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Consider the projection $\pi:\mathbb{F}_n \rightarrow \mathbb{P}^1$. Is it true that the following exact sequence $$0 \rightarrow T_{\pi} \rightarrow T_{\mathbb{F}_n} \rightarrow \pi^*T_{\mathbb{P}^1} \rightarrow 0$$ splits? Since $n=0$ case is trivial, I would like to see what happens when $n \ge 1$. I thought there might be some standard source for this but I could not find it on top of my head.

(Edited): I found a way to check it by using brute-force method from toric charts, but would like to find a simple intrinsic way of doing it. For instance, the Ext space for this exact sequence has positive dimension by using Hirzebruch-Riemann-Roch theorem on $\mathbb{F}_n$, so what makes tangent bundle correspond to the zero element of the Ext?

Changho Han
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    Why is this downvoted? – David Lui Sep 26 '22 at 04:04
  • I have no clue. – Changho Han Sep 26 '22 at 17:45
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    I guess the extension class of the tangent bundle sequence for the projectivization of a vector bundle $E$ on a smooth variety $X$ is induced by the traceless part of the Atiyah class of $E$ (I don't know a reference for this, but I am sure this is known). This implies that the extension class of the Hirzebruch surface is nontrivial if $n > 0$. – Sasha Sep 26 '22 at 19:04
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    Seconded - I find it unbelievable that this gets downvotes. I upvoted it, btw. – Cranium Clamp Sep 27 '22 at 00:31

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