Let $X$ be an irreducible affine variety (integral, reduced scheme of finite type over an algebraically closed field $\Bbbk$ *of characteristic zero*) of dimension $n$. The well-known Noether Normalization Lemma states that there is a finite morphism $\pi:X\to\mathbb A_\Bbbk^n$.

Assume now that I have an irreducible codimension one subvariety $Y\subseteq X$.

**Edit:** Also assume that the generic point of $Y$ is a regular point of $X$. Even more, you may assume that the generic point of $Y$ is in the normal locus of $X$ or even in the nonsingular locus of $X$. In fact, in cases that interest me, $Y$ is completely contained in the nonsingular locus of $X$.

I think it should be possible to choose a Noether Normalization as above with the property that $Y$ is not in the ramification locus, i.e. there is some $p\in\pi(Y)$ with $|\pi^{-1}(p)|=\deg(\pi)$. In other words: I want to choose $\pi$ such that it is unramified at the generic point of $Y$.

The question is: Can I do this? Assuming that I can: Do you know a proof or a reference for this statement? Even under additional assumptions, an affirmative answer would be very much appreciated.

Thanks a lot.

**Edit:** I realize now, after soberly considering Cantlog's comments, that I do indeed wish that $\deg\pi|_Y=\deg\pi$ holds in my situation, with $\pi$ unramified at the generic point of $Y$. This can also be expressed as the requirement that the inertia index of $\pi$ at the generic point of $Y$ is equal to $\deg\pi$. Clearly, this is not possible in general, but it will be possible in some cases. If you can name sufficient conditions for such a Normalization to exist, I would be very grateful.