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This is extension of my previous question Moduli space about $CP^{N-1}$ and $T^* CP^{N-1}$., meaning of $\mathcal O(-1)$ in algebraic geometry? .

What i have been considered are followings First briefly state the problem, i want to find the moduli space of \begin{align} |\phi_1|^2 + |\phi_2|^2 - |\phi_3|^2 - |\phi_4|^2 =r \end{align} where $\phi$ is complex and the moduli space is the solutions of above equation with quotient of $U(1)$. (I already know the answer for this moduli space $\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$.

First checking its dimension, each $\phi$ has 2 real dimension, and constriant(above equation) and quotient of $U(1)$ gives each 1 degrees of freedom, thus \begin{align} 8-1-1=6 \end{align} which gives 6 real dimension, $i.e$ 3 complex dimension. For $\mathcal O(-1) \oplus \mathcal O(-1)$ over $CP^1$. each taughtological line bundle has $1$ complex dimension and $CP^1$ has 1 complex dimension thus its space has consist of 3 complex dimension.

Nextly to construct, local trivialization, divide this space as $U_0$ and $U_1$. where $U_0 = \{ z_0 = \frac{\phi_4}{\phi_3} \in \mathbf{C}, \,\ \phi_3 \neq 0\}$. and $U_1 = \{ z_1 = \frac{\phi_3}{\phi_4} \in\mathbf{C}, \,\ \phi_4 \neq 0 \}$.

For $U_0$, from the equation \begin{align} &|\phi_1|^2 + |\phi_2|^2 -(1+ |z_0|^2)|\phi_3|^2 =r \\ & |\phi_1|^2 = r + (1+ |z_0|^2)|\phi_3|^2 -|\phi_2|^2 \end{align} where the $\phi_1$ is constrcuted by $\phi_2$, $\phi_3$ freely. Thus i see following local trivalization \begin{align} \Phi_{U_0} : E \rightarrow &U_0 \times C^2 \\ & \left( \frac{\phi_4}{\phi_3}, (\phi_2, \phi_3) \right)=\left( z_0, (\phi_2, \phi_3) \right) \end{align}

And do similar things on $U_1$ i see \begin{align} \Phi_{U_1} : E \rightarrow &U_1 \times C^2 \\ & \left( \frac{\phi_3}{\phi_4}, (\phi_2, \phi_4) \right) = \left( z_1, (\phi_2, \phi_4) \right) \end{align}

The next task which left to me is find the transition map between $\Phi_{U_0}$ and $\Phi_{U_1}$.

I know for $\mathcal O(k)$ over $CP^1$ cases; $i.e$, by taking local trivialization for $U_0 \times C$, and $U_1 \times C$. Then formulating inverse relation and their transition order, we can find its transition function. see shanyuji-Geometry lecture note 5,

Can you give me some explict transition function for the above case($U_i \times C^2$)? $i.e$ what i known from above, similar with reference set $z_0 = \frac{1}{z_1}$. which is exchange of $\phi_3 \leftrightarrow \phi_4$. but in this case still $\phi_2$ exists as a free variable. Thus i failed up constructing transition function betwen them....

Any comment or answer will be helpful in extending understanding of this topics. Thanks

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