A construction of the Deligne-Mumford orbifold
Joel W. Robbin
University of Wisconsin, Madison, USADietmar A. Salamon
ETH Zürich, Switzerland

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 with 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