  Let $S$ be a complex projective surface. Lefschetz originally proved
Lefschetz $(1, 1)$--Theorem by studying a Lefschetz pencil of hyperplane
sections of $S$ and the Abel--Jacobi mapping. In this paper, we attack
Lefschetz $(1, 1)$--Theorem by constructing certain two-parameter families of
twice hyperplane sections of $S$ and then applying the topological Abel--Jacobi
mapping. Our geometric constructions would give an inductive approach and some
insight for higher dimensional cases. We prove a strong tube theorem which
generalizes Schnell's tube theorem to integral homology groups for complex
projective curves and then obtain a Jacobi-type inversion theorem. In the end,
we give a geometric description for the deformation space of an elementary
vanishing cycle over a generic net.
