Super Atiyah classes and obstructions to splitting of supermoduli space

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.