Let $f: X \to Y$ be a morphism between $k$-schemes where $Y$ is irreducible with unique generic point $\eta$. Set $F := f^{-1}(\eta)$ as the generic fiber and consider a coherent sheaf $\mathcal{G}$ on $X$.
Let assump that $\mathcal{G}$ has follwing property:
The restriction $\mathcal{G} \vert _F$ is invertible, so locally free of rank $1$.
Notice that since $F$ is generally not an open subset the "restriction" is the pullback $i_F^*\mathcal{G}$ for canonical inclusion morphism $i_F: F \to X$.
I would like to know how to prove that the property above is extendabl in follwing sense: the claim is that there exist an open $F \subset U \subset _o X$ subscheme of $X$ such that the restriction $\mathcal{G} \vert _U$ remains invertible.
My Ideas: Maybe to go pointwise in the sense by considering an arbitrary point $x \in F$ and trying to find an open neighbourhood $U_x$ of it with desired property. Then take as $U $ the union of all $U_x$. I see the advantage of this approach because we can work locally and therefore assuming that $X$ is affine. Does it work or do I need here another better approach?