# Moduli space of curves and

## Idea

The moduli space of algebraic curves/Riemann surfaces is a sort of space of parameters parametrizing algebraic curves of given genus.

Deligne and Mumford have found a nontrivial compactification of a moduli space of Riemann surfaces of fixed genus which is a Deligne-Mumford stack, see at Deligne-Mumford compactification.

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 $\mathcal{M}_{g,n}$ which plays an important role in the mathematical study of Gromov-Witten invariants and of conformal blocks.

## References

• David Mumford, Towards an enumerative geometry of the moduli space of curves, Arithmetic and geometry, Vol. II, Birkhäuser Boston, Boston, MA, 1983, pp. 271–328, MR85j:14046

• Pierre Deligne, David 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 $\mathcal{M}_{g,n}$ 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.

Revised on January 13, 2014 14:05:53 by Urs Schreiber (89.204.155.62)