# Constructing a very ample line bundle on a projective bundle

Let $$X$$ be a smooth complex projective variety and $$p:Y\to X$$ be a locally trivial in analytic topology $$\mathbb CP^k$$-bundle. Suppose we have a line bundle $$L$$ on $$Y$$, restricting to $$\mathcal O(1)$$ on $$\mathbb CP^k$$-fibres.

Question. Is it true that there is a line bundle $$L'$$ on $$X$$ such that $$p^*L'\otimes L$$ is very ample on $$Y$$?

I think one could be able to prove this using Serre's (+ Kodaira?) vanishing, but I can't prove it so far.

Idea. Here is an idea of how one could try to solve this. So, first all, one can take $$L''$$ such that $$p^*L''\otimes L$$ is ample on $$Y$$. Next, one can try to use Kodaira vanishing, it says that $$K_Y\otimes p^*L''\otimes L$$ has zero higher cohomology. Now, we could take the line bundle $$K_Y\otimes (p^*L''\otimes L)^{k+2}$$, and this bundle will restrict to each fibre as $$\mathcal O(1)$$. By taking $$L''$$ positive enough, it should be possible to make $$K_Y\otimes (p^*L''\otimes L)^{k+2}$$ ample. I think that since this bundle is ample and its higher cohomology vanish, by Grothendiek-Riemann-Roch it will have a lot of sections (especially if $$L''$$ is very ample and has a lot of sections). This seems to be not far from proving that $$K_Y\otimes (p^*L''\otimes L)^{k+2}$$ is very ample... And I guess this bundle is $$p^*L'\otimes L$$ for some $$L'$$?

• If I understand you correctly, you are assuming $Y$ is algebraic and $p$ is a morphism in the category of algebraic varieties. Then, it is fairly easy to reduce to the case when $Y=\mathbb{}^k\times X$ and then the assertion is trivial. – Mohan Dec 3 '19 at 18:02
• Dear Mohan, thanks for this comment! You understand me correctly. But I am not familiar with the notation $Y=^k\times X$. Do you think you could write a complete answer? If you do, would you be so kind, please, to write an answer as an answer and not as a comment? – aglearner Dec 3 '19 at 18:10
• @aglearner: I suppose Mohan meant to write $\mathbb CP^k\times X$, but possibly used his own macro for $\mathbb CP$ which the system did not recognize, and hence ignored... :) – Sándor Kovács Dec 4 '19 at 3:02

Since $$R^ip_*L=0$$ for $$i>0$$, by semi-continuity theorem, you see that $$p_*L$$ is a vector bundle of rank $$k+1$$. You can twist by a sufficiently ample bundle $$L'$$ on $$X$$ to make it globally generated. Thus, you have $$O_X^m\to p_*L\otimes L'$$ surjective and thus you get an embedding $$Y\subset X\times \mathbb{P}^{m-1}$$. Further, the $$O(1)$$ of the second factor restricts to $$L\otimes p^*L'$$ on $$Y$$. Now, taking a very ample line bundle $$M$$ on $$X$$, $$M\otimes O(1)$$ is very ample on $$X\times\mathbb{P}^{m-1}$$ and then its restriction $$L\otimes p^*(L'\otimes M)$$ is very ample on $$Y$$.