Towards an enumerative geometry of the moduli space of twisted curves and r-th roots

Details Towards an enumerative geometry of the moduli space of twisted curves and r-th roots Towards an enumerative geometry of the moduli space of twisted curves and r-th roots

2006-07-13

arxiv.org/abs/math/0607324v4

Mathematics

Algebraic Geometry

35 pages, revised version, to appear in Compositio Mathematica

Scientific paper

The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture 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 math.AG/0603687, we construct a compact 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 r-th root in K-theory.

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