# Is an irreducible affine variety isomorphic to some affine hypersurface?

I'm reading the basics of birational geometry in Shafarevich's "Basic Algebraic Geometry, 1" third edition. In theorem 1.8 he proves that

Every irreducible affine variety $$X\subseteq \mathbb{A}^n$$ is birationally equivalent to some hypersurface $$Y\subseteq \mathbb{A}^m$$ for some $$m$$. (In fact, $$m=n+1$$)

I'm wondering whether the same statement still holds true or not if we replace "birationally equivalent" with "isomorphic". So, my question is:

Is every irreducible affine variety $$X\subseteq \mathbb{A}^n$$ isomorphic to some affine hypersurface $$Y\subseteq \mathbb{A}^m$$ for some $$m$$?

I don't think this is true (it seems "too good to be true"), but I don't know how can I disprove such statement. Can you provide me with an example of an affine variety which is never isomorphic to an affine hypersurface? (I can work out the details myself, I just can't think of an elementary way or example to do this). I'd be surprised if the statement is true, if so, how could I prove it? (it should be an important result in such case)

Regards

It is not true, because for a hypersurface $$Y \subset \mathbb{A}^m$$ one has $$\dim T_y(Y) \le m = \dim(Y) + 1,$$ which is not always true for an affine variety.