Why the sharp degree thresholds wait for the duality pair

Let $k$ be a field and let $C$ be a smooth proper geometrically integral curve over $k$ (Curves over a field) of genus $g=g(C)$ (Genus via the Euler characteristic). This page proves the Euler-characteristic form of Riemann-Roch, $l(D)-i(D)=h^0(C,{\mathcal{O}}_C(D))-h^1(C,{\mathcal{O}}_C(D))={\mathrm{deg}}_k(D)+1-g,$ with $i(D)\ge 0$, in the notation $l(D)=h^0(D)$ and $i(D)=h^1(D)$ of The Riemann-Roch dimension l(D) and The index of speciality i(D) (Riemann-Roch as l minus i, Riemann-Roch for curves: the Euler-characteristic form), and it proves the fixed-direction Serre vanishing theorem Vanishing of H^1 in a fixed ample direction: for a fixed finite $k$-morphism $\phi :C\to {\mathbb{P}}_k^1$ and a fixed effective divisor $A$ with ${\mathcal{O}}_C(A)\cong {\phi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}_k^1}(1)$, and for every divisor $D_0$ on $C$, there is an integer $n_0=n_0(D_0,\phi )$ such that $H^1(C,{\mathcal{O}}_C(D_0+nA+E))=0$ for every $n\ge n_0$ and every effective divisor $E$, equivalently $h^1(D)=0$ for every divisor $D\ge D_0+n_{0}A$. Through Riemann's theorem for sufficiently positive divisors and Sufficiently positive divisors in a fixed direction are nonspecial this yields the exact count $l(D)={\mathrm{deg}}_k(D)+1-g$ and nonspeciality, again only for $D\ge D_0+n_{0}A$: the bound is a bound along one fixed ample direction, it depends on $D_0$ and on $\phi$, and it is not a bound in ${\mathrm{deg}}_k(D)$.

The classical degree thresholds are not available on this page and must not be quoted from it. Each of the following rests on the identification $i(D)=l(K_C-D)$ supplied by Serre duality, in the pair on residues, Serre duality and the full Riemann-Roch theorem that follows this page:

1. the canonical identities ${\mathrm{deg}}_k{K}_C=2g-2$ and $h^0(C,K_C)=g$ for a canonical divisor $K_C$;
2. vanishing $i(D)=0$, equivalently $l(D)={\mathrm{deg}}_k(D)+1-g$, for every divisor of degree greater than $2g-2$ — not merely along one fixed ample direction;
3. base-point-freeness of every invertible sheaf of degree at least $2g$;
4. very ampleness of every invertible sheaf of degree at least $2g+1$.

In the vocabulary of the index of speciality (The index of speciality i(D)) the missing ingredient is exactly a description of the dual space of $H^1(C,{\mathcal{O}}_C(D))$: nothing on this page identifies that space with the space of sections of a complementary invertible sheaf, defines a canonical divisor, or proves a threshold in terms of ${\mathrm{deg}}_k(D)$ alone. The batch-8 items cor-canonical-degree-two-g-minus-two, cor-h0-canonical-differentials-genus, cor-h1-line-bundle-vanishes-degree-over-two-g-minus-two, thm-degree-two-g-line-bundle-basepoint-free and thm-degree-two-g-plus-one-line-bundle-very-ample are the destinations of that material in the following duality pair. This remark does not use those results; it records that the present page proves only the fixed-direction form stated above.

The Axiom of Choice is inherited here from the suppliers named above and is not otherwise used: the remark selects nothing and adds no choice principle of its own (The Axiom of Choice).