Let $D$ be the disk in $\mathbb C^2$ and let $D^\times$ be the puncturned one $D-\{0\}$. Given a holomorphic morphism $f:D^\times \to\mathbb P^n$, we want to know whether we can extend it to the whole disk $D$. If $f$ is a holomorphic function, then this is done by Hartogs's extension theorem. But here $f$ is a morphism, so $f$ may not have an extension to $0$, for example the inverse (rational) map of the blow-up ${Bl}_0 \mathbb P^2 \to\mathbb P^2$. I want to know in some sense if this is the only circumstance which we cannot extend. More precisely, I guess the following is true:
If there exists a sequence $(p_i)$ tends to $0$ such that $f(p_i)=x$ for all $i$ and some constant point $x\in \mathbb P^n$, then $f$ can be extended to $0$.
This is related to my former question, but they are independent. The relation is, if the answer to that question is positive, then it provides a holomorphic morphism wich cannot be extended, hence gives negative answer to this one.
Actually this is wrong. A simple counter-example is $$f:\mathbb C^2-\{0\} \to \mathbb P^2$$ $$(z,w)\mapsto[z:w:zw]$$ Easy to see the line $\{z=0\}$ is sent to $[0:1:0]$ and $\{w=0\}$ is sent to $[1:0:0]$. So, we cannot extend it to the whole $\mathbb C^2$.