4
$\begingroup$

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...).

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ Great answer, thanks $\endgroup$ – Gaussian Sep 6 '20 at 12:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.