Let $\mathfrak X\to \mathbb P^1$ be a projective family, where $\mathfrak X\subset \mathbb P^n\times\mathbb P^1$. Then we have the relative Fano variety of lines $F(\mathfrak X/\mathbb P^1)$; besides, using Segre embedding, $\mathfrak X\subset \mathbb P^n\times \mathbb P^1\subset\mathbb P^{2n+1}$, so we can talk about $F(\mathfrak X)$, the Fano variety of lines in the usual sense.
My question is:
Is $F(\mathfrak X/\mathbb P^1)$ isomorphic to $F(\mathfrak X)$?
(Because the lines in Segre embedding only lies in one fibre, I think it is reasonable to guess they are the same)