Normalization bijective on smooth points?

Question

Suppose we take an algebraic variety $X$ over $\mathbb{C}$ (I assume reduced). Is the normalization $$\pi:\tilde{X} \to X$$ always bijective on the smooth points of $X$?

Answer 1

Yes!
The key point is that the smooth points of $X$ are regular and a fortiori normal.
Now the set of normal points of a variety is open [ if you want to show-off tell your rival that this is a property valid for all quasi-excellent schemes :-)] and since normalization does not change any normal open subset of $X$, you can conclude that normalization is not only bijective but even an isomorphism over the smooth points of $X$.