Descendents on local curves: Stationary theory

The stable pairs theory of local curves in 3-folds (equivariant with respect to the scaling 2-torus) is studied with stationary descendent insertions. Reduction rules are found to lower descendents when higher than the degree. Factorization then yields a simple proof of rationality in the stationary case and a proof of the functional equation related to inverting q. The method yields an effective determination of stationary descendent integrals. Particular descendent series in the cap geometry play a special role and are calculated exactly using the stable pairs vertex and an analysis of the solution of the quantum differential equation for the Hilbert scheme of points of the plane.