Obstructions to deforming curves on a prime Fano 3‐fold

We prove that for every smooth prime Fano 3‐fold V, the Hilbert scheme of smooth connected curves on V contains a generically non‐reduced irreducible component of Mumford type. We also study the deformations of degenerate curves C in V, i.e., curves C contained in a smooth anticanonical member of V. We give a sufficient condition for C to be stably degenerate, i.e., every small (and global) deformation of C in V is contained in a deformation of S in V. As a result, by using the Hilbert‐flag scheme of V, we determine the dimension and the smoothness of at the point C], assuming that the class of C in is generated by together with the class of a line, or a conic on V.