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Let $ X $ and $Y$ be noetherian schemes, and $f: X\to Y$ a morphism of finite type. Suppose $L$ is an ample invertible sheaf on $X$.

I want to know whether $L^n$ is very ample on $X$ relative to $Y$ for sufficiently large $n$.

From Hartshorne chapter II theorem 7.6, we know this assertion is true when $Y$ is affine. For general case, I want to choose a finite open affine cover of $Y$. But we know an ample invertible sheaf resticted to an open subet $U\subset X$ may not ample over $U$. Therefore, I suspect this assertion may be wrong. Can someone give me a counterexample or give it a proof? Thanks a lot.

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  • $\begingroup$ Being very ample on $X$ is a stronger condition than being very ample relative to a morphism, so shouldn't this be true automatically (given that $L$ is ample hence some $L^n$ is very ample)? $\endgroup$
    – Tabes Bridges
    Commented Mar 31, 2023 at 23:32
  • $\begingroup$ Sorry, I can't get your point……You mean this assertion is ture? @Tabes Bridges $\endgroup$
    – ZhouQi
    Commented Apr 1, 2023 at 0:09
  • $\begingroup$ Is very ample not a relatitive notion? What is meaning of "being very ample on $X$" without a morphism? @Tabes Bridges $\endgroup$
    – ZhouQi
    Commented Apr 1, 2023 at 0:12
  • $\begingroup$ "Very ample on $X$" means "very ample relative to the structure map $X \to \operatorname{Spec}(k)$." $\endgroup$
    – Tabes Bridges
    Commented Apr 1, 2023 at 1:48
  • $\begingroup$ My point is that your hypothesis of $L$ ample on $X$ implies that $L^n$ is very ample, hence induces a closed immersion $X \to \mathbb P_k^n$. If this map is a closed immersion, then it remains a closed immersion when restricted to the fibers of $f$, showing that $L^n$ is very ample relative to $Y$. $\endgroup$
    – Tabes Bridges
    Commented Apr 1, 2023 at 1:50

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The confusion in the comments is because Hartshorne uses a definition that disagrees with the definition in EGA, as Hartshorne notes when he introduces his definition. If you use the definition in EGA, then a sufficiently positive tensor power of a relatively ample invertible sheaf on a Noetherian scheme is also relatively very ample. However, if you use the definition in Hartshorne's book, this is not true. Hartshorne includes an exercise calling attention to this disparity, and he immediately follows it by an exercise that shows, nonetheless, that a sufficiently positive tensor power of the "twist" of a relatively ample invertible sheaf on $X$ by the pullback of a sufficiently ample invertible sheaf on $Y$ does satisfy Hartshorne's definition of "relatively very ample." So Hartshorne is quite "up front" about the disparity.

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