I'd like to compute the Chern number of the tautological bundle of $\mathbb{CP}^1$. Consider $L \subset \mathbb{C}^2\times \mathbb{CP}^1$ given by $$ L \;\; =\;\; \left \{(w_1, w_2, [z_1, z_2]) \; | \; z_1w_2 = z_2w_1 \right \}. $$
I want to compute $c_1(L)$ by considering splitting $\mathbb{CP}^1$ into two neighborhoods $U_1$ and $U_2$ where $\mathbb{CP}^1 = U_1\cup U_2$ and $U_1\cap U_2 := C$ where $C$ is diffeomorphic to $\mathbb{S}^1$. If we then pick symplectic trivializations of $L$ with $U_1$ and $U_2$, then by computing the degree of the parameterization of $C$ we then obtain the Chern class.
Pick $U_1$ and $U_2$ to be \begin{eqnarray*} U_1 & = & \{[z, 1] \; : \; |z| \leq 1\} \\ U_2 & = & \{[1,z] \; : \; |z| \leq 1\}. \end{eqnarray*}
It's then apparent that $U_1 \cap U_2 = C = \left \{ \left[e^{2\pi i\theta},1 \right] \; : \; 0\leq \theta < 1 \right \}$. We can then pick parameterizations $\Phi_i: U_i \times \mathbb{C} \to L$ given by: \begin{eqnarray*} \Phi_1\left ([z,1], w\right ) & = & (wz, w, [z,1]) \\ \Phi_2\left ([1,z], w\right ) & = & (w, wz, [1,z]). \end{eqnarray*}
Because the Chern class is independent of the choice of trivializations we simply need to compute the degree of the transition between these mappings. This is given by \begin{eqnarray*} \left (\Phi_1^{-1}\circ \Phi_2\right )([1,z], w) & = & \Phi_1^{-1}\left (w,wz, [1,z] \right ) \\ & = & \Phi_1^{-1}\left (\frac{wz}{z}, wz, \left [ \frac{1}{z}, 1\right ] \right ) \\ & = & \left (\left [\frac{1}{z},1\right ], wz\right ) \\ & = & \left ([1,z], wz\right ) \end{eqnarray*}
Showing that $w\to wz$ in this transition. This however implies that along the boundary of the two regions $C$, that $\theta \to e^{2\pi i \theta}$.
The problem I see is that the degree of this map $C\to \mathbb{S}^1$ is just $1$, but I've been told that $c_1(L) = -1$. Is there an error in my computation or is there something I'm missing? What is the correct value of the Chern number? Also, is there a straightforward way to take this result and generalize this to the Chern number for the tautological bundle of $\mathbb{CP}^{n-1}$?