Nonsingular affine and projective curves as Riemann surfaces

(The affine curve is a nice topological space.) The set $C\subseteq {\mathbb{C}}^N$ is closed, so as a subspace of the Hausdorff second-countable space ${\mathbb{C}}^N$ it is itself Hausdorff and second countable by [F5] and [F6]; every connected component of $C$ is open by [F7], hence a nonempty connected Hausdorff second-countable space.

(The projective space is compact and second countable.) The quotient map $q:{\mathbb{C}}^{N+1}\setminus \{0\}\to {\mathbb{C}\mathbb{P}}^N$ has continuous restriction to $S^{2N+1}$ by [F3], it is surjective because every nonzero vector is a positive multiple of a unit vector, so ${\mathbb{C}\mathbb{P}}^N$ is compact by [F4]; its finitely many standard affine charts are homeomorphic to ${\mathbb{C}}^N$ by [F1], hence second countable by [F6], and a finite union of open second-countable subspaces is second countable, since the union of their countable bases is a countable base.

(The affine curve has a holomorphic atlas.) At each $p\in C$ the local parameter supplied by [F1] is a chart from a neighbourhood of $p$ onto a plane domain; over two such neighbourhoods the transition map between local parameters is holomorphic by [F1]; hence these charts form a holomorphic atlas on $C$, and restricting the charts to a connected component exhibits that component as a Riemann surface in the sense of [F2].

(The projective space is Hausdorff.) Let $[\zeta ]\mathrm{\ne }[\eta ]$ in ${\mathbb{C}\mathbb{P}}^N$. If some $U_i$ contains both, then the chart homeomorphism $U_i\to {\mathbb{C}}^N$ separates them, since ${\mathbb{C}}^N$ is Hausdorff and $U_i$ is open. Otherwise the supports of $\zeta$ and $\eta$ are disjoint. Put $S=\{i:{\zeta }_i\mathrm{\ne }0\}$ and define $g([\xi ])=({\sum }_{i\in S}\mathrm{\mid }{\xi }_i{\mathrm{\mid }}^2)\mathrm{/}({\sum }_{k=0}^N\mathrm{\mid }{\xi }_k{\mathrm{\mid }}^2)$. This is well defined under nonzero complex scaling; its composite with the quotient map is continuous, so it is continuous by [F3]. Since $g([\zeta ])=1$ and $g([\eta ])=0$, the open sets $\{g>2\mathrm{/}3\}$ and $\{g<1\mathrm{/}3\}$ are disjoint neighbourhoods of the two points.

(The conic is a nonsingular affine instance.) For $f=x^2+y^2-1$ one has $\mathrm{\nabla }f=(2x,2y)$, which vanishes only at the origin, and $(0,0)\mathrm{\notin }{C}_0$ because $-1\mathrm{\ne }0$; hence the Jacobian of the single equation has rank $1=N-1$ at every point of $C_0$ with $N=2$, so $C_0$ satisfies the hypothesis of steps 1.1 and 2.1 and is a complex curve whose components are Riemann surfaces.

(The projective curve has a holomorphic atlas.) Let $C\subseteq {\mathbb{C}\mathbb{P}}^N$ be the zero set of homogeneous polynomials $F_1,\dots ,F_r$ and suppose it is nonsingular in the Jacobian-rank sense in each standard affine chart, i.e. each piece $C\cap U_i$ is, under the chart identification, an affine curve satisfying the hypothesis of step 2.1; by step 2.1 each piece therefore carries the local-parameter atlas, and the charts coming from two different affine charts are compatible because the standard chart transitions are holomorphic and, by [F1], all local parameters of the curve transform holomorphically; hence these charts form one holomorphic atlas on $C$, and its connected components are Riemann surfaces.

(The projective curve is compact, Hausdorff and second countable.) The curve $C\subseteq {\mathbb{C}\mathbb{P}}^N$ is closed, because its preimage under the quotient map is the zero set of the continuous functions $F_i$ on ${\mathbb{C}}^{N+1}\setminus \{0\}$ by [F3]; being a closed subset of the compact space ${\mathbb{C}\mathbb{P}}^N$ of step 1.2 it is compact by [F5], and being a subspace of the Hausdorff second-countable space ${\mathbb{C}\mathbb{P}}^N$ of steps 1.2 and 2.2 it is Hausdorff and second countable by [F5]; hence, with the atlas of step 3.2, each component is a Riemann surface.

(Conclusion.) Steps 2.1 and 3.1 give the affine curves, in particular the nonsingular conic $x^2+y^2=1$, the local-parameter complex structure, and steps 3.2 and 4.1 do the same for a projective curve while proving that it is compact in its analytic topology; in each case the components inherit a holomorphic atlas, Hausdorffness and second countability, so they are Riemann surfaces.