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)
Related question: Any quasi-affine variety is isomorphic to a closed subvariety in $\mathbb{A}^n\setminus \mathbb{A}^m$ for some $m,n$.
Regards
Best Answer
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.