Suppose we have the curve $y^2=h(x)=x(x-1)(x-2)$ in $\mathbb{C}^2$. Then this is clearly not compact and I saw two definitions for compactifying this and I want to prove that they are equivalent.
Method $1$: Consider the homogenization of the curve given by $y^2z=x(x-z)(x-2z)$. Now just take the zero set in $\mathbb{C}P^2$. Now this is compact since it is a closed subset of a compact set.
Method 2: This is the method I saw in the book "Algebraic curves and Riemann surfaces" by Rick Miranda. The definition used here is two take two patches and glue them by an isomorphism.
So the curve we consider is $w^2=z^4h(\frac{1}{z})=z(1-z)(1-2z)$. Now we have an open set $V=\{(z,w): z \neq 0\}$ and an open set $U=\{(x,y): x \neq 0\}$ and they are isomorphic via a map $\phi(z,w)=(\frac{1}{x},\frac{y}{x^2})$. Now wee can glue these curves using this isomorphism.
Now I am really not sure if these two definitions are equivalent. Any help is appreciated. Thanks.