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In this paper we formulate sufficient conditions involving only objects naturally associated to $q$, in order for the fiber over $q$ to be finite. Assume that $F^{-1}(l)$ is 1-connected for the generic complex line $l$ containing $q$, and $F^{-1}(l)$ has finitely many components whenever $l$ is an exceptional line through $q$. Using arguments from topology and differential geometry, we establish a sharp estimate on the size of $F^{-1}(q)$. 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