Normalization of the projective closure of affine plane curve over $\mathbb{C}$ I am trying to understand how to do explicit calculations for finding the normalization of a plane curve. The intuition is somewhat clear to me: "separate" the singularities or smooth them out (for curves at least). However, in practice, finding the integral closure of a ring in its field of fractions is not so clear.
I was reading through this thread and the comments, but I couldn't quite figure out how to see that you need to need to add "x" amount of points above a singularity.
Example: Take the curve defined on the affine plane $y^2 = x^6 - 1$. It is smooth, and its projective closure is the curve defined by $Y^2Z^4 = X^6 - Z^6$ in the coordinates $[X,Y,Z]$ on $\mathbb{P}^2$. When $Z = 0$, we get $X = 0$ and $Y$ is free, so we add one point at infinity, $[0,1,0]$. By calculating the Jacobian or looking at the affine patch where $Y \neq 0$, we see that the projective closure is singular at this point. We are supposed to get two points from this one point in the normalization (since we know it is a hyperelliptic curve and there should be no ramification at $\infty$ in this case), but how am I supposed to see it?
If I were to do it algebraically I would have to look at the integral closure of $\left(\frac{k[x,z]}{(x^6 - z^6 - z^4)}\right)_{(\bar{x},\bar{z})}$ in $k(\bar{x},\bar{y})$, and this doesn't look too enticing. If I were to go this route, how do I know how many points I have to add? Is it the rank of the integral closure as a module over the original ring?

Directly finding the integral closure is pretty tricky in general. There are lots of other ways to do this. Probably the simplest is by blowing up.
All known algorithms for resolving curve singularities are described in detail in the first part of this book by Kollar. It's very readable. (The book gets harder in the later sections, but you don't need that material.) Take a look at the method that uses Newton polygons, for example. I can't find a good exposition online, but for an illustration of the power of Newton polygons, see this blog post.
If you're interested in computational methods for Riemann surfaces more generally, there is this book.
