Normalization of Curve $x-y^n=0$

Question

Let $X\subset \mathbb P^2$ be the curve (over $\mathbb C$) defined by $x-y^n=0$. We can view it as points of the form $$\{(x,\sqrt[n] x)\}$$ in $\mathbb A^2$ and then take closure in $\mathbb P^2$. Here $x$ takes all the $n$ roots. So it is natrual to consider points of the form $$\{([p:q],[1:\sqrt[n] \frac{p}{q}:(\sqrt[n] \frac{p}{q})^2:\ldots:(\sqrt[n] \frac{p}{q})^{n-1}])\}$$ in $\mathbb P^1\times \mathbb P^{n-1}$ defined by \begin{cases} (x_1/x_0)^n=p/q\\ x_1/x_0=x_2/x_1\\ x_2/x_1=x_3/x_2\\ \ldots\\ x_{n-1}/x_{n-2}=x_{n-1}/x_{n-2}\\ \end{cases} We denote it by $X'$.

I want to show that:

$X'$ is the normalization of $X$

It looks reasonable, and I have checked it is true for $n$ small. But is there some smart way to prove this for any $n$? And is there any intuitive way to explain this?

More generally, is it always true that we can normalize an arbitrary plane curve in this way?

Thanks in advance.

Answer 1

Unfortunately the answer is no, this will not yield the normalization of $X$; quite simply because $X'$ is not a curve unless $n=2$.

Let me illustrate for $n=5$. The homogenized equations are the following: \begin{align*} z x_2 &= x_1^2 \\ z^2 x_3 &= x_1^3 \\ z^3 x_4 &= x_1^4 \\ z^4 x_0 &= x_1^5 \end{align*} Now at any point with $z=x_1=0$, i.e. $p=[0:x_0:0:x_2:x_3:x_4]$, we have $p\in X'$. This means $\Bbb P^3\subseteq X'$ and consequently, $\dim(X')\ge 3$. It should be quite easy to see that $\dim(X')\ge n-1$ holds in the general case.

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That said, I thought it beneficial to remark that there is a deterministic way to normalize a curve through successive blow-up operations and it might be a good idea to go through this process for the curve $z^{n-1}x-y^n$ and see what happens. This procedure is well-understood theoretically and algorithmically.

You should familiarize yourself with the theory first, but if you are interested in concrete computations, I recommend to play around with singular's resolve function in their online interface;

https://www.singular.uni-kl.de:8003/

For example, the code

LIB"resolve.lib";
LIB"reszeta.lib";
ring R=0,(x,y,z),dp;
ideal I=z6x-y7; 
list L=resolve(I);
presentTree(L)

will suggest that the curve can be desingularized by $n-2$ blowups. It might be a good exercise to try and prove that this is indeed the case.