Geometry of moduli spaces of higher spin curves

ABSTRACT: Roughly speaking, the moduli space of higher spin curves parametrizes equivalence classes of pairs (C, L) where C is a smooth genus g algebraic curve and L is a line bundle on it whose r-th tensor power is isomorphic to the canonical bundle of the curve. The aim of the talk is to discuss important geometrical properties of these spaces under different points of view: one possible compactification together with the description of the rational Picard group, their birational geometry in some low genus cases and their relation with some special locus inside the classical moduli spaces of curves.