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The orbifold cohomology of the moduli of genus-two curves. (English) Zbl 1115.14018

Jarvis, Tyler J. (ed.) et al., Gromov-Witten theory of spin curves and orbifolds. AMS special session, San Francisco, CA, USA, May 3–4, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3534-3/pbk). Contemporary Mathematics 403, 167-184 (2006).
From the introduction: There is now an algebraic version of Chen and Ruan’s orbifold cohomology defined for smooth Deligne-Mumford stacks by D. Abramovich, T. Graber and A. Vistoli [in: Orbifolds in mathematics and physics, Contemp. Math. 310, 1–24 (2002; Zbl 1067.14055)]. In that same paper, the authors determine this ring with integral coefficients for the stacks \({\mathcal M}_{1,1}\) and \(\overline{{\mathcal M}}_{1,1}\) of one-pointed nonsingular (respectively stable) genus-one curves. The purpose of this note is to give an in-depth treatment of the exercise of computing the stringy Chow ring with rational coefficients for the stack \({\mathcal M}_2\) of nonsingular genus-two curves. I have also computed, by an approach similar to the one used in this paper, the ring with integral coefficients for \({\mathcal M}_2\) and \(\overline{{\mathcal M}}_2\). Those results will appear in subsequent publications.
For the entire collection see [Zbl 1091.14002].

MSC:

14H10 Families, moduli of curves (algebraic)
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

Citations:

Zbl 1067.14055
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