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

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