Special and nonspecial divisors

Assume the Axiom of Choice inherited from proper-cohomology finiteness and the Riemann-Roch theorem (The Axiom of Choice). Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) with genus $g=g(C)$ (Genus via the Euler characteristic), and let $D$ be a divisor on $C$ (Divisors on a smooth proper curve) with index of speciality $i(D)=h^1(C,{\mathcal{O}}_C(D))$ and $l(D)=h^0(D)$ (The index of speciality i(D), The Riemann-Roch dimension l(D)).

Call $D$ nonspecial when $i(D)=0,\text{that is, when}{H}^1(C,{\mathcal{O}}_C(D))=0,$ and call it special otherwise, so that $D$ is special exactly when $i(D)\ge 1$. These are the classical names for the two cases of the index of speciality: vanishing first cohomology, and nonvanishing first cohomology.

The current Riemann-Roch as l minus i gives $l(D)-i(D)={\mathrm{deg}}_k(D)+1-g,$ and $i(D)\ge 0$, so the Riemann inequality $l(D)\ge {\mathrm{deg}}_k(D)+1-g$ always holds. Consequently, $D\text{ nonspecial}⟺l(D)={\mathrm{deg}}_k(D)+1-g⟺\text{the Riemann inequality is an equality for }D,$ and $D\text{ special}⟺l(D)>{\mathrm{deg}}_k(D)+1-g,$ because the difference $l(D)-({\mathrm{deg}}_k(D)+1-g)=i(D)$ is exactly the index of speciality. Thus a divisor is special precisely when its space of sections is larger than the Riemann inequality requires, and the equality case of that inequality is the vanishing of $H^1(C,{\mathcal{O}}_C(D))$.

Speciality is a property of the divisor class and not of the individual divisor: the index of speciality depends only on the linear equivalence class of $D$ (The index of speciality i(D)), so linearly equivalent divisors are special or nonspecial together. For the zero divisor the definition gives $i(0)=h^1(C,{\mathcal{O}}_C)=g,$ the genus of $C$ (The index of speciality i(D), Genus via the Euler characteristic), so $0$ is nonspecial exactly when $g=0$ and special exactly when $g\ge 1$; equivalently, the equality reading $l(0)=1={\mathrm{deg}}_k(0)+1-g$ holds exactly when $g=0$, in agreement with $l(0)=1$ for the zero divisor (The Riemann-Roch dimension l(D)).

The finite-valued cohomology dimensions and Riemann-Roch identity used here are supplied by the current Finite-dimensionality of the Riemann-Roch space, The index of speciality i(D), Genus via the Euler characteristic and Riemann-Roch as l minus i. Their source files are present in the working tree; their mathematical status remains the status recorded in their frontmatter.