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A negative right-hand side does not contradict Riemann-Roch

Counterexample

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

Let $k$ be a field, let $m \geq 1$ be an integer, let $P_{k}^{1}$ be the projective line over $k$ with point at infinity $\infty$ and coordinate $t$ , and put $D : = - m [\infty]$ . Then $\deg_{k} (D) = - m < 0, l (D) = 0, i (D) = m - 1,$ so Riemann-Roch on $P_{k}^{1}$ ( $g = 0$ ) reads $0 - (m - 1) = 1 - m = \deg_{k} (D) + 1 - g;$ for $m = 2$ this is the identity $0 - 1 = - 1 = - 2 + 1$ . The explicitly refuted readings are:

" $l (D) = \deg_{k} (D) + 1 - g$ for every divisor $D$ ": at $D = - 2 [\infty]$ one has $l (D) = 0$ while $\deg_{k} (D) + 1 - g = - 1$ , and a negative integer is not the dimension of any $k$ -vector space;
"there exist $\deg_{k} (D) + 1 - g$ independent sections": for every $m \geq 2$ the number $1 - m$ is negative, so it cannot count sections, and indeed $l (D) = 0$ ;
"the Riemann inequality produces sections": the inequality $l (D) \geq 1 - m$ is vacuous for $m \geq 2$ , since its right-hand side is negative.

There is no contradiction with the vanishing $l (D) = 0$ for $\deg_{k} (D) < 0$ (No sections in negative degree): that corollary asserts exactly the value $l (D) = 0$ computed here and is proved without the Riemann inequality, because its nonpositive lower bound cannot ensure a nonzero section. The Euler characteristic $l (D) - i (D) = 1 - m$ is an integer that may be negative, and the compensation is supplied by $H^{1}$ , whose dimension $i (D) = m - 1$ is not zero as soon as $m \geq 2$ ; the right-hand side of Riemann-Roch is therefore not itself the dimension of a space of sections. At the boundary $m = 1$ , where $\deg_{k} (D) + 1 - g = 0$ and $l (D) = i (D) = 0$ , the first two numerical readings are consistent. The third reading still fails: the bound $l (D) \geq 0$ does not ensure a nonzero section. Thus $m = 2$ is the first failure of the first two readings and the first negative right-hand side; the third reading fails already at $m = 1$ .

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, and it also cited the same-page example ex-riemann-roch-projective-line-divisor. Both are examples-page items: the first is homed on the examples page cohomology-of-quasi-coherent-sheaves-on-affine-and-projective-schemes-examples, and an examples-page item may not be consumed by another item, while the second is an example on this very page and likewise cannot carry the load. The two citations are replaced here by the published A-page corollaries Global sections of projective twists and Top cohomology of projective twists (the same values at $n = 1$ , with the monomial count $d + 1$ for $d \geq 0$ and the rank $(\binom{- d - 1}{1}) = m - 1$ for $d = - m \leq - 2$ ) together with the A-page suppliers Divisors on the projective line are classified by degree and The Picard group of the projective line; every promised claim is preserved.

The current divisor-to-line-bundle route uses Divisors on the projective line are classified by degree and The Picard group of the projective line, while The Riemann-Roch dimension l(D) and The index of speciality i(D) identify the two cohomology dimensions. The published projective-line cohomology corollaries give their explicit values.

Facts & Assumptions

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

[F1]

Projective-line data: $P_{k}^{1}$ is a smooth proper geometrically integral curve over $k$ (Curves over a field) of genus $0$ ; the point at infinity has residue degree $[κ (\infty) : k] = 1$ ; 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 $\deg_{k} O (1) = 1$ ; and every divisor $D^{'}$ on $P_{k}^{1}$ is linearly equivalent to $\deg_{k} (D^{'}) [\infty]$ , the degree homomorphism on divisor classes being an isomorphism (Divisors on the projective line are classified by degree).

[F2]

Picard group: the degree homomorphism induces an isomorphism $Pic (P_{k}^{1}) \to Z$ under which the class of $O_{P_{k}^{1}} (d)$ corresponds to $d$ , so every invertible sheaf on $P_{k}^{1}$ is isomorphic to $O (d)$ for a unique integer $d$ (The Picard group of the projective line).

[F3]

Divisors and degree: a divisor on a smooth proper curve is a finite formal $Z$ -linear combination of closed points, and $\deg_{k} (D) = \sum_{x} n_{x} [κ (x) : k]$ is a group homomorphism on the divisor group (Divisors on a smooth proper curve, Degree divisor proper curve).

[F4]

Dimensions: $l (D) = \dim_{k} L (D) = h^{0} (C, O_{C} (D))$ and $i (D) = h^{1} (C, O_{C} (D)) = \dim_{k} H^{1} (C, O_{C} (D))$ are nonnegative integers, and a dimension over a field is nonnegative by definition (The Riemann-Roch dimension l(D), The index of speciality i(D), Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

[F5]

Riemann-Roch: for every divisor $D$ on the curve one has $l (D) - i (D) = \deg_{k} (D) + 1 - g$ with $g = g (C)$ and $i (D) \geq 0$ , with equality $l (D) = \deg_{k} (D) + 1 - g$ exactly when $i (D) = 0$ (Riemann-Roch as l minus i).

[F6]

Negative degree: if $\deg_{k} (D) < 0$ then $L (D) = 0$ and $l (D) = 0$ , by the effective-divisor argument; the Riemann inequality is not used, since its right-hand side is nonpositive and cannot ensure a nonzero section (No sections in negative degree).

[F7]

Explicit cohomology of the twists at $n = 1$ : $H^{0} (P_{k}^{1}, O (d)) ≅ k [x_{0}, x_{1}]_{d}$ for $d \geq 0$ , with monomial basis $x_{0}^{a} x_{1}^{d - a}$ ( $0 \leq a \leq d$ ) of size $d + 1$ , and $H^{0} = 0$ for $d < 0$ ; $H^{1} (P_{k}^{1}, O (d)) = 0$ for $d > - 2$ and is free of rank $(\binom{- d - 1}{1}) = - d - 1$ for $d \leq - 2$ when $k \neq 0$ (Global sections of projective twists, Top cohomology of projective twists).

[F8]

Riemann inequality: $l (D) \geq \deg_{k} (D) + 1 - g$ for every divisor $D$ (The Riemann inequality).

[F9]

Riemann inequality vacuity and speciality: the inequality is an equality exactly when $i (D) = 0$ , so for $i (D) \geq 1$ the divisor is special and the inequality is strict (The index of speciality i(D)).

[F10]

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, while The Riemann-Roch dimension l(D) and The index of speciality i(D) identify the cohomology dimensions. These interfaces give $O_{P_{k}^{1}} (- m [\infty]) ≅ O_{P_{k}^{1}} (- m)$ and the readings $l (D) = h^{0} (O (D))$ , $i (D) = h^{1} (O (D))$ used at steps 1.1 and 2.1.

[F11]

The Axiom of Choice is available and is inherited only through the cohomology, divisor and Riemann-Roch suppliers recorded above; the computation below evaluates explicit formulas at the given divisor and selects nothing (The Axiom of Choice).

Proof

technique · compute both sides of Riemann-Roch at $D=-m[\infty]$ from the explicit cohomology of the twists on $\mathbb P^1_k$, then read off the failure of the equality and existence readings for $m\ge2$

1.1F1F2F3F10

The divisor and its degree; the attached sheaf. By [F1] the residue degree of infinity is $[κ (\infty) : k] = 1$ , so by [F3] the degree of $D = - m [\infty]$ is $\deg_{k} (D) = - m \cdot [κ (\infty) : k] = - m < 0$ . Still by [F1], $O (1) ≅ O (\infty)$ , and by the current divisor/Picard route [F10] dualizing and tensoring give $O (- m) ≅ O (1)^{\otimes (- m)} ≅ O (- m [\infty]) = O (D)$ ; equivalently, both classes equal $- m$ under the isomorphism $Pic (P_{k}^{1}) ≅ Z$ of [F2] with $O (1) \mapsto 1$ .

2.1F4F6F7step 1.1

Sections and index of speciality of $D$ . By [F7] with $d = - m$ : the group $H^{0} (P_{k}^{1}, O (- m))$ vanishes because $- m < 0$ , and $H^{1} (P_{k}^{1}, O (- m))$ vanishes for $m = 1$ (as $- 1 > - 2$ ) while for $m \geq 2$ it is free of rank $(\binom{m - 1}{1}) = m - 1$ ; thus $h^{0} (O (- m)) = 0$ and $h^{1} (O (- m)) = m - 1$ for every $m \geq 1$ . By the isomorphism $O (D) ≅ O (- m)$ of step 1.1 and the definitions [F4] of $l$ and $i$ , $l (D) = h^{0} (O (D)) = h^{0} (O (- m)) = 0, i (D) = h^{1} (O (D)) = h^{1} (O (- m)) = m - 1.$ Independently [F6] gives $l (D) = 0$ directly from $\deg_{k} (D) < 0$ of step 1.1, so the two computations of $l (D)$ agree.

3.1F1F5step 1.1step 2.1

Riemann-Roch at $D$ . By [F5] applied to the divisor $D$ of step 1.1, with $g = g (P_{k}^{1}) = 0$ from [F1] and the values of step 2.1, $l (D) - i (D) = 0 - (m - 1) = 1 - m = - m + 1 = \deg_{k} (D) + 1 - g,$ an identity for every $m \geq 1$ ; at $m = 2$ it reads $0 - 1 = - 1 = - 2 + 1$ . The right-hand side $1 - m$ is negative exactly when $m \geq 2$ , and equals $0$ at the boundary $m = 1$ .

3.2F4F6F8F9step 1.1step 2.1

The refuted readings and the role of $H^{1}$ . For every $m \geq 2$ step 2.1 gives $l (D) = 0$ while $\deg_{k} (D) + 1 - g = 1 - m < 0$ ; since a dimension over $k$ is nonnegative ([F4]), the equality $l (D) = \deg_{k} (D) + 1 - g$ fails, and the negative number $1 - m$ cannot be the number of independent sections in $L (D)$ . The Riemann inequality [F8] reads $0 \geq 1 - m$ , which is true for $m \geq 2$ but vacuous: its right-hand side is negative, so it guarantees no nonzero section, in agreement with $l (D) = 0$ . The discrepancy is exactly the index of speciality: by step 2.1, $i (D) = m - 1 \geq 1$ for $m \geq 2$ , so by [F9] $D$ is special, the Riemann inequality is strict, and the $h^{1}$ term $0 - (m - 1) = 1 - m$ supplies the negative compensation. For $m = 1$ one has $i (D) = 0$ , $l (D) = 0$ and $1 - m = 0$ , so the first two numerical readings hold, but $l (D) \geq 0$ still ensures no nonzero section. The third reading therefore fails already at $m = 1$ , while the first two fail first at $m = 2$ , explicitly by $D = - 2 [\infty]$ with $l (D) = 0$ , $i (D) = 1$ and $0 - 1 = - 1$ .

4.1F4F6F11step 1.1step 2.1step 3.1step 3.2∎

Conclusion and choice accounting. Step 1.1 computes $\deg_{k} (- m [\infty]) = - m$ and identifies the attached sheaf with $O (- m)$ ; step 2.1 computes $l (D) = 0$ and $i (D) = m - 1$ ; step 3.1 verifies the Riemann-Roch identity $0 - (m - 1) = - m + 1$ ; and step 3.2 exhibits the failure of the readings " $l (D) = \deg_{k} (D) + 1 - g$ " and "there exist $\deg_{k} (D) + 1 - g$ sections" for every $m \geq 2$ , first at $D = - 2 [\infty]$ where the right-hand side is $- 1$ , while the negative-degree vanishing $l (D) = 0$ of [F6] remains consistent because the $H^{1}$ term compensates. No contradiction with the Riemann inequality arises: the inequality gives no positive lower bound in this family: it is $0 \geq 0$ at $m = 1$ and $0 \geq 1 - m$ with negative right-hand side for $m \geq 2$ . The Axiom of Choice is inherited only through the suppliers of [F11]; the computation selects nothing beyond the given field and integer $m$ .

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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)