factorization morphism between projectives spaces

Question

Please help me with this doubt:

Let $f:\mathbb{P}^1 \rightarrow \mathbb{P}^2$ be a non-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 morphisms $f:\mathbb{P}^1 \rightarrow X$ where $X$ is a projective variety. I am looking for conditions, for example if $X$ is smooth along the image ... I don't know. Please help me, thanks.

Answer 1

In general, no such factorization exists. Consider a morphism given by three general homogeneous polynomials of degree three.Then, the image is of degree 3 and $f$ is not an embedding. For any factorization as above, one sees that $\deg g$ divides 3, but $\deg g$ is a square and thus $g$ must be an isomorphism and thus so $f$ is an embedding, a contradiction.