# A construction of the Deligne-Mumford orbifold

### Joel W. Robbin

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

ETH Zürich, Switzerland

## Abstract

The Deligne–Mumford moduli space is the space $M_{g,n}$ of isomorphism classes of stable nodal Riemann surfaces of arithmetic genus $g$ with $n$ 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