blow-up of $\mathbb{P}^5$ as a projective bundle

Question

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

Answer 1

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.

Answer 2

Edit. It turns out that what follows is simply the realization of a blowing up of $\mathbb{P}^5$ as a $\mathbb{P}^2$-bundle over $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^1$. This is the blowing up that the OP already knows about, not the blowing up that the OP is asking about.

For three copies of $\mathbb{P}^1$, say $\mathbb{P}(A)$, $\mathbb{P}(B)$ and $\mathbb{P}(C)$, with respective universal invertible quotients, $$q_A:A\otimes_k\mathcal{O}_{\mathbb{P}(A)}\to \mathcal{O}_{\mathbb{P}(A)}(1), $$ $$q_B:B\otimes_k\mathcal{O}_{\mathbb{P}(B)}\to \mathcal{O}_{\mathbb{P}(B)}(1), $$ $$q_C:C\otimes_k\mathcal{O}_{\mathbb{P}(C)}\to \mathcal{O}_{\mathbb{P}(C)}(1), $$ on $X = \mathbb{P}(A)\times \mathbb{P}(B)\times \mathbb{P}(C)$, form the rank 3, locally free sheaf $$\mathcal{E} = \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(A)}(1) \oplus \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(B)}(1)\oplus \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(C)}(1), $$ with the associated quotient, $$ q: (A\oplus B \oplus C)\otimes_k \mathcal{O}_X \to \mathcal{E}.$$ This quotient defines an induced morphism, $$ f :\mathbb{P}_X(\mathcal{E}) \to \mathbb{P}(A\oplus B\oplus C).$$ The three obvious quotients, $$ r_A : \mathcal{E} \to \text{pr}_{\mathbb{P}(A)}^*\mathcal{O}_{\mathbb{P}(A)}(1), \ \ r_B : \mathcal{E} \to \text{pr}_{\mathbb{P}(B)}^*\mathcal{O}_{\mathbb{P}(B)}(1), \ \ r_C : \mathcal{E} \to \text{pr}_{\mathbb{P}(C)}^*\mathcal{O}_{\mathbb{P}(C)}(1), $$ which in turn defines three sections, $$ s_A: X\to \mathbb{P}_X(\mathcal{E}), \ \ s_B : X\to \mathbb{P}_X(\mathcal{E}),\ \ s_C:X\to \mathbb{P}_X(\mathcal{E}), $$ of the projection to $X$. The pairwise spans of these three sections are sub-$\mathbb{P}^1$-bundles, $$ \mathbb{P}_X(\mathcal{E}_{B,C}), \mathbb{P}_X(\mathcal{E}_{A,C}), \mathbb{P}_X(\mathcal{E}_{A,B}). $$ The images of these three subbundles are contracted under $f$ to the three $\mathbb{P}^3$s, $$\mathbb{P}(B\oplus C), \ \ \mathbb{P}(A\oplus C), \ \ \mathbb{P}(A\oplus B).$$ Thus $f$ is a birational, projective morphism that is an isomorphism over the complement of $$\mathbb{P}(B\oplus C) \cup \mathbb{P}(A\oplus C) \cup \mathbb{P}(A\oplus B).$$