  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.
