Timeline for Automorphism group of conic bundle fixing the base
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 30, 2023 at 10:48 | comment | added | Jason Starr | Even if the discriminant is smooth of high degree, there could still be nontrivial automorphisms: think about blowing up the restriction over a high degree hypersurface of a (constant) cross-section of a product conic bundle. If you add the hypothesis that the family is minimal, that will probably reduce the automorphism group to a finite group (by forcing the global sections of the tangent bundle to vanish). | |
| Jun 29, 2023 at 5:55 | comment | added | TCiur | I'm sorry, I meant to use $PGL_3$ instead. Also, if the discriminant curve is smooth and sufficiently high genus, it looks like this group is either trivial or $C_2$. Is that correct? | |
| Jun 29, 2023 at 5:52 | history | edited | TCiur | CC BY-SA 4.0 |
Wrong dimension for projective linear group: should be 3 not 2
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| Jun 28, 2023 at 10:53 | comment | added | Jason Starr | For a proper, geometrically connected, smooth curve of arithmetic genus $0$, the least degree of a very ample invertible sheaf is at most $2$, and the dual of the dualizing sheaf is a very ample invertible sheaf of degree $2$ whose associated closed immersion is in $\mathbb{P}^2$. Thus, the automorphism group is a closed subgroup of $\text{Aut}(\mathbb{P}^2)$, but this is $\textbf{PGL}_3$, not $\textbf{PGL}_2$. Perhaps that is your mistake. | |
| Jun 28, 2023 at 10:01 | comment | added | abx | The fiber of a conic bundle at the generic point $\eta $ is not in general isomorphic to $\mathbb{P}^1_{k(\eta )}$, hence its automorphism group is not $\operatorname{PGL}(2,k(\eta )) $. | |
| Jun 28, 2023 at 4:08 | history | asked | TCiur | CC BY-SA 4.0 |