I know that one-point compactification of $\mathbb{C}$ gives us Reimann sphere. Also, Riemann sphere is homeomorphic to complex projective line $\mathbb{P}^1(\mathbb{C}) = S^{3}/{\sim}$ where $x \sim \lambda x$ for $|\lambda| = 1$.
Now consider the following way of compactification of $\mathbb{C}$ to get Riemann sphere:
As a set we take the quotient $q \colon \mathbb{C} \amalg \mathbb{C} \to \mathbb{P}^1(\mathbb{C})$ of the disjoint union $\mathbb{C} \amalg \mathbb{C} = \mathbb{C} \times \{0,1\}$ by the equivalence relation $\sim$ given by: $(z,i) \sim (w,j)$ if and only if (($i = j$ and $z = w$) or $(i \neq j$ and $zw = 1$)). For $z \neq 0$ the element $(z,1)$ is equivalent to $(1/z, 0)$. The set $\mathbb{C} \times \{0\} \amalg \{(0,1)\}$ is a set of representatives for $\sim$, and so $q$ is a bijection from that set to $\mathbb{P}^1(\mathbb{C})$. Hence we can write $\mathbb{P}^1(\mathbb{C}) = \mathbb{C} \amalg \{\infty\}$.
We give $\mathbb{P}^1(\mathbb{C})$ the quotient topology: for $U$ a subset of $\mathbb{P}^1(\mathbb{C})$ we let $U$ be open if and only if $q^{-1} U \subset \mathbb{C} \amalg \mathbb{C}$ is open. Here $\mathbb{C} \amalg \mathbb{C} = \mathbb{C} \times \{0,1\}$ has the product topology, where $\{0,1\}$ has the discrete topology.
We also have a notion of holomorphic functons. Later we will describe “regular” functions on “open” subsets of $\mathbb{P}^1(\mathbb{C})$ for a topology to be defined. On $\mathbb{C}$ these regular functions will just be the polynomial ones.
A nice visualisation of $\mathbb{P}^1(\mathbb{C})$ is as follows. We define $D_0 = \{z \in \mathbb{C} : |z| \leq 1\}$ in the first chart $\mathbb{C} \times \{0\}$, and $D_{\infty} = \{z \in \mathbb{C} : |z| \leq 1\}$ in the second chart $\mathbb{C} \times \{1\}$. Then the quotient map $q$ glues these two discs along their boundaries via the map $z \mapsto z^{-1}$. As a topological space $\mathbb{P}^1(\mathbb{C})$ is isomorphic to the two-dimensional sphere $S^2$. See Wikipedia (Riemann sphere) for more information.
I don't understand this new method of compactification of $\mathbb{C}$.
Then a similar method is used to find the compactification of the curve $$ C = \{(a,b) \in \mathbb{C}^2 : b^n = a^{n-1} - 1\} $$ to get $\overline{C} = (C \amalg C')/{\sim}$ where $$ C' = \{(c,d) \in \mathbb{C}^2 : d^n = c - c^n\} $$ and $(a,b) \sim (a^{-1}, ba^{-1})$ as follows:
$$ \require{AMScd} \begin{CD} \overline{C} @. := @. (C @. \amalg @. C') @. \hspace{-0.5em}/{\sim} \\ @VxVV @. @VxVV @. @VVuV \\ \mathbb{P}^1(\mathbb{C}) @. = @. (\mathbb{C} @. \amalg @. \mathbb{C}) @. \hspace{-0.5em}/{\sim'} \end{CD} $$ This map $x$ is well-defined since if $a \neq 0$ then the point $(a,b)$ on the first chart is mapped to $a$ on the first chart of $\mathbb{P}^1(\mathbb{C})$, and the corresponding point $(a^{-1}, a^{-1} b)$ is mapped to $a^{-1}$ on the second chart of $\mathbb{P}^1(\mathbb{C})$, and these points coincide in $\mathbb{P}^1(\mathbb{C})$.
What is the general theory behind this? I don't have knowledge of manifolds/complex manifolds.