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Towards an enumerative geometry of the moduli space of twisted curves and rth roots

Part of: Curves

Published online by Cambridge University Press:  01 November 2008

Alessandro Chiodo*
Institut Fourier, UMR CNRS 5582, UFR de Mathématiques, Université de Grenoble 1, BP 74, 38402 Saint Martin d’Hères, France (email:
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The enumerative geometry of rth roots of line bundles is crucial in the theory of r-spin curves and occurs in the calculation of Gromov–Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In a previous paper, we constructed a compact moduli stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford’s formula for the Chern character of the Hodge bundle to the direct image of the universal rth root in K-theory.

Research Article
Copyright © Foundation Compositio Mathematica 2008


Partially supported by the Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme, MEIF-CT-2003-501940.