I am wondering if the blow-up of $\mathbb{P}^5$ along three disjoints $\mathbb{P}^1$ (say in generic position) can be understood as a projective bundle over some nice (Fano?) variety.


If one considers instead the blow-up of $\mathbb{P}^5$ along the union of three $\mathbb{P}^3$ in generic position, then it is easily proved that it is the projectivization of $\mathcal{O}(-1,0,0) \oplus \mathcal{O}(0,-1,0) \oplus \mathcal{O}(0,0,-1)$ over $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$.


So I thought that something similar could be true if the union of the three $\mathbb{P}^3$ was replaced by the union of three lines. I thought it could also be a $\mathbb{P}^2$-bundle over $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$. 


At least the Picard rank ($4$) and the rank of the K-theory ($18$) are the same for a $\mathbb{P}^2$-bundle over $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$ and for the blow-up of $\mathbb{P}^5$ along $3$ disjoint lines.


Many thanks in advance.


**EDIT** : Following the suggestion of Borisov, I ask a weaker a question : Is this blow-up of $\mathbb{P}^5$ along three disjoint lines (in generic position) $K$-equivalent to a $\mathbb{P}^2$-bundle over $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$?



blow-up of $\mathbb{P}^5$ as a projective bundle
Consider the six points $p\_0=(1:0:0:0:0:0)$, $p\_1=(0:1:0:0:0:0)$, and similarly $p\_2,\dotsc,p\_5$. Then one can take the 3 lines $\overline{p\_0p\_1}$, $\overline{p\_2p\_3}$, and $\overline{p\_4p\_5}$, which are in general position. These lines are toric subvarieties of $\mathbb{P}^5$ with the standard toric structure, so the blow-up $X\to\mathbb{P}^5$ along the three lines is a toric variety. Using toric geometry, one can compute the cone $\text{NE}(X)$ of effective curves, its extremal rays and the associated contractions. It turns out that $X$ is weak Fano, i.e. $-K\_X$ is nef and big, and $\text{NE}(X)$ has 6 extremal rays: 3 correspond to the 3 blow-ups, while the other 3 are small and have intersection zero with $K\_X$. In particular, $X$ has no projective bundle structure.
The loci $E\_i$ ($i=1,2,3$) of the 3 small rays are the transforms of the 3-dimensional linear subspaces of $\mathbb{P}^5$ containing 2 of the 3 lines, and they are pairwise disjoint. Each $E\_i$ is isomorphic to the blow-up of $\mathbb{P}^3$ along 2 lines, and it is a $\mathbb{P}^1$-bundle over $\mathbb{P}^1\times \mathbb{P}^1$. The small contraction restricts on $E\_i$ to the projection to $\mathbb{P}^1\times \mathbb{P}^1$.
One can flop these 3 small rays, and get a new smooth weak Fano variety $X'$ (still toric). This $X'$ is a $\mathbb{P}^2$-bundle over $\mathbb{P}^1\times\mathbb{P}^1\times \mathbb{P}^1$, more precisely: 
$$X'=\mathbb{P}\_{\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1}\left(\mathcal{O}(1,1,0)\oplus\mathcal{O}(1,0,1)\oplus\mathcal{O}(0,1,1)\right).$$
(Attention: I am using Grothendieck's convention for Proj, while it seems to me that Libli uses the opposite one.) The $\mathbb{P}^2$-bundle has 3 special sections $S\_i$ ($i=1,2,3$) with normal bundle $\mathcal{O}(-1,-1,0)\oplus\mathcal{O}(-1,0,-1)$ and so on. Each $S\_i$ is the locus of a small extremal ray of $X'$, having intersection zero with $K\_{X'}$, corresponding to one of the 3 flops.


