I don't fully understand how the gluing of the affine parts of a toric variety exactly works. I have a hard time developing a common sense or any intuition how to tell the result of a gluing morphism immediately and I would appreciate any help in this direction. I'm missing the concrete detailed math behind that as well, would be cool if someone can give more details here. The two examples I am looking at:
In both cases, we have the original lattice $N=\mathbb{Z}^2$ and the dual $M=\mathbb{Z}^2$.
Projective Space
Let's start with the given fan of $X=\mathbb{P}^2$: We can concretely calculate the toric varieties $X_{\sigma_i}$ and then glue them together along the walls $\tau_{ij} = \sigma_i \cap \sigma_j$. In general we have
$X_{\sigma} = \text{Spec} (R_{\sigma}) = \text{Spec}(\{f\in \mathbb{C}[z_1^{\pm 1},z_2^{\pm 1}]: \text{supp } f \subset S_{\sigma} = \check{\sigma} \cap M\})$
where $\check{\sigma}$ is the dual of $\sigma\subset N$ in $M$. So looking at $\sigma_1$ we get $\check{\sigma}_1 = \text{Cone}(-e_1, -e_1+e_2)$ which converts to $R_{\sigma_1} = \mathbb{C}[z_1^{-1},z_1^{-1}z_2]$ and thus $X_{\sigma_1} = \mathbb{C}^2_{(z_1^{-1}, z_1^{-1}z_2)}$ (the coordinates are noted as subscripts).
Analogously we can calculate $X_{\sigma_0} = \mathbb{C}^2_{(z_1, z_2)}$ and $X_{\sigma_2} = \mathbb{C}^2_{(z_2^{-1}, z_1z_2^{-1})}$.
Let's start by gluing $X_{\sigma_0}$ and $X_{\sigma_1}$ along $(z_1,z_2) \mapsto (z_1^{-1}, z_1^{-1}z_2)$. How can I picture this gluing morphism? In my understanding we get that $(z_1,z_2)_{\sigma_0} \sim (z_1^{-1},z_1^{-1}z_2)_{\sigma_1}$, where $\sim$ is the equivalence relation of the gluing morphism. What are the steps to get that $(X_{\sigma_0} \cup X_{\sigma_1}) / \sim$ is the same as $\mathbb{P}^2 \setminus \{(0:0:1\}$?
What I thought about is the following: As we are matching $z_1$ and $z_1^{-1}$ we need $\mathbb{P}^1$, $\mathbb{C}$ is not sufficient anymore. Therefore I would write the above glued variety as $\{((t_0:t_1), z_2): \frac{t_0}{t_1}=z_1\}\simeq \mathbb{P}^1 \times \mathbb{C}$. However two questions arise from this set: First, why aren't we taking $\frac{t_1}{t_0}=z_1$ and second where did $z_1^{-1}z_2$ go?
Assume I somehow managed this part, I get completely lost with the next gluing! Do I now have to glue $X_{\sigma_2}$ along two rays, do I need to glue all affine patches pairwise and then glue everything together or can I glue $X_{\sigma_2}$ directly with $(X_{\sigma_0} \cup X_{\sigma_1}) / \sim$ and if so, how can I do that? I would really appreciate a detailed answer because everywhere in the literature this seems to be "trivial".
Hirzebruch surface
The calculation of each affine patch is no problem for me. We get $$ X_{\sigma_1} = \mathbb{C}^2_{(z_1,z_2)} \\ X_{\sigma_2} = \mathbb{C}^2_{(z_1,z_2^{-1})} \\ X_{\sigma_3} = \mathbb{C}^2_{(z_1^{-1},z_1^{-r}z_2^{-1})} \\ X_{\sigma_4} = \mathbb{C}^2_{(z_1^{-1},z_1^rz_2)} \\ $$
How can I glue them together? As before we can glue $X_{\sigma_1}$ with $X_{\sigma_2}$ along the second coordinate (that means we identify $(z_1,z_2)_{\sigma_1}$ with $(z_1,z_2^{-1})_{\sigma_2}$) and $X_{\sigma_3}$ with $X_{\sigma_4}$ along the second coordinate as well.
For simplification define $X_I:=(\bigcup_{i\in I}X_{\sigma_i}) / \sim $
These two gluings give $$X_{12} = \{(z_1, (t_0:t_1)): \frac{t_0}{t_1}=z_2\}\simeq \mathbb{P}^1 \times \mathbb{C}$$ and $$X_{34} = \{(z_1^{-1}, (s_0:s_1)): \frac{s_0}{s_1}=z_1^rz_2\}\simeq \mathbb{P}^1 \times \mathbb{C}$$
Again the (small) question, can we choose $\frac{t_1}{t_0}=z_2$ or $\frac{s_1}{s_0}=z_1^rz_2$ as well?
So far so good. We can now glue those two varieties along the first coordinate and that's where I get stuck. I would suggest something of a similar form $\{((u_0:u_1), (s_0:s_1)):\: ...\}$. However the solutions seems to be $$ X_{1234} = \{((\lambda_0:\lambda_1:\lambda_2), (\mu_0:\mu_1)) : \lambda_0\mu_0^r = \lambda_1\mu_1^r\} $$
How is this equation derived and what is the intuition behind that? In general, what happens if we don't have the ability to glue some $f(z)$ and $f^{-1}(z)$ together (in above examples it was $f(z)=z_1$, $f(z)=z_2$ or $f(z)=z_1^rz_2$ ) for a resulting $\mathbb{P}^1$? How would be the approach in those cases? I would be glad if someone can clarify those questions.