  We show a version of the DT/PT correspondence relating local curve counting
invariants, encoding the contribution of a fixed smooth curve in a Calabi-Yau
threefold. We exploit a local study of the Hilbert-Chow morphism about the
cycle of a smooth curve. We determine, via Quot schemes, the global
Donaldson-Thomas theory of a general Abel-Jacobi curve of genus $3$.
