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moduli space of curves

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 gg with nn marked points g,n\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=1g = 1, n=1n =1 is the moduli stack of elliptic curves 1,1= ell\mathcal{M}_{1,1}= \mathcal{M}_{ell}.

Properties

Orbifold Euler characteristic

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

χ( g,1)=ζ(12g). \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 nB_n, this says equivalently that

χ( g,1)=B 2g2g. \chi(\mathcal{M}_{g,1}) = -\frac{B_{2g}}{2g} \,.

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

χ( 1,1)=112 \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}

for the orbifold Euler characteristic of the moduli space of elliptic curves.

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 g,n\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)