Let us consider the points 
$$p\_1=[1:0:...:0],p\_2 = [0:1:...:0],...,p\_{n-2} =[0:...:0:1],\\ 
p\_{n-1}=[1:1:...:1]\in\mathbb{P}^{n-3}$$ 
and the blow-up $X = Bl\_{p\_1,...,p\_{n-1}}\mathbb{P}^{n-3}$. 


Furthermore, consider 
$$p\_1 = ([0:1],...,[0:1]), p\_2 = ([1:0],...,[1:0]), p\_3=([1:1],...,[1:1])\in (\mathbb{P}^1)^{n-3}$$
and the blow-up $Y = Bl\_{p\_1,p\_2,p\_3}(\mathbb{P}^1)^{n-3}$. Note that for the Picard numbers we have 
$$\rho(X) = n = \rho(Y).$$
Does there exist a small $\mathbb{Q}$-factorial transformation $f:X\dashrightarrow Y$?



Blow-ups of $\mathbb{P}^{n-3}$ and $(\mathbb{P}^1)^{n-3}$
Here is an explicit way to construct a small modification.


Consider the points $p\_1,...,p\_{n-3}$. We have $n-3$ codimension two linear subspaces $H\_{i\_1,...,i\_{n-4}}^{n-5} = \left\langle p\_{i\_1},...,p\_{i\_{n-4}}\right\rangle$. For any choice of $i\_1,...,i\_{n-4}$ we define $\{j\_1,j\_2\} = \{0,...,n-3\}\setminus\{i\_1-1,...,i\_{n-4}-1\}$. Then, the projection from $H\_{i\_1,...,i\_{n-4}}^{n-5}$ is the rational map
$$ 
\begin{array}{lccc}
 \pi\_{i\_1,...,i\_{n-4}}: & \mathbb{P}^{n-3} & \dashrightarrow & \mathbb{P}^1 \\ 
 & \left[x\_0:...:x\_{n-3}\right] & \mapsto & [x\_{j\_1}:x\_{j\_2}]
\end{array} 
$$
We get a rational map 
$$ 
\begin{array}{lccc}
 g: & \mathbb{P}^{n-3} & \dashrightarrow & (\mathbb{P}^1)^{n-3} \\ 
 & x=\left[x\_0:...:x\_{n-3}\right] & \mapsto & (\pi\_{1,...,n-4}(x),...,\pi\_{2,...,n-3}(x))
\end{array} 
$$
Note that the hyperplane $W = \left\langle p\_1,...,p\_{n-3}\right\rangle = \{x\_{n-3}=0\}$ is mapped by $g$ to the point $q\_1=([1:0],...,[1:0])\in (\mathbb{P}^1)^{n-3}$. Furthermore, this is the only divisor contracted by $g$. Therefore, blowing-up $q\_1\in (\mathbb{P}^1)^{n-3}$ we get a small transformation $g\_1:X\_{n-3}^{n-3} = Bl\_{p\_1,...,p\_{n-3}}\mathbb{P}^{n-3}\dashrightarrow Y\_1^{n-3} = Bl\_{q\_1}(\mathbb{P}^1)^{n-3}$ mapping $\widetilde{W}$ (the strict transform of $W$) to the exceptional divisor $E\_{q\_1}$, while the exceptional divisors $E\_{p\_1},...,E\_{p\_{n-3}}$ are mapped to the strict transforms of the $n-3$ divisors in $(\mathbb{P}^1)^{n-3}$ obtained by fixing one the factors.


Furthermore, $g([0:...:0:1]) = ([0:1],...,[0:1])$ and $g([1:...:1]) = ([1:1],...,[1:1])$. Let $\mathcal{U}\subset X\_{n-3}^{n-3}$ and $\mathcal{V}\subset Y\_1^{n-3}$ be the two open subsets on which $g\_1$ is an isomorphism. Now, by applying the universal property of the blow-up we get that $g\_{1|\mathcal{U}}$ lifts to an isomorphism $f:Bl\_{p\_{n-2},p\_{n-1}}\mathcal{U}\rightarrow Bl\_{q\_2,q\_3}\mathcal{V}$. Since $g\_1$ is an isomorphism in codimension one we conclude that $f$ induces a small transformation $f:X = Bl\_{p\_1,...,p\_{n-1}}\mathbb{P}^n\dashrightarrow Y=Bl\_{q\_1,q\_2,q\_3}(\mathbb{P}^1)^{n-3}$ mapping $E\_{p\_{n-2}}$ to $E\_{q\_2}$, and $E\_{p\_{n-3}}$ to $E\_{q\_3}$.


