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Sorry for my bad English.

Let $X$ be scheme over $Y$, and $\mathscr{L}$ be invertible sheaf on $X$.

By definition of Hartshorne's algebraic geometry (p120),

$\mathscr{L}$ is very ample relative to $Y$, if there is an immersion $i:X\to \mathbb{P}^r_Y$ for some $r$, such that $i^{*}(\mathscr{O}(1))\cong \mathscr{L}$.

On the other hand, Hartshorne's proof of Iv.Prop.3.1(b) (p307) ,

let $D$ be divisor on curve $X$, $D$ is very ample iff correspond morphism $X\to \mathbb{P}^r_k $ is closed immersion.

Why is tihs? Thanks.

Yos
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  • Does this answer your question? https://math.stackexchange.com/questions/315273/how-should-i-think-about-very-ample-sheaves – Marktmeister Jan 05 '21 at 17:55
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    I think the point is just that $X$ is proper in that proposition, so any immersion into projective space is automatically closed. – Niven Jan 06 '21 at 00:06
  • @Niven Thank you very much. I found this Hartshorne's I.Ex.6.5. – Yos Jan 06 '21 at 03:41

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