Assume $f: \mathbb {CP}^n \dashrightarrow\mathbb {CP}^m$ is a rational map, and $B$ is the fundamental locus (means on which $f$ cannot define). Take $b\in B$. Is it possible that there exists $u\neq v$ in the target, such that the closures of $f^{-1}(u)$ and $f^{-1}(v)$ are both tangent to a same line $L\simeq\mathbb P^1$ passing through $b$ at $b$?
I guess this is not possible. If $B$ is smooth, it would be clear as we can blowup along $B$ to get a morphism, say $\overline f:\overline {\mathbb {CP}^n} \to \mathbb {CP}^m$. $\overline f^{-1}(u)$ and $\overline f^{-1}(v)$ will approach to the same point $\overline b$ at the exceptional divisor, as $f^{-1}(u)$ and $f^{-1}(v)$ approach to $b$ in the same direction. But this is not so clear if $B$ is not smooth.