Let $f:X\to Y$ be a projective morphism between two normal quasi-projective varieties, and $L$ a $f$-ample line bundle on $Y$. Then the claim is: There is a compactication $\bar{f}:\overline{X}\to\overline{Y}$ and a ** globally ample** line bundle $\overline{L}$ on $\overline{X}$, (i.e., $\overline{L}$ is ample on $\overline{X}$) such that $\overline{f}|_X=f$ and $\overline{L}|_{\overline{X}}\sim_f L$, where $\overline{X}$ and $\overline{Y}$ are normal projective varieties.

Note that if we forget about the line bundle $L$, then getting an $\overline{f}$ satisfying the given properties is not so hard. Indeed there exist (normal) projective varieties $X'$ and $\overline{Y}$ be such that $X\hookrightarrow X'$ and $Y\hookrightarrow\overline{Y}$ are open immersions. Then $f:X\to Y$ gives rise to a rational map $f:X'--->\overline{Y}$. Let $\overline{X}$ be normalization of the graph of this rational map with projection morphisms $g:\overline{X}\to X'$ and $\bar{f}:\overline{X}\to\overline{Y}$. Clearly $g$ is an isomorphism over the open set $X\subset X'$; so identifying $X$ with $g^{-1}X$ we see that $\bar{f}|_{\overline{X}}=f$.

I don't know how to find a ** globally ample line bundle** on $\overline{X}$ which will restrict to a

**on $X$. I realize that this is somewhat of an elementary question, so I apologies in advance for posting it here. But I need to know the proof of this result in order to use it confidently. So any help with references for these sort of ''elementary'' things, or hints on how to prove it will be greatly appreciated!**

*relatively ample line bundle*Quick note: $\overline{L}|_{\overline{X}}\sim_f L$ means $\overline{L}|_{\overline{X}}\cong L\otimes f^*M$ for some line bundle $M$ on $Y$.