# nLab moduli space of curves

## Moduli space of curves and its Deligne-Mumford compactification

### Idea

Moduli space of algebraic curves/Riemann surfaces is sort of space of parameters parametrizing algebraic curves of given genus, see moduli space for a more abstract philosophy. Deligne and Mumford have found a nontrivial compactification of a moduli space of Riemann surfaces of fixed genus which is a Deligne-Mumford stack.

There is also a decorated version of curves with marked points, and of the corresponding compactified moduli space of stable curves of genus $g$ with $n$ marked points ${ℳ}_{g,n}$ which plays important role in the mathematical study of Gromov-Witten invariants and of conformal blocks.

Related $n$Lab entries: moduli space, moduli stack, mapping class group, enumerative geometry, Mumford class, Deligne-Mumford stack

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• Pierre Deligne, D. Mumford, The irreducibility of the space of curves of given genus . Publications Mathématiques de l’IHÉS (Paris) 36: 75–109 (1969) numdam
• John Harer, The cohomology of the moduli space of curves, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.
• Ib Madsen, Michael Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843–941, MR2009b:14051, doi, math.AT/0212321
• G. Mondello, Combinatorial classes on ${ℳ}_{g,n}$ are tautological, Int. Math. Res. Not. 44 (2004), 2329-–2390, MR2005g:14056, doi, math.AG/0303207
• Gabriele Mondello, Riemann surfaces, ribbon graphs and combinatorial classes, in: Handbook of Teichmüller theory. Vol. II, 151–215, IRMA Lect. Math. Theor. Phys., 13, Eur. Math. Soc., Zürich, 2009; draft with index: pdf, arxiv version math.AG/0705.1792, MR2010f:32012
• Alastair Hamilton, Classes on compactifications of the moduli space of curves through solutions to the quantum master equation, Lett. Math. Phys. 89 (2009), no. 2, 115–130.