2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72800Here 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).The final version of this paper will appear in Trans. Amer. Math. SocAlgebraic Geometry14H10, 14E08text