  We explicitly show how the chiral superstring amplitudes can be obtained
through factorisation of the higher genus chiral measure induced by suitable
degenerations of Riemann surfaces. This powerful tool also allows to derive, at
any genera, consistency relations involving the amplitudes and the measure. A
key point concerns the choice of the local coordinate at the node on degenerate
Riemann surfaces that greatly simplifies the computations. As a first
application, starting from recent ansaetze for the chiral measure up to genus
five, we compute the chiral two-point function for massless Neveu-Schwarz
states at genus two, three and four. For genus higher than three, these
computations include some new corrections to the conjectural formulae appeared
so far in the literature. After GSO projection, the two-point function vanishes
at genus two and three, as expected from space-time supersymmetry arguments,
but not at genus four. This suggests that the ansatz for the superstring
measure should be corrected for genus higher than four.
