Fix a base field $k$. Let $i:C(\simeq\mathbb{P}^1) \hookrightarrow \mathbb{P}^n_k$ be a rational curve on a projective space. Is there a formula about $i^*T_{\mathbb{P}^n}$?
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2$\begingroup$ That depends on the curve. $\endgroup$– SashaCommented Dec 5, 2022 at 11:51
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$\begingroup$ If $C$ has degree $1$, is there a formula about $i^∗T_{\mathbb{P}^n}$? $\endgroup$– user145752Commented Dec 5, 2022 at 12:51
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2$\begingroup$ If $C$ is a line, then $i^*T_{\mathbb{P}^n} \cong \mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus (n-1)}$. $\endgroup$– SashaCommented Dec 5, 2022 at 13:22
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$\begingroup$ Could you please give me a reference? I would like to understand how to show it. $\endgroup$– user145752Commented Dec 6, 2022 at 3:06
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2$\begingroup$ @user145752 the normal bundle of a line in P^n is n-1 copies of O(1). Now $ i^* \mathbb{P}^n $ sits in between O(2) and the normal bundle, so has to be the direct sum because of vanishing Ext.. $\endgroup$– Cranium ClampCommented Dec 6, 2022 at 6:12
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