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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.

In this paper, we study the spectrum of the complex Hillwhere n k ∈ Z ≥0 with max n k ≥ 1 and z 0 ∈ C is chosen such that q(x; τ) has no singularities on R. For any fixed τ ∈ iR >0 , we give a necessary and sufficient condition on (n 0 , n 1 , n 2 , n 3 ) to guarantee that the spectrum σ(L) is

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Let X be a closed Riemann surface. When X is embedded into a projective space, the first rational cohomology group can be concretely obtained from the monodromy in the family of its smooth hyperplane sections by C. Schnell's tube mapping. We generalize this result to the first integral homology group by relating the tube mapping with the topological Abel-Jacobi mapping. By making use of the mapping class group action, we prove that all tube classes constructed from the elementary vanishing cycles form a cofinite subgroup of the first integral homology group of X.