Abstract
In this chapter we construct the moduli space \(\overline{M}_{g,n}\) of stable n-pointed curves of genus g and look at its structure from various points of view. First we construct \(\overline{M}_{g,n}\) as an analytic space, and then we show that this analytic space has a natural structure of algebraic space. After a utilitarian introduction to orbifolds and stacks, in particular to Deligne–Mumford stacks, we then show that \(\overline{M}_{g,n}\) is just a coarse reflection of a more fundamental object, the moduli stack \(\overline{\mathcal{M}}_{g,n}\) of stable n-pointed curves of genus g, and we show, using in an essential way the results of Chapter X, that \(\overline{\mathcal{M}}_{g,n}\) is a Deligne–Mumford stack. After discussing some basic constructions, such as normalization and quotient by a finite group, in the context of Deligne–Mumford stacks, we close by interpreting the fundamental constructions of projection and clutching as morphisms of moduli spaces, and by observing that contraction and stabilization give an isomorphism of stacks between \(\overline{\mathcal{M}}_{g,n+1}\) and the “universal curve” \(\overline{\mathcal{C}}_{g,n}\) over \(\overline{\mathcal{M}}_{g,n}\).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Arbarello, E., Cornalba, M., Griffiths, P.A. (2011). The moduli space of stable curves. In: Geometry of Algebraic Curves. Grundlehren der mathematischen Wissenschaften, vol 268. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69392-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-69392-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42688-2
Online ISBN: 978-3-540-69392-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)