The dimension of sections of D differs from that of K minus D by degree plus one minus genus. This is a compact note, but the quantifiers and hypotheses stay on the page.
The mathematical object
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.
A reliable calculation names domain and codomain. The notation \(\mathsf{data}\mapsto\mathsf{claim}\) is harmless only after both \(\operatorname{dom}\) and \(\operatorname{cod}\) have been fixed.
One explicit computation
A worked instance is useful here because it exposes every index that the compressed statement hides.
Why the identity matters
The formula is reusable precisely because it says which pieces are structural and which belong only to the worked example.
Where it can fail
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.
A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.