The setup: say I have some rational projective variety $X$ of dimension $n$ over $\mathbb{C}$ such that the map $$ X \dashrightarrow \mathbb{P}^n $$ is given by some linear series $\mathcal{L}$.
My question: Is there a general way to write down the inverse map as a linear series $\mathcal{L}'$ in terms of $\mathcal{L}$?
Note that I don't a priori have equations for $X$, but I may know $\mathcal{L}$ pretty explicitly, e.g., maybe $X$ is a threefold and the map to $\mathbb{P}^3$ is double projection from a conic: $\mathcal{L} = |K_X - 2C|$.