Timeline for Singular Del Pezzo of degree 2
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2022 at 18:52 | comment | added | Sasha | @HU: Ah, OK, I missed that point. In this case, the linear system $|-K_{\tilde{X}} -E|$ (where $\tilde{X}$ is the blowup, $K_{\tilde{X}}$ is its canonical class, and $E$ is exceptional divisor) provides $\tilde{X}$ with a structure of a conic bundle over $\mathbb{P}^1$. | |
| Jun 8, 2022 at 18:37 | comment | added | H U | @Sasha over an algebraically closed field it is but not over a general field | |
| Jun 8, 2022 at 17:20 | comment | added | Sasha | @HU: Any weak del Pezzo surface of degree $2$ is a blowup (in many different ways) of a plane in 7 points in almost general position. | |
| Jun 8, 2022 at 14:44 | comment | added | H U | @Sasha For the singular cubic case the rational map above will map $X$ birationally into a plane. I was wondering if one could do something similar for Del pezzos of degree 2? | |
| Jun 8, 2022 at 14:34 | comment | added | Sasha | @HU: In the cubic surface example, the blowup of a singular point is a weak del Pezzo surface that has a natural morphism to $\mathbb{P}^2$ (and this morphism itself is the blowup of 6 points lying on a conic). In the degree 2 case the blowup of the singular point is also a weak del Pezzo, so what do you want to understand about it? | |
| Jun 8, 2022 at 14:21 | comment | added | Will Sawin | But this map is well-defined only on the blow-up of the singular point. | |
| Jun 8, 2022 at 13:51 | comment | added | H U | @Sasha No I don’t want to blow up. For example on a cubic surface $X:x_{0}f_{2}+f_{3}$ with a singular point at $(x_{0},\dotsc,x_{3})=(1,0,0,0)$ one can “project from the singular point” by constructing a map to $\mathbb{P}^{2}$ by sending $(x_{0},\dotsc,x_{3})\mapsto (x_{1},x_{2},x_{3})$. | |
| Jun 8, 2022 at 13:43 | comment | added | Sasha | That depends on what do you mean. If you have a del Pezzo of higher degree, "projecting from a point" usually means just blowing up this point. And of course, you can blowup a singular point on a del Pezzo surface of degree 2. | |
| Jun 8, 2022 at 13:41 | history | asked | H U | CC BY-SA 4.0 |