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moduli space of curves in nLab

Moduli space of curves

Context

Complex geometry

Moduli space of curves

Examples

Idea

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

Deligne-Mumford 69 a nontrivial compactification of a moduli space of Riemann surfaces of fixed genus which is a Deligne-Mumford stack, since called the 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 ℳ 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 ℳ 1,1=ℳ ell\mathcal{M}_{1,1}= \mathcal{M}_{ell}.

Properties

Over the complex numbers

This is reviewed for instance in (Madsen 07, section 1.1).

Homotopy type and cohomology

The homotopy type of the Riemann moduli spaces ℳ g\mathcal{M}_g (i.e. of the geometric realization of the orbifold, hence essentially of the delooping of the mapping class group) is essentially unknown. Even its orbifold cohomology over the rational numbers is fully known only for g≤4g \leq 4.

However, in a stable range it has been fully computed in (Madsen-Weiss 02), proving what was previously conjectured by Mumford's conjecture. A review is in (Madsen 07).

Orbifold Euler characteristic

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

χ(ℳ g,1)=ζ(1−2g). \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(ℳ 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(ℳ 1,1)=−112 \chi(\mathcal{M}_{1,1}) = -\frac{1}{12}

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

Relation to Teichmüller space

Over the complex number then the moduli space of curves is a quotient of Teichmüller space. (Hubbard-Koch 13)

Examples

g=0g = 0, n=0n = 0

By the Riemann mapping theorem?, ℳ 0,0≃*\mathcal{M}_{0,0}\simeq \ast is the point.

g=1g = 1, n=0n = 0

ℳ 1,0≃ℝ 2\mathcal{M}_{1,0} \simeq \mathbb{R}^2

g=1g = 1, n=1n = 1

ℳ 1,1=ℳ ell\mathcal{M}_{1,1} = \mathcal{M}_{ell}, the moduli stack of elliptic curves.

g≥2g \geq 2, n=0n = 0

For genus g≥2g\geq 2 the moduli stack of complex structures is equivalently that of hyperbolic metrics. This way a lot of hyperbolic geometry is used in the study of ℳ g≥2,n\mathcal{M}_{g \geq 2, n}.

References

Original articles:

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
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

On moduli spaces of curves/moduli spaces of Riemann surfaces with emphasis on their orbifold-structure:

Lizhen Ji, Shing-Tung Yau, Transformation Groups and Moduli Spaces of Curves, International Press of Boston 2011 (ISBN:9781571462237)

Discussion of the orbifold cohomology of the moduli stack is in

John Harer, The cohomology of the moduli space of curves, Lec. Notes in Math. 1337, p. 138–221. Springer, Berlin, 1988.

The stable rational orbifold cohomology of the moduli stack was computed in

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

exposition of this is in

Ib Madsen, Moduli spaces from a topological viewpoint, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) (pdf)