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.

As for the first part of your question, I believe your description is valid. Indeed, they are precisely what I described in the above.

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.


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