  Multi-loop interaction amplitudes in the theory of the closed, oriented
superstrings are obtained by the integration of local amplitudes which are
represented by a sum of the spinning string local amplitudes. The last local
amplitudes are given explicitly through super-Schottky group parameters and
interaction vertex coordinates on the $(1|1)$ complex supermanifold. The
integration is ambiguous under those replacements of the integration variables
which admix Grassmann variables to the boson ones. So the calculation is guided
by a preservation of local symmetries of the superstring. The obtained
amplitudes are free from divergences and consistent with the world-sheet
symmetries. The vacuum amplitude and 1-, 2- and 3-point amplitudes of massless
states vanish once the integration over certin modular variables and
interaction vertex coordinates.
