The infinitesimal Abel-Jacobi mapping and moving the \({\mathcal O}(2)+{\mathcal O}(-4)\) curve (Q922603)

A smooth rational curve in a smooth projective surface S either contracts or moves in S. An example is given that the analogous statement for a smooth projective threefold is not true. In fact consider the family of smooth quintic hypersurfaces V in \({\mathbb{P}}_ 4\) containing a smooth plane conic C with a double structure. It is shown using the Abel-Jacobi mapping that for generic V and any \(m\geq 1\) no deformations of mC in V exist. On the other hand it is easily seen using a criterion of Laufer, that C does not contract.