Timeline for A birational morphism of a finite cover to itself
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2019 at 5:52 | comment | added | abx | @Jérémy Blanc: Oops, of course, thanks. I guess I got confused with the degree of the branch locus. | |
| Jun 14, 2019 at 19:31 | comment | added | Jérémy Blanc | @abx I think you mean del Pezzo surface of degree $2$, not $4$. | |
| Jun 5, 2019 at 15:49 | comment | added | Sheng Meng | @abx Thanks for the precise example! Indeed, any projective variety of dimension $n$ has a finite surjective morphism to $\mathbb{P}^n$. | |
| Jun 5, 2019 at 14:30 | review | Close votes | |||
| Jun 11, 2019 at 3:05 | |||||
| Jun 5, 2019 at 14:13 | comment | added | abx | Take $Y=\mathbb{P}^2$, $X=$ a del Pezzo surface of degree 4. There is a double covering $\pi : X\rightarrow \mathbb{P}^2$ and a birational morphism $\tau :X\rightarrow \Bbb{P}^2$ obtained by blowing up 7 points in general position. | |
| Jun 5, 2019 at 11:25 | comment | added | Sheng Meng | @NickL The original question has many other assumptions, I just wonder what happens if removing all the assumptions. I would appreciate you copy this comment to an answer, I guess you mean $X:=$ blowup of $\mathbb{P}^2$, right? It has finite morphism to $\mathbb{P}^2$. | |
| Jun 5, 2019 at 11:15 | comment | added | Sheng Meng | @NickL If $\pi$ is isomorphic, then $\tau:X\to Y\cong X$ is a birational automorphism which has to be an automorphism. | |
| Jun 5, 2019 at 11:03 | comment | added | Sheng Meng | @NickL Thanks, I see! | |
| Jun 5, 2019 at 10:59 | comment | added | Nick L | The blow down of a $(-1)$-curve is a counter example. Maybe there is an error in the question, because the statement in brackets doesn't make sense i.e. there is no relation between $\pi$ and $\tau$. | |
| Jun 5, 2019 at 10:54 | history | asked | Sheng Meng | CC BY-SA 4.0 |