Let $X$ be a smooth, proper variety, and $f: \mathbb{P}^1 \to X$ be a finite morphism. Then it is claimed that $f^* T_X$ contains $T_{\mathbb{P}^1}$ as a subsheaf. However, I did not see the reason.
I think we have exact sequence $$f^* \Omega_{X} \to \Omega_{\mathbb{P}^1} \to \Omega_{\mathbb{P}^1/X} \to 0.$$
Taking the dual sheaf, we have
$$0 \to (\Omega_{\mathbb{P}^1/X})^\vee \to T_{\mathbb{P}^1} \to f^*T_{X}.$$
Then the problem because why $(\Omega_{\mathbb{P}^1/X})^\vee=0$?
