Nicola Pagani: Chen-Ruan cohomology of moduli of curves

In 1969 Deligne and Mumford defined moduli spaces of smooth genus g curves with n marked points and their compactification, as smooth algebraic stacks. In the eighties, Mumford started a research program on the enumerative geometry and intersection theory on these moduli spaces. This led to several results and conjectures on the cohomology ring and on the Chow ring. Motivated by string theory, Chen and Ruan in 2001 and Abramovich, Graber and Vistoli in 2003 defined the orbifold cohomology and its algebraic analogue, the stringy Chow ring, respectively. This is meant to be the degree zero part of the small quantum cohomology ring for orbifolds.

In the first part of this seminar, we review the definition of Chen-Ruan cohomology. We explain what we mean by computing the Chen-Ruan cohomology for moduli spaces of curves, and then we describe the results that we obtained in the first non-trivial case: the moduli spaces of genus 1 curves with marked points and its compactification. In the second part, we try to give more technical details, and to study moduli of curves of higher genus. If there is time, we will spend a few words about the quest of defining an orbifold Tautological Ring, taking inspiration from Faber’s conjectures and Faber-Pandharipande’s definition.