# Transition functions for the tautological bundle

Define the tautological bundle over $CP^1$ to be $\tau = \{[a_1, a_2], (z_1, z_2) \in CP^1\times\mathbb{C}^2 | \exists \lambda \in \mathbb{C} \;\text{such that} \;\lambda (z_1,z_2) = (a_1, a_2) \}.$

Then $\tau$ trivializes over open sets $U_i = \{[a_1,a_2] | a_i \neq 0\}, i=1,2$ where $(\pi, \phi_i):\pi^{-1}(U_1) \rightarrow CP^1\times\mathbb{C}$ is given by $\phi_i([a_1,a_2]) = a_i.$ Note that $(\pi, \phi_1)^{-1}:U_1\times\mathbb{C} \rightarrow \pi^{-1}(U_1)$ is given by $([a_1,a_2], w) \rightarrow ([a_1, a_2], (w, \frac{a_2}{a_1}w)).$

Thus the transition function $T_{12}$ is given by $[a_1, a_2] \rightarrow \frac{a_2}{a_1}.$ Up to homotopy we need only look at the map on the equator of $CP^1 = S^2.$ Since we can identify $U_1$ with $S^2 \setminus \infty = \mathbb{C}$ by $[a_1,a_2] \rightarrow \frac{a_2}{a_1},$ the transition function $T_{12}$ appears to be a degree one map $S^1 \rightarrow S^1.$

Is this right? If so, why is this bundle always denoted $O_\mathbb{P}(-1)?$ Shouldn't it be called $O_\mathbb{P}(1)?$

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