$ \newcommand\R{\mathbb R} \newcommand\C{\mathbb C} \DeclareMathOperator\Cone{Cone} $I'm trying to establish a good notation to avoid confusion when we glue toric varieties from affine pieces. A simple example would be the toric variety given by the fan $\Sigma=\{\tau,\sigma_1,\sigma_2\}$ in $\mathbb R$ with cones $$ \tau = \{0\},\quad \sigma_1=\Cone(1)=\R_{\ge 0},\quad \sigma_2=\Cone(-1)=\R_{\le 0}. $$ The semigroup algebras of $\sigma_1,\sigma_2$ are $$ \C[S_{\sigma_1}] = \C[\chi^1],\quad\C[S_{\sigma_2}]=\C[\chi^{-1}]. $$ To avoid confusion I introduce new independent variables $x_1, x_2$ and introduce isomorphism $$ \C[S_{\sigma_1}] = \C[\chi^1]\cong \C[x_1],\quad\C[S_{\sigma_2}]=\C[\chi^{-1}] \cong \C[x_2]. $$ The affine pieces are $U_{\sigma_1},U_{\sigma_2}\cong\C$ glued along $U_\tau\cong\C^*$. I'd like to say that one copy of $\C$ is parametrized by the variable $x_1$, the other by the variable $x_2$ and we need to glue along $x_1\sim (x_2)^{-1}$ since under the introduced isomorphisms we have $\chi^1 = (\chi^{-1})^{-1}$.

In the end I'd end up with something like $$ X_\Sigma = \left( \C_{x_1} \sqcup \C_{x_2} \right)\big/ \left( x_1\sim (x_2)^{-1}\right)_{x_1,x_2\neq 0}, $$ where $\C_{x_1}$, $\C_{x_2}$ is supposed to suggest we denote the elements of the two copies $x_1$, $x_2$, respectively. Then we can see that this is exactly how the charts $\{(x_1:1)\},\{(1:x_2)\}\subseteq\C\mathbb P^1$ are glued, so $X_\Sigma=\C\mathbb P^1$.

Is this a good way to deal with gluing toric varieties? I don't want to involve many homomorphisms formally so I'm looking for a good way and notation to get this message through without too much technicalities but still leave no confusion to what is happening.

In the literature most authors don't bother to introduce seperate variables for each affine piece and just say the gluing is given by the embeddings of the $\C$-algebras, which I don't think is fair for people just starting with these kind of constructions. That's why I'm looking for a practical way to make the gluing process transparent.

  • $\begingroup$ I think your desires are conflicting. You don't want to involve many homomorphisms, but you want all the details spelled out rather than waved away. All the details are in the maps - they define the gluing rule! The pdf I emailed you a few months ago has quite a few explicit examples of the gluing construction, including this one (Example 2.29) for which I gave every detail. Once you've carried out the details with the maps, then you will be able to see what the gluing relations are quite quickly, but the reason for it still comes from the maps. $\endgroup$ – Ragib Zaman Mar 27 '14 at 11:13
  • $\begingroup$ @RagibZaman Indeed I see this conflict and I'm looking for a compromise. I want to focus more on how the gluing is obtained from the fan in a more or less computational matter instead of focusing on how we arrive there using the correspondence of points and maximal ideals. Introducing seperate variables and keeping track of how they relate seems to manage this without talking about maximal ideals at all, doesn't it? $\endgroup$ – Christoph Mar 27 '14 at 11:24

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