + 1, 0)$ . By [F3] applied with $n = 1$ , $H^{1} (P_{k}^{1}, O (d)) = 0$ for $d > - 2$ , that is for $d \geq - 1$ , while for $d \leq - 2$ its rank is $(\binom{- d - 1}{1}) = - d - 1$ ; hence $h^{1} (O (d)) = \max (- d - 1, 0)$ . Combining with step 1.1, $l (D) = \max (d + 1, 0), i (D) = \max (- d - 1, 0)$ for every integer $d$ .

3.1F4F6step 1.1step 2.1

Riemann-Roch as an identity between explicit numbers. If $d \geq - 1$ then $- d - 1 \leq 0$ , so $i (D) = \max (- d - 1, 0) = 0$ , while $d + 1 \geq 0$ gives $l (D) = \max (d + 1, 0) = d + 1$ ; then $l (D) - i (D) = d + 1$ . If instead $d \leq - 2$ then $d + 1 < 0$ , so $l (D) = \max (d + 1, 0) = 0$ , while $- d - 1 \geq 1$ gives $i (D) = - d - 1$ ; then $l (D) - i (D) = 0 - (- d - 1) = d + 1$ . In both cases, by step 1.1, $l (D) - i (D) = d + 1 = \deg_{k} (D) + 1 - g,$ which is [F6] with $g = 0$ and $\deg_{k} (D) = d$ ; the two cases $d \geq - 1$ and $d \leq - 2$ exhaust $Z$ and meet no other, and for $d \leq - 2$ the section space $l (D) = 0$ is compensated by $i (D) = - d - 1 > 0$ , while at $d = - 1$ both dimensions and the right-hand side are zero, so the right-hand side stays correct even where it is negative.

3.2F1F4F6F7F8step 2.1

Nonspecialty and the complete linear system. By [F7] the divisor $D$ is nonspecial exactly when $i (D) = \max (- d - 1, 0) = 0$ , that is exactly when $- d - 1 \leq 0$ , i.e. $d \geq - 1$ : the divisors $d [\infty]$ of degree at least $- 1$ are exactly the nonspecial ones, and by [F6] the same threshold is the equality case $l (D) = d + 1$ of the Riemann inequality, while for $d \leq - 2$ one has $l (D) = 0 > d + 1$ and $D$ is special. For the complete linear system, if $d \geq 0$ then $D = d [\infty]$ is an effective divisor and $D$ is linearly equivalent to itself, so $D \in ∣ D ∣$ and $∣ D ∣$ is nonempty; if $d < 0$ and $D^{'} = \sum_{x} n_{x} [x]$ is effective with $D^{'}$ linearly equivalent to $D$ , then $\deg_{k} (D^{'}) = \deg_{k} (D) = d < 0$ by the well-definedness of the degree homomorphism on classes [F1], while effectivity gives $n_{x} \geq 0$ and hence $\deg_{k} (D^{'}) = \sum_{x} n_{x} [κ (x) : k] \geq 0$ by [F4] and $[κ (x) : k] \geq 1$ , a contradiction; so $∣ D ∣$ is empty for $d < 0$ . This agrees with [F8], under which $∣ D ∣$ is in bijection with the $k$ -lines in $L (D)$ : by step 2.1, $l (D) = \max (d + 1, 0)$ is at least $1$ exactly for $d \geq 0$ .

4.1F2F3F9F10step 1.1step 2.1step 3.1step 3.2∎

Assembly and choice accounting. Step 2.1 computes $l (D) = \max (d + 1, 0)$ and $i (D) = \max (- d - 1, 0)$ ; step 3.1 verifies $l (D) - i (D) = d + 1 = \deg_{k} (D) + 1 - g$ for every integer $d$ by an exhaustive case check on $d \geq - 1$ versus $d \leq - 2$ ; step 3.2 identifies the nonspecial divisors with the degrees $d \geq - 1$ and shows $∣ D ∣$ nonempty exactly for $d \geq 0$ , in agreement with the section dimension. The sheaf identification $O (D) ≅ O (d)$ and the identity $l (D) = h^{0} (O (D))$ of step 1.1 use the current interfaces [F9]; the numerical values of steps 2.1 and 3.1 depend on them only through that identification, and the cohomology values themselves are the published A-page corollaries [F2] and [F3]. The Axiom of Choice enters only through the suppliers recorded in [F10]; every value above is computed from explicit formulas, and no family of objects is selected.

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