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

Abstract

Let H_ℙ3_S denote the Hilbert scheme of smooth connected curves in ℙ3. We consider maximal irreducible closed subsets W ⊂ H_ℙ3_S whose general member C is contained in a smooth cubic surface and investigate the conditions for W to be a S component of (H_ℙ3_S)red. We especially study the case where the dimension of the S tangent space of H_ℙ3_S at [C] is greater than dim W (≥ 4 deg(C)) by one. We compute obstructions to deforming C in ℙ3 and prove that for every W in this case, H_ℙ3_S is non-reduced along W and W is a component of (H_ℙ3_S)red.