  The first obstruction to splitting a supermanifold S is one of the three
components of its super Atiyah class, the two other components being the
ordinary Atiyah classes on the reduced space M of the even and odd tangent
bundles of S. We evaluate these classes explicitly for the moduli space of
super Riemann surfaces ("super moduli space") and its reduced space, the moduli
space of spin curves. These classes are interpreted in terms of certain
extensions arising from line bundles on the square of the varying (super)
Riemann surface. These results are used to give a new proof of the
non-projectedness of ${\mathfrak{M}}_{g,1}$, the moduli space of super Riemann
surfaces with one puncture.
