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

### Hirokazu Nasu

Kyoto University, Japan

## 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.