Deformations of curves on a higher dimensional algebraic variety and their obstructions

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In this project, we investigated infinitesimal deformations of curves on a higher dimensional algebraic variety and their obstructions, and non-reduced components of the Hilbert scheme. As a result, we have proved a conjecture due to Kleppe and Ellia, which is concerned with non-reduced components of the Hilbert scheme of space curves, in the case where a general member of the components are quadratically normal. We also study the deformations of degenerate curves on a higher dimensional scroll, and construct a family of curves, which have a first order deformation not liftable to the second order deformation.