Currently (background: I'm preparing to write a thesis in mathematical physics) I'm quite often encountering a certain method for constructing projective manifolds, where the space is specified by just giving the following data:
- A set of coordinates
- A list of scaling relations for those coordinates
- The Stanley-Reisner ideal (specifying which coordinates must not be zero at the same time).
Example:
For example, we could take coordinates called $z_0, z_1, z_2$ and $\lambda$.
Impose the scaling relations
$$ \begin{array}{cccc}
z_0 & z_1 & z_2 & \lambda \\
\hline
1 & 1 & 1 & 0 \\
0 & 1 & 1 & -1
\end{array} $$
(meaning $(z_0 : z_1 : z_2 : \lambda) = (\Lambda z_0 : \Lambda z_1 : \Lambda z_2 : \lambda) = (z_0 : \Lambda z_1 : \Lambda z_2 : \Lambda^{-1} \lambda)$) and take the SR ideal $\langle z_0z_1z_2, z_0\lambda, z_1z_2 \rangle$.
Note that this is a way to describe the blow-up of $\mathbb P^2$ in $(1:0:0)$, this can be seen by defining $\sigma(z_0,z_1,z_2,\lambda) = (z_0 : z_1 \lambda : z_2 \lambda) \in \mathbb P^2$.
Questions:
- What do you call such a space? My google-fu is failing and I have never seen this in a math book, I think.
- (Does it possibly have something to do with toric varieties?)
- Where can I read about such things?
- What is the relation to (products of) weighted projective spaces?
(Seeing that e.g. the blow-up of $\mathbb P^2$ in a point can be written as the solution of a certain equation in $\mathbb P^2 \times \mathbb P^1$. And that apparently homology classes / line bundles over such a space look exactly like those over a product of weighted projective spaces.)
