  The genus-dependence of multi-loop superstring amplitudes is bounded at large
orders in perturbation theory using the super-Schottky group parametrization of
supermoduli space. Partial estimates of supermoduli space integrals suggest an
exponential dependence on the genus when the integration region is restricted
to a single fundamental domain of the super-modular group in the super-Schottky
parameter space. Bounds for N-point superstring scattering amplitudes are
obtained for arbitrary N and are shown to be consistent with exact results
recently obtained for special type II string amplitudes for orbifold or
Calabi-Yau compactifications. It is suggested that the generic estimates, which
imply the validity of superstring perturbation theory in the weak-coupling
limit, might be used to determine scattering amplitudes at strong coupling
because of the S-duality of type II and heterotic string theories.
Non-perturbative effects are consistent with these estimates, based on a sum
over closed surfaces, because they can be derived from an additional
contribution to the sum over surfaces corresponding to the insertion of
Dirichlet boundaries.
