A construction of the Deligne-Mumford orbifold

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 $g$ 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.