Consider a vector bundle $\pi:E\to\mathbb{C}P^{1}$. We know that we can make the decomposition $$ E=\mathcal{O}(q_{1})\oplus\mathcal{O}(q_{2})\oplus\cdots\oplus\mathcal{O}(q_{n}) \text{ , } q_{i}\in\mathbb{Z} $$ I am wondering when the total space of this vector bundle can be described as a toric variety.
An answer to this question claims without reference or proof that the total space is simply the weighted projective space $\mathbb{P}(q_{1},q_{2},\cdots,q_{n})$, which is clearly a toric variety. However to my knowledge, weighted projective spaces are only defined for strictly positve weights, which is not necessarily the case for the weights appearing in $E$.
With this in mind, I guess my quesiton boils down to the following sub-questions.
- Is there a good notion of weighted projective space when some of the weights are negative or zero?
- Is the total space of $E$ such a weighted projective space, or does this only hold for positive weights?
- If yes, how can I see that the total space of $E$ is such a weighted projective space?