# Ampleness of a line bundle on the generic fibre

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:

1. Does there exist an open subset $$U$$ of $$S$$ such that $$\mathscr{L}\otimes U$$ is ample on $$f^{-1}(U)$$?
2. 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...).

## 1 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.

• Great answer, thanks – Gaussian Sep 6 '20 at 12:43