Let $f:X\rightarrow Y$ be a morphism of finitely generated schemes over a $\mathbb{C}$, and $L$ be a line bundle on $X$ which is ample on all closed-fibers. I.e. for every closed point $y\in Y$, the bundle $L_y$ on $X_y$ is ample.
Does it follow that $L$ is relatively ample over $f$?
Note that I am not assuming that $f$ is a proper map, otherwise the above is true and can be proved by the theorem on formal functions.
A very related (though not exactly equivalent) question: can you chech the property of being a quasi-affine morphism on closed points?
Thank you!