A construction of the Deligne-Mumford orbifold

Abstract

The Deligne–Mumford moduli space is the space ${\overline{\mathcal{M}}}_{g,n}$ 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.

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