  Superstring perturbation theory is traditionally carried out by using
picture-changing operators (PCO's) to integrate over odd moduli. Naively the
PCO's can be inserted anywhere on a string worldsheet, but actually a
constraint must be placed on PCO insertions to avoid spurious singularities.
Accordingly, it has been long known that the simplest version of the PCO
procedure is valid only locally on the moduli space of Riemann surfaces, and
that a correct PCO-based algorithm to compute scattering amplitudes must be
based on piecing together local descriptions. Recently, "vertical integration"
was proposed as a relatively simple method to do this. Here, we spell out in
detail what vertical integration means if carried out systematically. This
involves a hierarchical procedure with corrections of high order. One might
anticipate such a structure from the viewpoint of super Riemann surfaces.
