The Riemann--Roch Space Bounds Allowed Poles

L(D) consists of rational functions whose divisor becomes effective after adding D. This is a compact note, but the quantifiers and hypotheses stay on the page.

Statement

On a smooth projective curve $C$, each closed point $p$ defines a valuation $\operatorname{ord}_p$. Divisors combine these local orders into a global bookkeeping device.

$$ L(D)=\{f\in k(C)^\times:\operatorname{div}(f)+D\ge0\}\cup\{0\} $$

The first display fixes the mathematical data. I label it $\mathsf{data}$ mentally, while the next is the $\mathsf{claim}$; the bridge between them is the displayed $\Longrightarrow$, not an automatic implication.

$$ \ell(D)=\dim_kL(D) $$

Worked algebra

A worked instance is useful here because it exposes every index that the compressed statement hides.

$$ C=\mathbf P^1,\quad D=d[\infty],\ d\ge0\Longrightarrow L(D)=\langle1,t,\ldots,t^d\rangle,\quad\ell(D)=d+1 $$

Conceptual compression

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad L(D)=\{f\in k(C)^\times:\operatorname{div}(f)+D\ge0\}\cup\{0\},\\[5pt] \mathsf{C}\;&:\quad \ell(D)=\dim_kL(D). \end{aligned} $$

Caveat

A divisor being degree zero does not make it principal. The difference is measured by the Picard group and, in degree zero, by the Jacobian.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \ell(D)=\dim_kL(D) \end{gathered}} $$

A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.

This article was posted on Mon 21 March 2022. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

The Riemann--Roch Space Bounds Allowed Poles by Vicfred

Statement

Worked algebra

Conceptual compression

Caveat