Uniqueness theorems for parametrized algebraic curves

Abstract:

Let ${L_1}, \ldots ,{L_n}$ be lines in ${\mathbb {P}^2}$ and let $f,g:{\mathbb {P}^1} \to {\mathbb {P}^2}$ be nonconstant algebraic maps. For certain configurations of lines ${L_1}, \ldots ,{L_n}$, the hypothesis that, for $i = 1, \ldots ,n$, the inverse images ${f^{ - 1}}({L_i})$ and ${g^{ - 1}}({L_i})$ are equal, not necessarily with the same multiplicities, implies that $f$ is identically equal to $g$.