Moduli space of curves and
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 with marked points which plays an important role in the mathematical study of Gromov-Witten invariants and of conformal blocks.
The special case of , is the moduli stack of elliptic curves .
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 (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 .
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 of is given by the Riemann zeta function at negative integral values as follows (Zagier-Harer 86):
By the expression of the Riemann zeta function at negative integral values by the Bernoulli numbers , this says equivalently that
For instance for (once punctured complex tori, hence complex elliptic curves) this yields
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)
By the Riemann mapping theorem?, is the point.
, the moduli stack of elliptic curves.
For genus 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 .
Original articles include
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
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
exposition of this is in
- Ib Madsen, Moduli spaces from a topological viewpoint, Proceedings of the International Congress of Mathematics, Madrid 2006 (2007) (pdf)
Gabriele Mondello, Combinatorial classes on 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)
The complex analytic structure and the relation to Teichmüller space is further discussed in
- Mathematical ideas and notions in quantum field theory – 5. The Euler characteristic of the moduli space of curves (pdf)
Étale homotopy type of moduli stacks of curves is discussed in
- Paola Frediani, Frank Neumann, Étale homotopy types of moduli stacks of algebraic curves with symmetries, K-Theory 30: 315-340, 2003 (arXiv:math/0404387)