Timeline for Small contractions as blow ups
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 29, 2019 at 7:39 | comment | added | Sándor Kovács | As I mentioned in an answer to mathoverflow.net/a/45193/10076, there is a pretty good way to understand what happens if $Y$ has klt singularities (incidentally, comparing to Jason's comment, a cone over a curve in $\mathbb P^2$ is klt iff its degree is at most $2$). What happens is that in that case there is a small contraction from a $\mathbb Q$-factorial model, so the existence of a small resolution depends on whether that (non-unique) model can be smooth. | |
| Jun 12, 2018 at 18:13 | comment | added | Sasha | @harajm: Take a product of a non $\mathbb{Q}$-factorial surface with something smooth. This will give you an example in dimension higher than 2. | |
| Jun 6, 2018 at 15:26 | comment | added | harajm | Thank you @Jason Starr for explaining that there are surfaces which are not $\mathbb{Q}$-factorial. If you write your comment as an answer I would accept it since it answers my question. However, I also added an edit to my question in order to get some more insights on when a higher dimensional non $\mathbb{Q}$-factorial variety is the target of a small contraction and how one maybe could go about producing small contractions by blowing-up. I would love to hear about what is known in this direction. If you like to share any insights I would be grateful. Thank you very much for your time. | |
| Jun 6, 2018 at 15:10 | history | edited | harajm | CC BY-SA 4.0 |
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| Jun 6, 2018 at 11:46 | comment | added | Jason Starr | Every projective cone $Y$ in $\mathbb{P}^3$ over a smooth curve $C$ of degree $d\geq 3$ in $\mathbb{P}^2$ is a normal, projective variety that is not $\mathbb{Q}$-factorial (the Picard group of $Y$ is just $\mathbb{Z}$, but the Weil divisor class group contains the Jacobian of $C$). Since $Y$ is a surface, every birational proper morphism $f:X\to Y$ has codimension $1$ exceptional set. So $f$ is not a small contraction. | |
| Jun 5, 2018 at 17:01 | review | First posts | |||
| Jun 5, 2018 at 17:21 | |||||
| Jun 5, 2018 at 15:45 | history | asked | harajm | CC BY-SA 4.0 |