A smooth conic with a rational point is a projective line

Assume the Axiom of Choice inherited from the current plane-genus, rational-point and cohomology suppliers.

Let $k$ be a field of characteristic not two, let $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ be a smooth plane conic that is a curve (integral of dimension one — the hypothesis under which Arithmetic genus of a plane curve applies), and let $p\in C$ be a $k$-rational point. Then:

1. ${\mathrm{deg}}_k[p]=[\kappa (p):k]=1$, so $D=[p]$ is a divisor of degree one (Degree divisor proper curve);
2. A genus-zero curve with a degree-one divisor is the projective line applies once $g(C)=0$: Arithmetic genus of a plane curve gives arithmetic genus $p_a(C)=\frac{(2-1)(2-2)}{2}=0$, which for a smooth curve is the genus, and a degree-one divisor is present, so $C\cong {\mathbb{P}}_k^1$;
3. under such an isomorphism the degree-one divisor $[p]$ corresponds to a degree-one divisor $[q]$ of ${\mathbb{P}}_k^1$, and the projective-line computation gives $l([q])=h^0(\mathcal{O}(1))=2$ and $i([q])=0$ (Global sections of projective twists); dimensions of cohomology are invariant under isomorphism, so $l([p])=2,i([p])=0:$ Riemann-Roch on $C$ reads $2-i([p])=1+1-0$, forcing $i([p])=0$, and the two-dimensional space $L([p])$ is spanned by $1$ and a coordinate function with a single simple pole at $p$, the coordinate of the isomorphism $C\cong {\mathbb{P}}_k^1$ supplied by the rational-point theorem.

The classical form of this computation is the projection parametrisation: for each line $\mathrm{\ell }$ through $p$ the residual intersection $C\cap \mathrm{\ell }$ pairs the second point of $\mathrm{\ell }\cap C$ with $p$, giving the pencil $\mathrm{\mid }[p]\mathrm{\mid }$ and the coordinate above; the reverse direction — that the quadratic Veronese image of ${\mathbb{P}}_k^1$ is such a conic — is the batch-6 examples-page item ex-quadratic-veronese-conic, which is not consumable here because examples-page items are leaves.

Scaffold repair, recorded for the owner. The frozen scaffold cited the examples-page items ex-quadratic-veronese-conic and ex-rational-parametrization-circle-conic. Both are leaves and cannot carry a load; the citations are replaced by the A-page rational-point theorem A genus-zero curve with a degree-one divisor is the projective line, the published A-page computation of $h^0(\mathcal{O}(1))=2$ Global sections of projective twists, and the local argument of items 1–3. Every promised numerical claim (${\mathrm{deg}}_k[p]=1$, $p_a=0$, $g=0$, $C\cong {\mathbb{P}}_k^1$, $l([p])=2$, $i([p])=0$, and the reading $2-i=1+1-0$) is preserved. The explicit line-pencil description of $\mathrm{\mid }[p]\mathrm{\mid }$ is recorded as the classical geometric picture rather than as a consumed claim, with its would-be supplier named above.

The current Arithmetic genus of a plane curve gives the arithmetic genus; smoothness identifies it with the curve genus. The current A genus-zero curve with a degree-one divisor is the projective line supplies the isomorphism, and the published projective-space cohomology result Global sections of projective twists supplies the section dimension. The proof transports cohomology through the isomorphism using the current cohomology and Riemann-Roch interfaces cited below.