Let $X$ be an affine algebraic curve with $0 \in X$ and $\tilde{X}$ the strict transform of $X$ w.r.t the blowup of $X$ at $0$. How to prove that $\pi \colon \tilde{X} \to X$ is finite? Is it even correct in general?
I'm thinking of something like the blow up of $V(y^2 - x^2 - x^3)$. If I consider $V(y^2 - x^2 - x^3, xs-yt)$ in $\mathbb{A}^2 \times \mathbb{P}^1$ and go to the chart $t=1$ I get the equation $s^2 - 1 - x$ where I easily see, that $s$ in integral over $k[x,y]/(y^2 - x^2 - x^3)$. On the chart $s=1$, I can't see it that easily or if I go to a chart where one of the points in the fiber over $0$ is missing I don't get a finite morphism.
My idea was to use relationship affine + proper = finite for morphism of schemes like here https://lovelylittlelemmas.rjprojects.net/proper-affine-finite/. But Im not very familiar with (non)-affine schemes.
So it is known that a blowing up is proper. And the closed immersion of the strict transform in the blow up is proper too? Because I can find a chart with all points of the fiber over $0$, $\pi$ must be affine too? and is then finite.
My questions are now:
1) Is this argumentation correct?
2) I actually only need the fact $\mathrm{Spec}(A[x]) \to \mathrm{Spec}(B[x])$ is closed, where $A$ and $B$ are the coordinate rings of (affine) $\tilde{X}$ resp. $X$, which follows from $\tilde{X} \to X$ universally closed. Is there a more elementary way to see this.
3) How can I easily prove that $\pi$ is quasi-finite, i.e. the fiber over $0$ finite? Can I see more easily, that $\pi$ is affine?
4) The whole argumentation shouldn't be true for blow ups at ideals with dimension > 0, otherwise I could finitely desingularize every variety. Would this be because $\pi$ is no longer affine?
5) What if the affine curve is not irreducible, i.e. $X$ not integral. There wouldn't be any problem with properness? I'm asking because Hartshorne Prop. 7.16 is asking for integral schemes to show that blowing up is proper, but in the proof he is referring to a Proposition where this property is not used.
The reason I'm interested in a finite blow-up is because I want to calculate the normalization this way. This question is actually a reformulation of this question, where there wasn't any answer.
Thank you very much!