Let $f:X\rightarrow Y$ be a surjective birational morphism of varieties. Suppose the center of the birational morphism is $Z$ and $f:f^{-1}(Z)\rightarrow Z$ is a $\mathbb{P}^n$-bundle. Consider the relative tangent sheaf $T\_f$. It is obviously torsion sheaf supported on $f^{-1}(Z)$. This torsion sheaf $det$ $T\_f|\_{f^{-1}(Z)}$ is a line bundle on $f^{-1}(Z)$. Can this line bundle be extended to whole of X as a line bundle?


What i am asking is how to define a relative ample line bundle in some canonical way for a birational morphism of above type? For a projective bundle for example the line bundle i am asking is : relative tangent bundle.



relative tangent sheaf
I am not sure I understand the second question, but the answer to the first one is no. Take for $f$ the blowing up of a smooth curve $C$ in $\mathbb{P}^3$. Then $f$ is the projective bundle $\mathbb{P}\_C(N^\*)\rightarrow C$, where $N$ is the normal bundle of $C$ in $\mathbb{P}^3$, and $\det(T\_f)=f^\*\!\det(N)(2)$. Since $\operatorname{Pic}(X)$ is spanned by $f^\*\mathcal{O}\_{\mathbb{P}^3}(1)$ and $\mathcal{O}\_X(f^{-1}(C))$, $\det (T\_f)$ is the restriction of a line bundle on $X$ if and only if $\det(N)$ is the restriction to $C$ of a line bundle on $\mathbb{P}^3$. Then any curve $C$ of degree $d$ and genus $g$ with $d \nmid 2g-2$ gives a counter-example. 


