# Coordinates on $\mathcal O_{\mathbb CP^1}(-n)$

The line bundle $$\mathcal O_{\mathbb CP^1}(-1) = \{(z,\ell)|~z \in \ell \}$$ is a submanifold of $$\mathbb C^2 \times \mathbb CP^1$$ with bundle map the projection. Thus we can the restrict the coordinates of $$\mathbb C^2 \times \mathbb CP^1$$ to coordinates of $$\mathcal O(-1)$$.

What about its powers? Are they also such submanifolds? I was thinking that maybe $$\mathcal O_{\mathbb CP^1}(-n) = \{((z_0^n, z_1^n),[z_0,z_1])\}\subset \mathbb C^2 \times \mathbb CP^1.$$

This would give the right transition functions $$(\frac {z_1}{z_0})^n$$. But I don't think that it is well defined, because the function $$[z_0: z_1] \mapsto (z_0^n, z_1^n)$$ is not injective (look at the roots of unity).

So how would one define coordinates on $$\mathcal O_{\mathbb CP^1}(-n)$$? I am especially interested in the case $$n=2$$.

One way of constructing coordinates on $$\mathcal{O}(n)$$ is to observe that $$\mathcal{O}(-1)$$ is a line bundle, so a local section $$s$$ over $$U \subset \mathbb{CP}$$ will trivialise the bundle over $$U$$.
Taking tensor powers of $$s$$ gives a local section of $$\mathcal{O}(-n) = \mathcal{O}(-1)^{\otimes(n)}$$; i.e. $$s^{\otimes n}$$ trivialises $$\mathcal{O}(-n)$$ over $$U$$.
This construction will show that in the intersection of coordinate charts on $$M$$, we get the desired transition functions, as the tensor product of two 1 dimensional matrices is in fact just scalar multiplication.
If you can't see what I mean, consider $$z$$ as a section over $$U$$, where $$U$$ the coordinate chart \begin{align} \phi: U \subset\mathbb{CP} &\to \mathbb{C} \\ [l:1] &\mapsto l \end{align} If $$z: U \to \mathcal{O}(-1)$$ is a local section $$z \in [l:1] \implies z(l)= c(l)(l, 1)$$ where $$c: \mathbb{C} \to \mathbb{C}$$. A tensor power of this section would then be $$c(l)^n(l^n, 1)$$ as required.