Hello everybody please help me with this doubt.. let $f:\mathbb{P}^1 \rightarrow \mathbb{P}^2$ no constant morphism ..is there any factorization of $f$ as $$\mathbb{P}^{1} \overset{h}{\rightarrow}\mathbb{P}^{2}\overset{g}{\rightarrow}\mathbb{P}^{2}$$ where $h$ is an embedding and $g$ is the finite type such that $c_{1}(f^{*}T_{\mathbb{P}^2})=c_{1}(h^{*}T_{\mathbb{P}^2})$ or maybe $f^{*}T_{\mathbb{P}^2}=h^{*}T_{\mathbb{P}^2}$..($c_1$ is the first chern class)..or is there a general result for morphism $f:\mathbb{P}^1 \rightarrow X$ where $X$ is a projective variety i am loking for conditions for example if $X$ is smooth along the image ..i dont know..please help me thanks.

  • $\begingroup$ I guess that P1 and P2 are the COMPLEX projective spaces? Please specify which kind of "morphisms" and "embeddings". Isn't that a question in algebraic geometry rather than differential topology? $\endgroup$ – Gael Meigniez Feb 11 at 7:27

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