Embedding varieties as divisors

Question

Let $X$ be a smooth affine algebraic variety. Does there necessarily exist an embedding into some affine space $A^n$ of codimension $1$?

I guess so. Next one I'm less sure.

Let $X$ be a complete intersection inside an affine space $A^m$. Does there exist a different embedding into another $A^n$ of codimension $1$?

jori · Accepted Answer · 2021-02-02 18:39:42Z

6

Not every smooth affine curve can be embedded into the affine plane so the answer is no.

answered Feb 2, 2021 at 18:39

jori

761 bronze badge

$\begingroup$ Oh wow, I feel stupid. Could you just give me an example of such a curve? Thanks so mcuh. $\endgroup$
– GiveMeDivisorsPlease
Commented Feb 2, 2021 at 19:21
5

$\begingroup$ A smooth curve embedded into $\mathbb{A}^2$ has trivial canonical bundle. Take any smooth projective curve $C$ of genus $\geq 2$ and general point $p\in C$, then $C\smallsetminus p$ is affine with non-trivial canonical bundle. $\endgroup$
– abx
Commented Feb 2, 2021 at 19:29
$\begingroup$ @abx Of course, thanks!! $\endgroup$
– GiveMeDivisorsPlease
Commented Feb 2, 2021 at 20:11
$\begingroup$ @abx Could you explain why the canonical bundle being nontrivial on $C$ globally implies that it is nontrivial on $C \smallsetminus p$ for a general $p$? $\endgroup$
– user2831784
Commented Feb 2, 2021 at 21:16
2

$\begingroup$ @user2831784: from the exact sequence of Picard groups for $C \setminus p \subset C$ one gets that if $K_{C \setminus p}$ is trivial, then $K = (2g-2)[p]$. This can not hold for general $p$, as if it does for points $p$ and $q$, then $[p] - [q]$ is $(2g-2)$-torsion in the Jacobian. $\endgroup$
– Evgeny Shinder
Commented Feb 2, 2021 at 21:42

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