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Structures of the disconnected part of higher genus superstring amplitudesrestricted to the hyper elliptic cases are investigated in the NSR formalism,based on the DHoker Phong and recent results. A set of equations, which we canregard as a basic tool to sum over the spin structures of any of g loop, Mpoint amplitudes systematically, is shown by using a classical result ofAbelian functions. We discuss structures of g loop, M point massless externalboson superstring amplitudes by assuming that the spin structure dependence ofany of the disconnected amplitudes is only on one kind of constants, the genusg Weierstrass Pe function valued at the summation of g number of half periodschosen out of 2g+1 half periods. This is a natural generalization of the caseof genus 1. This assumption will be validated by a conjectured theorem whichstates that the spin structure dependent part of any string amplitude will benaturally decomposed into two parts. One is composed of manifestly modularinvariant functions of positions of inserting operators, and the other is thepolynomial of Pe function constants related to the moduli of Riemann surfacesonly. It is shown that this is actually the case for any M for g=1, and M=1,2,3for any g. Due to a technical problem, our consideration is at presentrestricted to the case that g(g+1)divided by 2 is odd. Example calculations areshown for the genus 2 by the method described here. In particular, our methodcorrectly reproduces biholomorphic 1 form of DHoker Phong result as for thefour point amplitudes of the disconnected parts.

On the pole structures of the disconnected part of hyper elliptic g loop M point super string amplitudes