Can a regular quasi affine variety (i.e. open subscheme of an affine variety) contain a (possibly singular) projective curve?
2
-
4$\begingroup$ No because if C is a projective curve in a quasi-affine variety which is open in an affine variety Y, then the immersion C-->Y is closed, hence C is affine, a contradiction. $\endgroup$– Matthieu RomagnyCommented Jul 13, 2022 at 9:29
-
2$\begingroup$ Admittedly, this question would be more suitable for MSE; at any rate, I provided a short answer. Feel free to migrate it. $\endgroup$– Francesco PolizziCommented Jul 13, 2022 at 9:51
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
4
I assume you work over an algebraically closed field.
It is well-known that, on an affine variety, regular functions separate points. The same is almost true for a quasi-affine variety, in the sense that, given any $x \in X$, there exist only finitely many points $y \in X$ such that all the regular functions in $X$ assume the same value at $x$ and $y$: this is a result by Goodman and Hartshorne, see Tony Pantev's answer to MO2083.
Now take a curve $C \subset X$, with $X$ quasi-affine. Since the support of $C$ is infinite, by the previous argument we can find two distinct points $x,\, y \in C$ and a regular function $f$ on $X$ such that $f(x) \neq f(y)$.
Restricting $f$ to $C$, we obtain a regular function on $C$ with the same property. But then $C$ cannot be projective, since projective varieties admit no non-constant regular functions.
answered Jul 13, 2022 at 9:49
-
$\begingroup$ Regarding point separation, if $U \subset X$ is open, and regular functions separate points on $X$, then surely they separate points on $U$, just because restriction of a regular function is regular? $\endgroup$ Commented Jul 14, 2022 at 7:20
-
$\begingroup$ In fact, it seems to me that you are right. If $X$ is embeddable in an affine scheme $X_1$, then the global sections in $X_1$ must separate the points in $X$. But, then, I do not understand the remark in Toni Pantev's answer linked above. $\endgroup$ Commented Jul 14, 2022 at 8:35
-
$\begingroup$ Tony was claiming the other direction, I think, if points are separated in some weak sense, the variety is quasi-affine, which seems nontrivial. $\endgroup$ Commented Jul 14, 2022 at 9:13
-
$\begingroup$ In other words, that weak separation implies strong separation? $\endgroup$ Commented Jul 14, 2022 at 9:15