A genus-zero curve need not be the projective line

Given: the Axiom of Choice inherited from the current smoothness, properness, plane-genus and rational-point suppliers; the field $k=\mathbb{R}$, the form $F=x^2+y^2+z^2$, the closed subscheme $C=V_{+}(F)\subseteq {\mathbb{P}}_{\mathbb{R}}^2$, and its base change $C_{\mathbb{C}}$ to $\mathbb{C}$.

Smoothness: for either $K=\mathbb{R}$ or $K=\mathbb{C}$, each of the three standard affine charts of $C_K$ is presented as $A_K=K[u,v]\mathrm{/}(1+u^2+v^2)$. Its Jacobian row is $(2u,2v)$, and in $A_K$ one has $1=(-u\mathrm{/}2)(2u)+(-v\mathrm{/}2)(2v)$. Thus no prime contains both entries; around every prime one of them is invertible, so the one-equation presentation is standard smooth there by Standard smooth presentations and locally standard smooth maps. By [F2] the structure map $C_K\to \mathrm{Spec}K$ is proper, hence of finite type by Proper morphisms, so the finite-type hypothesis in Smooth morphisms via local standard smooth presentations holds. Therefore every chart, and hence $C_K$, is smooth over $K$.

Properness: for either $K=\mathbb{R}$ or $K=\mathbb{C}$, ${\mathbb{P}}_K^2$ is proper over $K$; the closed immersion $C_K\hookrightarrow {\mathbb{P}}_K^2$ is proper, and its composition with the structure morphism is proper (Finite-dimensional projective space is proper over every base, Closed immersions are proper, Properness survives composition, Proper morphisms).

Geometric integrality: over $\mathbb{C}$ the form $F$ has no factorisation into linear forms. Any proper factorisation of a homogeneous quadratic over a field has two degree-one factors; writing each as its degree-one homogeneous part plus a constant, the degree-zero and degree-one parts of their product force both constants to vanish. Thus it suffices to test homogeneous linear factors. If $F=L_1{L}_2$, then restricting to $z=0$ and using unique factorisation in $\mathbb{C}[x,y]$ lets us rescale and order the factors as $L_1=x+iy+\alpha z$ and $L_2=x-iy+\gamma z$. Their product has $xz$, $yz$, and $z^2$ coefficients $\alpha +\gamma$, $i(\gamma -\alpha )$, and $\alpha \gamma$, respectively. Equality with $F$ requires $\alpha +\gamma =0$, $i(\gamma -\alpha )=0$, and $\alpha \gamma =1$; the first two force $\alpha =\gamma =0$ in characteristic zero, contradicting the third. Thus $F$ is irreducible in $\mathbb{C}[x,y,z]$; its principal ideal is prime by the finite-variable UFD lemma, so $C_{\mathbb{C}}$ is reduced and irreducible. It is nonempty since $[i:0:1]\in C_{\mathbb{C}}$. The extension $\mathbb{C}\mathrm{/}\mathbb{R}$ is algebraic ($\mathbb{C}\mathrm{/}\mathbb{R}$ has power basis $1,i$ and degree $2$) and $\mathbb{C}$ is algebraically closed (The complex numbers are algebraically closed), so $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$ (An algebraic closure of a field). Hence $C_{\mathbb{C}}$ is the geometric fibre defining geometric integrality of $C$ over $\mathbb{R}$; for $C_{\mathbb{C}}$ over $\mathbb{C}$, take the algebraic closure to be $\mathbb{C}$ itself. The fibres are integral in the sense of Geometric properties of fibres, giving the geometric-integrality assertions in Curves over a field (Geometric fibres and geometric points, Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes).

Scheme chain dimension: for $K\in \{\mathbb{R},\mathbb{C}\}$, the three standard affine charts of $C_K$ have coordinate ring $A_K=K[u,v]\mathrm{/}(1+u^2+v^2)$ (Relative projective space from standard charts). By [F3], $F$ is irreducible over $\mathbb{C}$ and therefore over $\mathbb{R}$; for either field $K$ its homogeneous ideal is prime by the UFD property of $K[x,y,z]$ (Every finite-variable polynomial ring over a field is a UFD, with prime irreducibles and principal height-one primes). Each chart ring is the degree-zero subring of the localization of this homogeneous domain at one coordinate, hence is a nonzero domain. The map $K[u]\to A_K$ is injective: if a nonzero $q(u)$ lay in $(1+u^2+v^2)$, then in $K[u][v]$ it would equal $(1+u^2+v^2)h$ for nonzero $h$, contradicting additivity of degree in $v$ (Over an integral domain, degrees add under multiplication of nonzero polynomials). Thus $u$ is transcendental over $K$, while $v$ is algebraic over $K(u)$ by $v^2+u^2+1=0$; the transcendence-degree tower formula gives ${\mathrm{trdeg}}_K\mathrm{Frac}(A_K)=1$ (Transcendence degree is additive in finite towers). The affine-domain dimension theorem gives Krull dimension $\dim A_K=1$ (Affine-domain dimension equals transcendence degree). The ring $A_K$ is Noetherian as a quotient of a finite-variable polynomial ring, so its spectrum is Noetherian (Finite-variable polynomial algebras over fields are Noetherian by finite generators, The spectrum of a Noetherian ring is a Noetherian topological space). Every nonempty irreducible closed subset of this spectrum has a prime ideal as its unique generic point, and distinct primes have distinct closures; thus chains of nonempty irreducible closed subsets have exactly the lengths of chains of prime ideals. The chart's chain dimension is therefore its Krull dimension, one (A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point, Krull dimension of a nonzero ring, Chain dimension and the empty-space convention). Each of the three charts is Noetherian. A descending chain of closed subsets of $C_K$ stabilizes after restriction to each chart; since the cover is finite, the maximum of those three stabilization indices works on all of $C_K$. Thus $C_K$ is Noetherian, and the open-cover dimension lemma gives its chain dimension as the supremum of the three chart dimensions, namely one (Dimension can be computed on an open cover).

Arithmetic genus: a curve $X=V_{+}(G)$ cut out by a nonzero homogeneous form of degree $d\ge 1$ has $p_a(X)=\frac{(d-1)(d-2)}{2}$; for $d=2$ this is $0$, and for a smooth curve $p_a=g$ (Arithmetic genus of a plane curve, Genus via the Euler characteristic).

Residue-field degrees: if $p$ is a closed point, an affine neighborhood $\mathrm{Spec}A$ of $p$ is of finite type over $\mathbb{R}$, and $p$ remains closed there, so $\kappa (p)=A\mathrm{/}{\mathfrak{m}}_p$ is finite over $\mathbb{R}$ (Curves over a field, A maximal ideal of an affine algebra has finite residue field over the base field). The field $\mathbb{R}$ is perfect because it has characteristic zero (Fields of characteristic zero, finite fields, and algebraically closed fields are perfect), so the finite extension $\kappa (p)\mathrm{/}\mathbb{R}$ is separable (Every algebraic extension of a perfect field is separable) and simple (A finite extension generated by elements all but possibly one of which are separable is simple). If $\kappa (p)=\mathbb{R}(\alpha )$, the minimal polynomial of $\alpha$ is irreducible and has degree $[\kappa (p):\mathbb{R}]$ (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element, A simple algebraic extension is its minimal-polynomial quotient and has power basis $1,a,\dots ,a^{n-1}$ and degree $n$); every irreducible real polynomial has degree $1$ or $2$ (An irreducible polynomial in $\mathbb{R}[x]$ has degree $1$ or $2$). If this degree were $1$, then $\kappa (p)=\mathbb{R}$ and the residue-field point would give an $\mathbb{R}$-morphism $\mathrm{Spec}\mathbb{R}\to C$ (Field-valued points and local-ring points).

No real points and even divisor degrees: every real sum of squares $x^2+y^2+z^2$ vanishes only at the origin, so $C(\mathbb{R})=\mathrm{\varnothing }$. Therefore [F6] excludes residue degree $1$ for every closed point, and every closed point has residue degree $2$. A divisor is a finite signed combination $D={\sum }_p{n}_p[p]$ of closed points and has degree ${\mathrm{deg}}_{\mathbb{R}}D={\sum }_p{n}_p[\kappa (p):\mathbb{R}]$ (Degree divisor proper curve, The residue field at a point of an affine scheme); hence ${\mathrm{deg}}_{\mathbb{R}}D=2{\sum }_p{n}_p$ is even.

The rational-point theorem: a smooth proper geometrically integral curve of genus $0$ over a field $K$ that admits a divisor of degree one (equivalently a $K$-rational closed point) is isomorphic to ${\mathbb{P}}_K^1$ (A genus-zero curve with a degree-one divisor is the projective line); and an isomorphism of $K$-schemes induces a bijection of $K$-rational points, while ${\mathbb{P}}_K^1$ has the $K$-point $[1:0]$.

The Axiom of Choice is assumed in the local-standard-smooth definition used in [F1], in the three properness results used in [F2], in the Noetherian-spectrum and irreducible-closed-subset correspondences used in [F4], and in the genus-zero rational-point theorem [F8]; the arithmetic-genus route in [F5] also inherits the properness suppliers. The factorisation, chart-dimension computations, and residue-field degree argument require no further choice principle, and the algebraic closure used here is the explicitly given $\mathbb{C}$ (The Axiom of Choice).