JournalsjemsVol. 8, No. 4pp. 611–699

A construction of the Deligne-Mumford orbifold

  • Joel W. Robbin

    University of Wisconsin, Madison, USA
  • Dietmar A. Salamon

    ETH Zürich, Switzerland
A construction of the Deligne-Mumford orbifold cover
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Abstract

The Deligne--Mumford moduli space is the space \bar\mathcal{M}_{g,n} of isomorphism classes of stable nodal Riemann surfaces of arithmetic genus gg with nn marked points. A marked nodal Riemann surface is stable if and only if its isomorphism group is finite. We introduce the notion of a universal unfolding of a marked nodal Riemann surface and show that it exists if and only if the surface is stable. A natural construction based on the existence of universal unfoldings endows the Deligne--Mumford moduli space with an orbifold structure. We include a proof of compactness. Our proofs use the methods of differential geometry rather than algebraic geometry.

Cite this article

Joel W. Robbin, Dietmar A. Salamon, A construction of the Deligne-Mumford orbifold. J. Eur. Math. Soc. 8 (2006), no. 4, pp. 611–699

DOI 10.4171/JEMS/69