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Oberseminar WiSe 2008/09 Zusammenfassung des Vortrags von N. Pagani

The orbifold tautological ring for moduli of genus 1 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 analogous, the stringy Chow ring, respectively. This is meant to be the degree zero part of the small quantum cohomology ring for orbifolds. In this seminar, we show some results on the orbifold cohomology of the moduli spaces of curves in the case g=1, both in the open and in the compact case. We discuss a possible extension of the tautological ring to an "orbifold tautological ring". We shall also try to generalize Witten and Faber's conjectures (stated for the usual cohomology) in the orbifold cohomology setting.