Local holomorphic charts on nonsingular complex algebraic curves

(A free coordinate exists.) If $N=1$, there are no defining functions or dependent coordinates: the sole ambient coordinate is free, and $C$ agrees locally with an open subset of $\mathbb{C}$ by the empty zero-set condition. Write $w\in {\mathbb{C}}^0$ for the unique empty coordinate tuple in this case. If $N\ge 2$, the Jacobian matrix of $F$ at $p$ has rank $N-1$, so some $(N-1)\times (N-1)$ minor is invertible; permute the ambient coordinates so that the columns of that minor are the last $N-1$ and write the permutation as the invertible linear change of names $z=(z_1,w)$, $w\in {\mathbb{C}}^{N-1}$, with $p=(p_1,p^{\mathrm{\prime }})$.

(Standard affine charts are homeomorphisms with holomorphic transitions.) For $i\in \{0,\dots ,N\}$ put $U_i=\{[\zeta ]:{\zeta }_i\mathrm{\ne }0\}\subseteq {\mathbb{C}\mathbb{P}}^N$ and $p_i(\zeta )=({\zeta }_k\mathrm{/}{\zeta }_i{)}_{k\mathrm{\ne }i}\in {\mathbb{C}}^N$; the section $s_i(w)=[w_0:\cdots :w_{i-1}:1:w_{i+1}:\cdots :w_N]$ satisfies $p_i\circ s_i=\mathrm{i}\mathrm{d}$ and $s_i\circ p_i={\mathrm{i}\mathrm{d}}_{U_i}$, so $p_i$ is a bijection. The preimage $q^{-1}(U_i)=\{\zeta :{\zeta }_i\mathrm{\ne }0\}$ is open and saturated, hence $U_i$ is open by the quotient topology [F5]. The restriction $q:q^{-1}(U_i)\to U_i$ is a quotient map because $U_i$ is open and saturated. Thus $p_i$ is continuous by [F5], since $p_i\circ q$ is the continuous coordinate-ratio map on $q^{-1}(U_i)$; $s_i$ is continuous because it is the quotient of the continuous map sending $w$ to its normalized coordinate vector. These coordinate formulas are holomorphic where their denominators do not vanish by [F6]; thus $p_i:U_i\to {\mathbb{C}}^N$ is a homeomorphism, and for $i\mathrm{\ne }j$ the transition $p_j\circ p_i^{-1}(w)=(w_k\mathrm{/}{w}_j{)}_{k\mathrm{\ne }j}$ is holomorphic on its domain $\{w:w_j\mathrm{\ne }0\}$.

(The curve is locally a graph.) If $N=1$, choose a connected plane neighbourhood $A$ of $p$ contained in the local open piece of $C$ from step 1.1; with $B={\mathbb{C}}^0$ and the unique map $\varphi :A\to B$, the piece $C\cap (A\times B)$ is its graph. If $N\ge 2$, apply [F1] with $m=1$, $n=N-1$, $a=p_1$, $b=p^{\mathrm{\prime }}$ to $F$ to obtain a plane domain $A$ containing $p_1$ (after shrinking to the connected component of the first coordinate of the neighbourhood), a domain $B\ni p^{\mathrm{\prime }}$ and a holomorphic $\varphi :A\to B$ with $F(z,w)=0⟺w=\varphi (z)$ on $A\times B$; because $C$ agrees near $p$ with $F^{-1}(0)$, one has $C\cap (A\times B)=\{(z,\varphi (z)):z\in A\}$.

(The free coordinate is a chart.) On the piece $V:=C\cap (A\times B)$ the projection $\pi (z,\varphi (z))=z$ is the restriction of the continuous coordinate function $z_1$ and has the continuous inverse $z\mapsto (z,\varphi (z))$ built from the holomorphic, hence continuous, $\varphi$; so $\pi :V\to A$ is a homeomorphism onto the plane domain $A$ with holomorphic inverse, i.e. a chart for a holomorphic atlas on $C$.

(Projective case.) If $p$ lies in a standard affine chart $U_i$ of ${\mathbb{C}\mathbb{P}}^N$, step 1.2 identifies $U_i$ with ${\mathbb{C}}^N$ by a homeomorphism whose transitions to any other standard affine chart are holomorphic, and the hypothesis of the lemma is imposed in these coordinates, so the construction of steps 1.1–3.1 applies verbatim in the chart and supplies the local parameter at $p$; the same chart transition is used when a second parameter comes from another affine chart.

(Transitions of local parameters are holomorphic.) Let $c_1(u)=q_1(u)$ and $c_2(v)$ be two such parameters, where $q_1(u)$ is the point whose free coordinate equals $u$; by step 2.1 every ambient coordinate of a point of the graph is either the free coordinate $u$ or a component of $\varphi (u)$, hence a holomorphic function of $u$, so the second free coordinate $w(u)$ and with it the transition $u\mapsto w(u)$ is holomorphic; if the two parameters come from different standard affine charts, then the passage between the two affine coordinate systems is the holomorphic transition of step 1.2, and a composite of holomorphic functions is holomorphic.

(Conclusion.) Steps 2.1 and 3.1 exhibit, in the affine case, a plane domain $A$ and a homeomorphism of a neighbourhood of $p$ in $C$ onto $A$ with holomorphic inverse, i.e. a chart given by one free ambient coordinate; step 4.1 gives the same in the projective case inside a standard affine chart, and step 5.1 shows that all such local parameters have holomorphic transition maps, so the local parameters form a holomorphic atlas on the pieces where they are defined.