I have a quick question concerning the ampleness of the following line bundle, it came up in an article I've been trying to read.
Suppose that $\pi: Y \rightarrow X$ is a morphism of complex projective varieties. Suppose that $L$ is a very ample line bundle on $X$, and $M$ is a very ample line bundle on $Y$. Why is it true that $\pi^{*}(L^{d}) \otimes M$ an ample line bundle on $Y$ for some large value of $d$?
Note that I am not saying that $\pi^{*}(L) \otimes M^{d}$ is ample for large $d$, but that $\pi^{*}(L^{d}) \otimes M$ is ample. I can verify the former case.
Thank you!