geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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 $\mathcal{M}_{1,1}= \mathcal{M}_{ell}$.
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):
By the expression of the Riemann zeta function at negative integral values by the Bernoulli numbers $B_n$, this says equivalently that
For instance for $g = 1$ (once punctured complex tori, hence complex elliptic curves) this yields
for the orbifold Euler characteristic of the moduli space of elliptic curves.
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
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