Arithmetic genus zero forces a smooth rational curve

Given: Full AC and an integral proper curve $C$ over the algebraically closed field $\mathbb{C}$.

Full AC is The Axiom of Choice, inherited from the normalization and proper-cohomology suppliers.

The normalization $\nu :\tilde{C}\to C$ is finite, $\tilde{C}$ is smooth and proper, and its genus is the geometric genus $g(C)$ (Geometric genus of a singular curve, Normalization of a reduced curve is finite, Normalization of an integral finite-type curve by gluing affine integral closures).

The normalization defect $\delta (C)$ is a nonnegative integer and $\delta (C)=\chi ({\mathcal{O}}_{\tilde{C}})-\chi ({\mathcal{O}}_C)=p_a(C)-g(C)$ (Normalization defect delta of a reduced curve, The normalization defect is an Euler characteristic and a weighted sum of local lengths, Genus and arithmetic genus of a curve, Geometric genus of a singular curve).

The normalization map is an isomorphism exactly when its defect sheaf is zero; in particular $\delta (C)=0$ implies $C$ is regular, and a regular curve over the perfect field $\mathbb{C}$ is smooth (The normalization defect is an Euler characteristic and a weighted sum of local lengths, Normalization defect delta of a reduced curve, Geometric genus of a singular curve).

The curve $C$ has a closed $\mathbb{C}$-rational point. Choose a nonempty affine open $U=\mathrm{Spec}A\subset C$. Since $C$ is an integral finite-type curve, $A$ is a nonzero finite-type $\mathbb{C}$-domain of dimension one. The zero ideal is proper, so In a nonzero commutative ring, every proper ideal is contained in a maximal ideal gives a maximal ideal $\mathfrak{m}\subset A$; Over an algebraically closed field, maximal ideals of an affine algebra are kernels of points identifies its residue field with $\mathbb{C}$. Thus the corresponding point is closed in $U$ and is not generic. Its closure in $C$ cannot be all of $C$, since its intersection with $U$ is just that one point whereas $U$ has dimension one. The only proper irreducible closed subsets of an integral curve are closed points, so this point is closed in $C$ and has residue field $\mathbb{C}$ (Curves over a field, The residue field at a point of an affine scheme).

For a closed point $p$ on a smooth proper curve, the divisor $[p]$ has degree $[\kappa (p):\mathbb{C}]$ (Divisors on a smooth proper curve, Degree divisor proper curve); hence the point from [F4] supplies a divisor of degree one. A smooth proper genus-zero curve with a degree-one divisor is isomorphic to ${\mathbb{P}}^1$ (A genus-zero curve with a degree-one divisor is the projective line).

For a quasi-compact separated scheme, a finite affine open cover computes the sheaf cohomology of every quasi-coherent module by its ordered Čech complex (Cech cohomology computes quasi-coherent cohomology on a separated scheme). This applies to the two standard affine opens of ${\mathbb{P}}_{\mathbb{C}}^1$ and its quasi-coherent structure sheaf; their intersection is $\mathrm{Spec}\mathbb{C}[z,z^{-1}]$.