Timeline for If a variety $X$ has finite automorphism group, is the same true for its $n$-fold self-products?
Current License: CC BY-SA 3.0
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 28, 2018 at 14:25 | comment | added | YCor | @AriyanJavanpeykar interesting, thanks! | |
| Feb 28, 2018 at 13:50 | comment | added | Ariyan Javanpeykar | @YCor If $X$ is of log-general type, then Aut$(X)$ is finite. Also, if $X$ is of log-general type, then $X^n$ is of log-general type. Now, if $X$ is an open of $\mathbb{A}^1$ with only finitely many automorphisms (i.e., there are $a$ and $b$ in $\mathbb{A}^1$ such that $X$ is an open of $\# \mathbb{A}^1\setminus\{a,b\}$), then $X$ is of log-general type and thus $X^n$ is of log-general type. Thus, $Aut(X^n)$ is finite. | |
| Feb 28, 2018 at 9:42 | comment | added | YCor | Yes but what about a curve which is not supposed singular, not supposed non-singular, not supposed affine, not supposed projective, not supposed non-projective. Standard name is "curve". Of course people tend to add implicit assumptions when they usually need it (irreducible, smooth, projective, etc,) | |
| Feb 28, 2018 at 3:53 | comment | added | Igor Rivin | @YCor you confuse me. An affine curve is usually called "an affine curve", and a singular projective curve "singular projective curve" (or "singular curve", being as "projective" tends to be the default) As to the question, where does the OP indicate that this is NOT what he wants? | |
| Feb 28, 2018 at 0:56 | comment | added | YCor | Apparently not in the question, and certainly not in most reference books of algebraic geometry such as Hartshorne etc. (And using it as default is not really practical... if a curve is assumed smooth projective, what can be said of an affine curve? or a singular projective curve? a curve which is not really a curve?... I wouldn't have noticed if the question were about smooth projective curves. Actually I'm curious about the question when $X$ is just the affine line minus a finite subset. | |
| Feb 28, 2018 at 0:30 | comment | added | Igor Rivin | @YCor Isn't that the default? | |
| Feb 27, 2018 at 22:23 | comment | added | YCor | "Two curves": you mean "two smooth projective curves" | |
| Feb 23, 2018 at 1:11 | history | answered | Igor Rivin | CC BY-SA 3.0 |