We prove that for genus greater than or equal to 5, the moduli space of super
Riemann surfaces is not projected (and in particular is not split): it cannot
be holomorphically projected to its underlying reduced manifold. Physically,
this means that certain approaches to superstring perturbation theory that are
very powerful in low orders have no close analog in higher orders.
Mathematically, it means that the moduli space of super Riemann surfaces cannot
be constructed in an elementary way starting with the moduli space of ordinary
Riemann surfaces. It has a life of its own.