Let $D$ be the disk in $\mathbb C^2$ and let $D^\times$ be the puncturned one $D-\{0\}$. Let $C_1$ and $C_2$ be two curves passing through the origin $0$, and $C_1\cap C_2=\{0\}$. We denote the puncturned ones $C_i-\{0\}$ by $C_i^\times $. My question is:
Does there exist a holomorphic morphism $f:D^\times \to\mathbb P^n$ such that $f|_{C_i^\times}$ is a constant map to $x_i$, with $x_1\neq x_2$?
If $f$ is a holomorphic function, then this is impossible by Hartogs's extension theorem. But here $f$ is a morphism, so $f$ may not have an extension to $0$. For example, the inverse map of the blow-up ${Bl}_0 \mathbb P^2 \to\mathbb P^2$ cannot extend to the origin. In some sense, I want to know that if this is the only circumstance which we cannot extend.
I suppose that I should first consider the baby case where $C_1$ is the $x$-axis and $C_2$ is the $y$-axis. I guess the $f$ does not exist, but how can I prove it?