Moduli of curves with principal and spin bundles : singularities and global geometry

Overview & Synopsis

The moduli space Mgbar of genus g stable curves is a central object in algebraic geometry. From the point of view of birational geometry, it is natural to ask if Mgbar is of general type. Harris-Mumford and Eisenbud-Harris found that Mgbar is of general type for genus g>=24 and g=22. The case g=23 keep being mysterious. In the last decade, in an attempt to clarify this, a new approach emerged: the idea is to consider finite covers of Mgbar that are moduli spaces of stable curves equipped with additional structure as l-covers (l-th roots of the trivial bundle) or l-spin bundles (l-th roots of the canonical bundle). These spaces have the property that the transition to general type happens to a lower genus. In this work we intend to generalize this approach in two ways: - a study of moduli space of curves with any root of any power of the canonical bundle; - a study of the moduli space of curves with G-covers for any finite group G. In order to define these moduli spaces we use the notion of twisted curve (see Abramovich-Corti-Vistoli). The fundamental result obtained is that it is possible to describe the singular locus of these moduli spaces via the notion of dual graph of a curve. Thanks to this analysis, we are able to develop calculations on the tautological rings of the spaces, and in particular we conjecture that the moduli space of curves with S3-covers is of general type for odd genus g>=13.