A construction of the Deligne-Mumford orbifold

• Autores: Dietmar Salamon, Joel W. Robbin
• Localización: Journal of the European Mathematical Society, ISSN 1435-9855, Vol. 8, Nº 4, 2006, págs. 611-699
• Idioma: inglés
• DOI: 10.4171/jems/69
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• Resumen

  • 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.