# Transition functions of toric projective bundle (Proposition in [Cox, Toric Varieties])

My reference: David Cox's "Toric Varieties"

My question is the proof of Proposition 7.3.3.

Proposition 7.3.3. The cones {$\sigma_i$ | $\sigma \in \Sigma$, $i = 0,\dots,r$} and their faces form a fan $\Sigma_{\mathcal{E}}$ in $N_{\mathbb{R}} \times \overline{N}_{\mathbb{R}}$ whose toric variety $X_\mathcal{E}$ is the projective bundle $\mathbb{P}(\mathcal{E})$.

I already proved the first part, that is, $X_\mathcal{E}$ is a fibration over $X_{\Sigma}$ with fiber $\mathbb(P)^r$.

That is, \begin{equation*} \begin{split} p^{-1}(U_{\sigma}) \simeq & \ U_{\sigma} \times X_{\Sigma_{\theta} , N_{\theta}}\\ \simeq & \ U_{\sigma} \times X_{\Sigma_{\theta} , \overline{N}}\\ \simeq & \ U_{\sigma} \times \mathbb{P}^r \end{split} \end{equation*}

where everything comes from the following short exact sequence and splitting theorem of fans:

\begin{equation*} 0\longrightarrow N_{\theta} = \ker(\overline{p}) \longrightarrow N \times \overline{N} \xrightarrow{\overline{p}} N \longrightarrow 0 \end{equation*}

My Question

I don't know how to prove the second part in the proof:

"Furthermore, working over an affine open subset of $X_{\Sigma}$, one sees that $X_{\Sigma}$ is obtained from $V_\mathcal{E}$ by the process described in 7.0."

How do I prove that? What should I check? And the author only say that "leave details as exercise", but the exercise is just "Complete the proposition 7.3.3". So, I have no idea to check this argument.

Thank you for your help!

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