complex geometry

# 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.

The special case of $g = 1$, $n =1$ is the moduli stack of elliptic curves $\mathcal{M}_{1,1}= \mathcal{M}_{ell}$.

## Properties

### Orbifold Euler characteristic

The orbifold Euler characteristic $\chi$ of $\mathcal{M}_{g,1}$ is given by the Riemann zeta function at negative integral values as follows (Zagier-Harer 86):

$\chi(\mathcal{M}_{g,1}) = \zeta(1-2g) \,.$

By the expression of the Riemann zeta function at negative integral values by the Bernoulli numbers $B_n$, this says equivalently that

$\chi(\mathcal{M}_{g,1}) = -\frac{B_{2g}}{2g} \,.$

For instance for $g = 1$ (once punctured complex tori, hence complex elliptic curves) this yields

$\chi(\mathcal{M}_{1,1}) = -\frac{1}{12}$

## 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.

• The moduli of curves pdf

The orbifold Euler characteristic of the moduli space of curves was originally computed in

• Don Zagier, John Harer, The Euler characteristic of the moduli space of curves, Inventiones mathematicae (1986) Volume: 85, page 457-486 (EUDML)

Reviews include

• Mathematical ideas and notions in quantum field theory – 5. The Euler characteristic of the moduli space of curves (pdf)

Revised on April 9, 2014 08:35:51 by Urs Schreiber (145.116.131.80)