THE BIRATIONAL GEOMETRY OF MODULI SPACES OF SPIN CURVES

Moduli spaces of spin curves parametrize roots of canonical bundles of curves (theta-characteristics), and are highly interesting
covers of the moduli space of curves. Their geometry is of interest both from an algebraic-geometric, as well as from a string-theoretic point of view. Recently, in joint work with A. Verra we have achieved a complete classification of these spaces from the point of view of birational geometry. For instance, the even spin moduli space is a unirational variety for g<8, of general type for g>8, and a very mysterious variety of Calabi-Yau type for g=8.