Cubic threefold and vanishing cycles on its hyperplane sections
For a complex projective manifold X, a vanishing cycle is a topological sphere on a smooth hyperplane section that is contracted to a point as the hyperplane section deforms and becomes tangent to X. What is a vanishing cycle on a singular hyperplane section? We will try to answer the question in the case when X is a smooth cubic hypersurface of the complex projective four-space. In particular, we compactify the parameter space of vanishing cycles on smooth hyperplane sections and interpret the boundary points by the Hilbert scheme of X and singularities on cubic surfaces.