A sufficiently positive divisor is nonspecial and Riemann-Roch counts its sections

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

Let $k$ be a field, let $C$ be a smooth proper geometrically integral curve over $k$ of genus $g=g(C)$ (Curves over a field, Genus via the Euler characteristic), let $\phi :C\to {\mathbb{P}}_k^1$ be a finite $k$-morphism, and let $A$ be an effective divisor on $C$ with ${\mathcal{O}}_C(A)\cong {\phi }^{\ast }{\mathcal{O}}_{{\mathbb{P}}_k^1}(1)$. Let $D_0$ be any divisor on $C$ and let $n_0=n_0(D_0,\phi )$ be an integer supplied for $D_0$ and $\phi$ by the fixed-direction vanishing theorem. Then for every $n\ge n_0$ and every effective divisor $E$ the divisor $D=D_0+nA+E$ is nonspecial, $H^1(C,{\mathcal{O}}_C(D))=0$ (Sufficiently positive divisors in a fixed direction are nonspecial), so Riemann-Roch computes its dimension exactly: $l(D)={\mathrm{deg}}_k(D)+1-g$ (Riemann-Roch as l minus i).

The concrete instance is the projective line. Take $C={\mathbb{P}}_k^1$, $\phi =\mathrm{i}\mathrm{d}$ and $A=[\mathrm{\infty }]$, so that ${\phi }^{\ast }\mathcal{O}(1)=\mathcal{O}(1)\cong \mathcal{O}([\mathrm{\infty }])$; then $g=0$ and, for every $d\in \mathbb{Z}$, $l(d[\mathrm{\infty }])=h^0({\mathbb{P}}_k^1,\mathcal{O}(d))=\{\begin{array}{ll}d+1,& d\ge 0,\\ 0,& d<0,\end{array}i(d[\mathrm{\infty }])=h^1({\mathbb{P}}_k^1,\mathcal{O}(d))=\{\begin{array}{ll}0,& d\ge -1,\\ -d-1,& d\le -2,\end{array}$ while ${\mathrm{deg}}_k(d[\mathrm{\infty }])=d\cdot [\kappa (\mathrm{\infty }):k]=d$. Consequently:

1. for $d\ge -1$ the divisor $d[\mathrm{\infty }]$ is nonspecial, $i(d[\mathrm{\infty }])=0$, and Riemann-Roch reads $l(d[\mathrm{\infty }])=d+1={\mathrm{deg}}_k(d[\mathrm{\infty }])+1-g$ — including the boundary case $d=-1$, where $l=i=0$ (the sheaf $\mathcal{O}(-1)$ has no sections);
2. for $d\le -2$ the divisor is special, with $l(d[\mathrm{\infty }])=0$ and $i(d[\mathrm{\infty }])=-d-1\ge 1$, and Riemann-Roch reads $0-(-d-1)=d+1$;
3. so in this family nonspeciality holds exactly for $d\ge -1$, and the exact threshold is the boundary value $d=-1$; the general statement only provides some threshold $n_0(D_0,\phi )$ depending on $D_0$ and on the fixed morphism, which this instance computes exactly.

The "sufficiently positive" threshold of the general statement is the fixed-direction one of the vanishing theorem; the example does not assert the universal bound "${\mathrm{deg}}_k(D)>2g-2$ implies $i(D)=0$", which requires Serre duality and belongs to the next pair.

Scaffold repair, recorded for the owner. The frozen scaffold cited the examples-page item ex-cohomology-o-d-projective-line-all-d (on another page) and ex-riemann-roch-projective-line-divisor (on this page) for the projective-line values $l(d[\mathrm{\infty }])=d+1$ and the nonspecial range. Both are examples and cannot carry a load; the citations are replaced by the published A-page suppliers Global sections of projective twists, Top cohomology of projective twists and Divisors on the projective line are classified by degree, together with The Picard group of the projective line for the identification $\mathcal{O}(d[\mathrm{\infty }])\cong \mathcal{O}(d)$ of the attached sheaves. Every promised numerical claim is preserved: $l(d[\mathrm{\infty }])=d+1$ for $d\ge 0$, nonspeciality exactly for $d\ge -1$, the boundary case $d=-1$ with $l=i=0$, and the fixed-direction restriction.

The current Sufficiently positive divisors in a fixed direction are nonspecial supplies the fixed-direction vanishing, and Riemann-Roch as l minus i supplies the dimension identity. The current projective-line divisor and Picard interfaces Divisors on the projective line are classified by degree and The Picard group of the projective line identify the attached twists; the published cohomology corollaries supply their dimensions.