On Riemann sphere ${\Bbb P}_{\Bbb C}^1 = {\mathrm{Spec}}\, {\Bbb C}[X] \cup {\mathrm{Spec}}\, {\Bbb C}[Y]$, where the patching is given by $X \mapsto \frac{1}{Y}$, we can define the trivial line bundle. That is, we glue two lines, of which one is ${\Bbb C}$ on each point $x \in {\mathrm{Spec}}\,{\Bbb C}[X]$ and another ${\Bbb C}$ on $y \in {\mathrm{Spec}}\,{\Bbb C}[Y]$ by the identity map. This is the trivial line bundle ${\cal O}_{{\Bbb P}_{{\Bbb C}}^1}$ and the total space $T_{{\cal O}_{{\Bbb P}_{{\Bbb C}}^1}}$ of ${\cal O}_{{\Bbb P}_{{\Bbb C}}^1}$ over ${\Bbb P}_{\Bbb C}^1$ is simply the product space ${\Bbb C} \times {\Bbb P}_{\Bbb C}^1$.
Next choose the constant $\alpha \in {\Bbb C}$ and patch the value $\alpha c$ on the fibre ${\Bbb C}$ over $x \in {\mathrm{Spec}}\,{\Bbb C}[X]$ with the value $c$ on the fibre ${\Bbb C}$ over $y \in {\mathrm{Spec}}\,{\Bbb C}[Y]$, where $x = 1/y$ as usual. This is again the trivial line bundle $L \cong {\cal O}_{{\Bbb P}_{\Bbb C}^1}$.
Q. What is the intuitive figure of the total space $T_{L}$ of $L$ over ${\Bbb P}_{{\Bbb C}}^1$? Is there any trivial model of $T_{L}$?
That is, how can I imagine visually the effect of the multiplication to the line ${\Bbb C}$ over each $x \in {\mathrm{Spec}}\,{\Bbb C}[X]$ as opposed to another ${\Bbb C}$ over each $y \in {\mathrm{Spec}}\,{\Bbb C}[Y]$ being the same?
Question: ". What is the intuitive figure of the total space $T_L$ of $L:=\mathcal{O}_C$ over $C:=\mathbb{P}^1_k$ where $k$ is the complex number field? Is there any trivial model of $T_L$?"
Answer: Let $L:=\mathcal{O}_C e$ and $L^*:=\mathcal{O}_C x$ with $x$ the dual basis of $e$. It follows
$$T_L:=Spec(Sym_{L}^*(L^*)):=Spec(\mathcal{O}_C[x]).$$
If $D(x_0):=Spec(k[t]), D(x_1):=Spec(k[1/t])$ and $\pi: T_L \rightarrow C$ is the canonical map it follows $\pi^{-1}(D(x_0))\cong Spec(k[t,x])$ and $\pi^{-1}(D(x_1)) \cong Spec(k[1/t,x])$. Hence $T_L \cong C\times_k \mathbb{A}^1_k$ is the trivial affine bundle with fiber $\mathbb{A}^1_{\kappa(x)}$ for any point $x\in C$. The map $\pi$ is the projection map $p_C: C \times_k \mathbb{A}^1_k \rightarrow C$ onto the $C$-factor. In general if you take the "trivial" sheaf $E:=\mathcal{O}_C\{e_1,..,e_n\}$ on $C$ it follows $\mathbb{V}(E^*):=Spec(Sym^*(E^*)) \cong C\times_k \mathbb{A}^n_k$ is the trivial rank $n$ geometric vector bundle on $C$ with fiber $\mathbb{A}^n_{\kappa(x)}$. You may do the same construction with any finite direct sum $E:=\oplus_i \mathcal{O}_C(d_i)$, and the corresponding geometric vector bundle $\mathbb{V}(E^*)$ is no longer (globally) trivial. It is locally trivial: for any open set $D(x_i)\subseteq C$ it follows $\pi^{-1}(D(x_i)) \cong D(x_i) \times_k \mathbb{A}^n_k$.