Why the following map isn't an holomorphic immersion?
$ F:\mathbb{P}^1 \rightarrow \mathbb{P}^2$ $[x:y] \rightarrow [x^3:xy^2:y^3] $
I have to check 3 things:
1)It's holomorphic. ok
2)It's injective. ok
3)Locally, for every p $\in$ X, one of the coordinates of $ F(p) \in F(X)$ gives a local chart for $F(X)$.
I think I have to look at $[1:0]$ to show that 3) is not true, but I don't know how to do it.