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

Publisher： Cambridge University Press

E-ISSN： 1570-5846|144|6|1461-1496

ISSN： 0010-437x

Source： Compositio Mathematica, Vol.144, Iss.6, 2008-11, pp. : 1461-1496

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Abstract

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.

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