In order to obtain an explicit description of the diffeomorphism, one can work as follows.

Take $B=\mathbb{C}^{n-1}$, with coordinates $t_1, \ldots, t_{n-1}$, and consider the complex space
$\mathcal{X}$ obtained glueing $\mathbb{P}^1 \times \mathbb{C} \times B$ with $\mathbb{P}^1 \times \mathbb{C} \times B$ by the identification of 

$(y_0, y_1, z, t_1, \ldots, t_{n-1})$ with $(y_0', y_1', z',t_1, \ldots, t_{n-1})$

if 

$z'=z^{-1}, \quad y_1'=y_1z^{-n}, \quad y_0'=y_0+y_1 \Sum_{i=1}^{n-1}t_iz^{-i}$.

Let us denote by $\pi \colon \mathcal{X} \to B$ the family obtained in this way.

Let now $T_k \subset B$ be the determinantal locus given by

rank \begin{bmatrix}
t_1 & \ldots & t_{k+1} \cr
t_2 & \ldots & t_{k+2} \cr
\cdot & \cdot & \cdot \cr
\cdot & \cdot & \cdot \cr
\cdot & \cdot & \cdot \cr
t_{n-k-1} & \ldots & t_{n-1}
\end{bmatrix} $\leq k$

Then, if $t \in T_k - T_{k-1}$ we have $X_t \cong \mathbb{F}_{n-2k}$.

By using Ehresmann theorem, one concludes that $\mathbb{F}_n$ and $F_{n-2k}$ are diffeomorphic.