Double ramification cycles on the moduli spaces of curves. (English) Zbl 1370.14029
A double ramification cycle DR\(_g(\mu, \nu)\) on the moduli space of curves parameterizes curves of genus \(g\) that admit a map to \(\mathbb P^1\) with specified ramification profile \(\mu\) over \(0\) and \(\nu\) over \(\infty\). The study of the cycle class of DR\(_g(\mu, \nu)\) is a classical topic, dating back to a question of Eliashberg in 2001. Hain first calculated the cycle class of DR\(_g(\mu, \nu)\) in the locus of curves of compact type [R. Hain, in: Handbook of moduli. Volume I. Somerville, MA: International Press; Beijing: Higher Education Press. 527–578 (2015; Zbl 1322.14049)], see also [S. Grushevsky and D. Zakharov, Proc. Am. Math. Soc. 142, No. 12, 4053–4064 (2014; Zbl 1327.14132); Duke Math. J. 163, No. 5, 953–982 (2014; Zbl 1302.14039); R. Cavalieri et al., J. Pure Appl. Algebra 216, No. 4, 950–981 (2012; Zbl 1273.14053)] for alternate calculations. In this remarkable paper the authors calculate the cycle class of DR\(_g(\mu, \nu)\) on the moduli space of Deligne-Mumford stable curves, which verifies a previous conjectural formula of Pixton and completely answers Eliashberg’s question. We remark that here DR\(_g(\mu, \nu)\) is defined via the virtual fundamental class of the moduli space of stable maps to rubber, and the resulting formula expresses the class of DR\(_g(\mu, \nu)\) as a sum over stable graphs. As applications, some Hodge integral calculations on the Deligne-Mumford moduli space are deduced from the formula.
Reviewer: Dawei Chen (Chestnut Hill)
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