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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.

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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.

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  • $\begingroup$ how $det(T_f)=f^*det (N)(-2)$?....$det T_f$ should be relatively ample right?? $\endgroup$ – S.D. Oct 8 '17 at 11:12
  • $\begingroup$ suppose $L$ be a line bundle restrics to $f^*det N$ ..then it is clear that restriction of $L$ along each fibre is trivial...is it enough to say it is pullback of a line bundle from below?? $\endgroup$ – S.D. Oct 8 '17 at 11:25
  • $\begingroup$ 1) Yes, sorry, it is $f^*\det(N)(2)$. Corrected. $\qquad$ 2) Yes, for a projective bundle $f:P\rightarrow C$ this is enough, because $\operatorname{Pic}(P)= f^*\operatorname{Pic}(C)\oplus \mathbb{Z}[\mathcal{O}_P(1)] $. $\endgroup$ – abx Oct 8 '17 at 15:58
  • $\begingroup$ sorry if i am missing an easy point... $X=Blow_C \mathbb{P}^3\rightarrow \mathbb{P}^3$ is not a projective bundle...so i asked $\endgroup$ – S.D. Oct 8 '17 at 19:18
  • $\begingroup$ $f^*\det(N)$ is a line bundle on $f^{-1}(C)$, which is a projective bundle over $C$. $\endgroup$ – abx Oct 8 '17 at 19:30

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