Let $f:X\rightarrow S$ be a morphism of regular schemes and $\mathscr{L}$ a line bundle over $X$. Suppose that $S$ is irreducible and let $\eta$ be its generic point.
Assume that $\mathscr{L}\otimes\kappa(\eta)$ is an ample line bundle on $X_{\eta}$. I have two questions:
- Does there exist an open subset $U$ of $S$ such that $\mathscr{L}\otimes U$ is ample on $f^{-1}(U)$?
- Does there exist an open subset $U$ of $S$ such that $\mathscr{L}\otimes\kappa(s)$ is ample on $X_s$ for all $s\in U$?
If necesary, one can add some more assumptions on $X$, $S$ (noetherianity…) and $f$ (properness…).
Best Answer
In EGA IV part three we find the following result:
Corollary 9.6.4: Let $f:X \to S$ be a proper and finitely presented morphisms of schemes and $\mathcal{L}$ a line bundle on $X$. Then the set $U \subset S$ of $s \in S$ such that $\mathcal{L}_s$ is relative ample for $f_s$ is open in $S$, and the restriction of $\mathcal{L}$ to $f^{-1}(U)$ is relative ample for $f:f^{-1}(U) \to U$.
This answers your second question because $\mathcal{L}_s$ being relatively ample for $f_s$ is equivalent to $\mathcal{L}_s$ being ample on $X_s$. Your first question follows by taking an affine open $V \subset U$ and applying part 3 of https://stacks.math.columbia.edu/tag/01VJ.