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) birational to a $\mathbb{P}^2$-bundle over $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$?

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 $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$?

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.

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) birational to a $\mathbb{P}^2$-bundle over $\mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1$?