  Here we investigate some birational properties of two collections of moduli
spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin
curves. In particular, we focus on vanishings of Hodge numbers of type (p,0)
and on computations of the Kodaira dimension. Our methods are purely algebraic
geometric and rely on an induction argument on the number of marked points and
the genus of the curves (cf. mathAG/9803001).
