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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.)
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    $\begingroup$ I guess that you are after toric varieties. You should most certainly look at Cox's book on the subject. $\endgroup$ – Mariano Suárez-Álvarez Apr 17 '15 at 9:01
  • $\begingroup$ I was just about to suggest the same book.... don't be frightened by it's massive size: it's a nice reading and, if I recall correctly, you'll likely not need anything past chapter 6. $\endgroup$ – A.P. Apr 20 '15 at 9:30

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