# factorization morphism between projectives spaces

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.

• 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? – Gael Meigniez Feb 11 at 7:27