I am trying to get used to work in the projective space. Therefore I wanted to know which tactics there are to imagine the projective space.
$$\mathbb{P^n}(k):= (k^{n+1}\backslash \{0\})/k^{*}$$
I like for example for the real projective space the idea of the sphere, but I am not sure if I got it right:
$\mathbb{P}(\mathbb{R})^2$ is the set of lines in $\mathbb{R}^3$ going through the origin. So if we put a Shere in $\mathbb{R}^3$ then each point in $\mathbb{P}(\mathbb{R})^2$ is described by two antipodal points on the sphere.
Then I saw that a parabola looks somehow like an ellipse on the sphere. (In the picture the red line. For the sake of lucidity I did not put the antipodal 'ellipse'.)
So can I imagine that the poles are zero and infinity?
I would be really happy if you could share your imgaination methods with me!
All the best, Luca
If I'm dealing with sheaves then I think of affine pieces with an n-1 dimensional piece at infinity. E.g. P1 as an affine line with point at infinity. If I'm working topologically, then I think of building it out a cell complex (see Hatcher's book).