When are vector bundles on toric varieties also toric varieties?

Let $X$ be a toric variety, and $\pi:E\to X$ a vector bundle, say of rank $2$. You can think of $X=\mathbb P^1$.

When is the total space of $E$, or of $P(E)$, a toric variety? What do I need in order to lift the torus action on $X$ to a torus action on $E$, or on $P(E)$, so to get an open orbit? I ask this because I was reading this thesis, and on page $2$ I found the following statement, which I do not understand:

If a vector bundle over a toric variety splits as a sum of line bundles, then its projectivization admits a toric variety structure".

I want to understand at least the case of $\mathbb P^1$, where every vector bundle splits. But I have no idea why the splitting is so relevant in general.

Solution 1:

For the case $X = \mathbb{P}^1$, it perhaps desirable to view $X$ as the weighted projective space $X = \mathbb{P}[1,1]$. Then if $E$ is the bundle $\mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_r)$, the total space of $E$ may be identified with the weighted projective space $\mathbb{P}[1,1,a_1, \ldots, a_r]$. The fan for this total space can be described as having generators $v_0, v_1, w_1, \ldots, w_r$ with the relation $v_0 + v_1 + \sum_i a_i w_i = 0$.

For the case of more general $X$, I'd imagine you can do something similar: whereby the fan of the total space can be obtained from the fan of the base by adding a number of generator(s) determined by the rank of the bundle and subjecting these generator(s) to new relation(s) which are modifications of the relations from the fan of the base.