# Existence of very ample line bundle $M$ s.t. $M \otimes L_i$ is very ample

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.)

• 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. Commented Oct 6, 2021 at 8:43
• @Rushy that looks like most of an answer to me - would you care to flesh it out and record it below? Commented Oct 6, 2021 at 9:05
• @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. Commented Oct 6, 2021 at 9:25
• Actually, turns out II.7.5d) has been asked before on this site, that simplifies things a little. Commented Oct 6, 2021 at 9:34

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 $$i$$.