Pullback of tangent bundle Hello everybody please help me with this doubt.
Let $$f:\mathbb{P^1} \rightarrow X$$ where $X$ is a projective non singular variety of dimension $n$. How could I compute the pullback $f^*T_X$? I know that by Grothendieck-Birchoff $$f^*T_X =\sum_{i=1}^nO(a_i)$$ but my problem here is try to find these $a_i$. Please help me - for example, in the case when $X=\mathbb{P}^2$ or $X=\mathbb{K}^3$ or maybe on another easy variety. I want to learn the technique with an example. Thank you.
Remark:
$f$ is a rational curve  i.e $f(\mathbb{P^1})$ birational with $\mathbb{P^1}$
 A: I will give a partial result in the case $X=\mathbb P^2$. I will also assume that $f$ has degree one onto its image. (Because other maps factor through such a map.)
Recall tangent sheaf of $X=\mathbb P^2$ fits into the exact sequence
$$0\to \mathcal{O}\to \mathcal{O}(1)^{\oplus 3}\to T_X\to 0.$$
Pullback via $f^*$, then take $c_1$, denote $d=\deg f(\mathbb P^1)$, one has $$c_1(f^*T_X)=3d.\tag{1}\label{1}$$ 
On the other hand, there is an associated exact sequence $$0\to T_{\mathbb P^1}\to f^*T_{X}\to N_f\to 0,$$
where $N_f$ is the normal sheaf associated to the map $f:\mathbb P^1\to X$. By the fact $T_{\mathbb P^1}=\mathcal{O}(2)$ and $(\ref{1})$, the exact sequence reduces to 
$$0\to \mathcal{O}_{\mathbb P^1}(2)\to f^*T_{X}\to \mathcal{O}_{\mathbb P^1}(3d-2)\to 0 \tag{2}\label{2}$$
whose extension class $e$ lives in $H^1(\mathbb P^1,\mathcal{E})$ where $\mathcal{E}=Hom(\mathcal{O}_{\mathbb P^1}(3d-2),\mathcal{O}_{\mathbb P^1}(2))\cong \mathcal{O}_{\mathbb P^1}(4-3d)$. Since $H^1(\mathbb P^1,O_{\mathbb P^1}(n))\neq 0$ only if $n=-2$, it follows that when $d\neq 2$, the exact sequence $(\ref{2})$ has to splits, so $$f^*T_X\cong \mathcal{O}_{\mathbb P^1}(2)\oplus\mathcal{O}_{\mathbb P^1}(3d-2).$$
When $d=2$, $f$ is an embedding and $f(\mathbb P^1)$ is a smooth conic, and it seems to me that $f^*T_X$ should be one of the following two cases: $$\mathcal{O}_{\mathbb P^1}(2)\oplus \mathcal{O}_{\mathbb P^1}(4), \text{when}\  e=0;$$
$$\mathcal{O}_{\mathbb P^1}(3)\oplus \mathcal{O}_{\mathbb P^1}(3), \text{when}\  e\neq0.$$
But I don't know which case it belongs to. Any comment is welcome!
