  The modular transformations of the $(1|1)$ complex supermanifolds in the
like-Schottky modular parameterization are discussed. It is shown that these
"supermodular" transformations depend on the spinor structure of the
supermanifold by terms proportional to the odd modular parameters. The above
terms are calculated in the explicit form. They are urgent for the divergency
problem in the Ramond-Neveu-Schwarz superstring theory and for calculating the
fundamental domain in the modular space. The supermodular transformations of
the multi-loop superstring partition functions calculated by the solution of
the Ward identities are studied. The above Ward identities are shown to be
covariant under the supermodular transformations. So the partition functions
necessarily possess the covariance under the transformations discussed. It is
demonstrated explicitly the covariance of the above partition functions at zero
odd moduli under those supermodular transformations, which turn a pair of even
genus-1 spinor structures to a pair of the odd genus-1 spinor ones. The brief
consideration of the cancellation of divergences is given.
