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Let H S P3 denote the Hilbert scheme of smooth connected curves in P 3 .W e consider maximal irreducible closed subsets W ⊂ H S3 whose general member C is contained in a smooth cubic surface and investigate the conditions for W to be a component of (H S3 )red. We especially study the case where the dimension of the tangent space of H S P3 at [C] is greater than dim W (≥ 4d eg(C)) by one. We compute obstructions to deforming C in P 3 and prove that for every W in this case, H S3 is non-reduced along W and W is a component of (H S P3 )red.

Obstructions to Deforming Space Curves and Non-reduced Components of the Hilbert Scheme