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Sorry for my bad English.

Let $X$ be projective variety over algebraically closed field $k$, and $L_1,\dots, L_n$ be any line bundles on $X$.

Now can we construct very ample line bundle $M$ on $X$ such that $M\otimes L_i$ is also very ample for $1\le i\le n$?

(In Mumford's "Abelian varieties" p.145 of new edition, he says this is possible.)

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  • $\begingroup$ You could take a look at exercise II.7.5 in Hartshorne's Algebraic Geometry. For large enough $m$, the $\mathcal L_i \otimes \mathcal{O}(m)$ are generated by global sections since $\mathcal O(1)$ is (very) ample, then II.7.5d) tells you that the $\mathcal L_i \otimes \mathcal{O}(m + 1)$ are very ample. $\endgroup$
    – Rushy
    Commented Oct 6, 2021 at 8:43
  • $\begingroup$ @Rushy that looks like most of an answer to me - would you care to flesh it out and record it below? $\endgroup$
    – KReiser
    Commented Oct 6, 2021 at 9:05
  • $\begingroup$ @KReiser, unfortunately I'm stuck with just my phone for the time being, and I'm not aware of a short proof of II.7.5d) (the proof I know is the one in Liu's book using the Segre embedding), but without this I feel like it wouldn't be a complete answer. $\endgroup$
    – Rushy
    Commented Oct 6, 2021 at 9:25
  • $\begingroup$ Actually, turns out II.7.5d) has been asked before on this site, that simplifies things a little. $\endgroup$
    – Rushy
    Commented Oct 6, 2021 at 9:34

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Let $X$ be your projective variety, and let $\mathcal O(1)$ be a very ample sheaf on $X$ associated to a closed immersion of $X$ into $\mathbb P_k^r$.

Because $\mathcal O(1)$ is ample (since it is very ample) and invertible sheaves are coherent, for every $i$ you have that there exists an integer $m_i > 0$ such that for all $m > m_i$, the sheaf $\mathcal L_i \otimes \mathcal O(m)$ is generated by global sections (here, $\mathcal L_i$ is the invertible sheaf corresponding to the line bundle $L_i$)

Let now $M$ be some integer larger than all the $m_i$. Then for all $i$, the sheaves $\mathcal L_i \otimes \mathcal O(M)$ are generated by global sections.

By exercise II.7.5d) in Hartshorne's Algebraic Geometry (see for example this post), the sheaves $\mathcal L_i \otimes \mathcal O(M + 1)$ are very ample.

Thus, if you take $B$ to be the very ample line bundle associated to the very ample sheaf $\mathcal O(M + 1)$, you have that $B \otimes L_i$ is very ample for all $i$.

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