2
$\begingroup$

Throughout, singular Del Pezzo means a surface with only isolated singularities and ample anti-canonical divisor.

Suppose $X$ is a singular Del Pezzo of degree 2 over a field $k$ where $\text{char}(k)\neq 2$. Assume $X$ has one singular point. Is it possible to project from this singular point? If so what is the image of this projection?

Improve this question
$\endgroup$
8
  • $\begingroup$ 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. $\endgroup$
    – Sasha
    Commented Jun 8, 2022 at 13:43
  • 2
    $\begingroup$ @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 $(x_{0},\dotsc,x_{3})\mapsto (x_{1},x_{2},x_{3})$. $\endgroup$
    – H U
    Commented Jun 8, 2022 at 13:51
  • $\begingroup$ But this map is well-defined only on the blow-up of the singular point. $\endgroup$
    – Will Sawin
    Commented Jun 8, 2022 at 14:21
  • $\begingroup$ @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? $\endgroup$
    – Sasha
    Commented Jun 8, 2022 at 14:34
  • 2
    $\begingroup$ @Sasha over an algebraically closed field it is but not over a general field $\endgroup$
    – H U
    Commented Jun 8, 2022 at 18:37

0

Reset to default

You must log in to answer this question.