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Orbifold Euler characteristics of the moduli spaces of smooth curves
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Numbers
$g$
$n$ 
$\chi^{\mathrm{orb}}(\mathcal M_{g,n})$
1
1:
-1/12
comment: The orbifold Euler characteristic of the moduli stack of elliptic curves is $-1/12$; the generic elliptic involution is included.
1
2:
1/12
1
3:
-1/6
1
4:
1/2
1
5:
-2
1
6:
10
1
7:
-60
1
8:
420
1
9:
-3360
1
10:
30240
1
11:
-302400
1
12:
3326400
2
0:
-1/240
2
1:
1/120
2
2:
-1/40
2
3:
1/10
2
4:
-1/2
2
5:
3
2
6:
-21
2
7:
168
2
8:
-1512
2
9:
15120
2
10:
-166320
2
11:
1995840
2
12:
-25945920
3
0:
1/1008
3
1:
-1/252
3
2:
5/252
3
3:
-5/42
3
4:
5/6
3
5:
-20/3
3
6:
60
3
7:
-600
3
8:
6600
3
9:
-79200
3
10:
1029600
3
11:
-14414400
3
12:
216216000
4
0:
-1/1440
4
1:
1/240
4
2:
-7/240
4
3:
7/30
4
4:
-21/10
4
5:
21
4
6:
-231
4
7:
2772
4
8:
-36036
4
9:
504504
4
10:
-7567560
4
11:
121080960
4
12:
-2058376320
5
0:
1/1056
5
1:
-1/132
5
2:
3/44
5
3:
-15/22
5
4:
15/2
5
5:
-90
5
6:
1170
5
7:
-16380
5
8:
245700
5
9:
-3931200
5
10:
66830400
5
11:
-1202947200
5
12:
22855996800
6
0:
-691/327600
6
1:
691/32760
6
2:
-7601/32760
6
3:
7601/2730
6
4:
-7601/210
6
5:
7601/15
6
6:
-7601
6
7:
121616
6
8:
-2067472
6
9:
37214496
6
10:
-707075424
6
11:
14141508480
6
12:
-296971678080
7
0:
1/144
7
1:
-1/12
7
2:
13/12
7
3:
-91/6
7
4:
455/2
7
5:
-3640
7
6:
61880
7
7:
-1113840
7
8:
21162960
7
9:
-423259200
7
10:
8888443200
7
11:
-195545750400
7
12:
4497552259200
8
0:
-3617/114240
8
1:
3617/8160
8
2:
-3617/544
8
3:
3617/34
8
4:
-3617/2
8
5:
32553
8
6:
-618507
8
7:
12370140
8
8:
-259772940
8
9:
5715004680
8
10:
-131445107640
8
11:
3154682583360
8
12:
-78867064584000
9
0:
43867/229824
9
1:
-43867/14364
9
2:
745739/14364
9
3:
-745739/798
9
4:
745739/42
9
5:
-7457390/21
9
6:
7457390
9
7:
-164062580
9
8:
3773439340
9
9:
-90562544160
9
10:
2264063604000
9
11:
-58865653704000
9
12:
1589372650008000
10
0:
-174611/118800
10
1:
174611/6600
10
2:
-3317609/6600
10
3:
3317609/330
10
4:
-23223263/110
10
5:
23223263/5
10
6:
-534135049/5
10
7:
12819241176/5
10
8:
-64096205880
10
9:
1666501352880
10
10:
-44995536527760
10
11:
1259875022777280
10
12:
-36536375660541120
11
0:
77683/5520
11
1:
-77683/276
11
2:
543781/92
11
3:
-5981591/46
11
4:
5981591/2
11
5:
-71779092
11
6:
1794477300
11
7:
-46656409800
11
8:
1259723064600
11
9:
-35272245808800
11
10:
1022895128455200
11
11:
-30686853853656000
11
12:
951292469463336000
12
0:
-236364091/1441440
12
1:
236364091/65520
12
2:
-5436374093/65520
12
3:
5436374093/2730
12
4:
-27181870465/546
12
5:
27181870465/21
12
6:
-244636834185/7
12
7:
978547336740
12
8:
-28377872765460
12
9:
851336182963800
12
10:
-26391421671877800
12
11:
844525493500089600
12
12:
-27869341285502956800
Definition
The number $\chi_{g,n}=\chi^{\mathrm{orb}}(\mathcal M_{g,n})$ is the orbifold Euler characteristic of the moduli stack of smooth connected complex curves of genus $g$ with $n$ distinct ordered marked points [1].
Parameters
$g$
—   genus ($g\geq0$)
$n$
—   number of marked points ($n\geq0$, with $2g-2+n>0$)
Formulas
(1)
The Bernoulli numbers use $t/(e^t-1)=\sum_{k\geq0}B_k t^k/k!$, in particular $B_2=1/6$ and $B_4=-1/30$.
(2)
For $g\geq1$ and $2g-2+n>0$, $\displaystyle\chi_{g,n}=(-1)^n\frac{(2g-1)(2g+n-3)!}{(2g)!}B_{2g}$, where $B_{2g}$ is a Bernoulli number [1].
(3)
For every stable pair $(g,n)$, $\displaystyle\chi_{g,n+1}=(2-2g-n)\chi_{g,n}$ [1].
(4)
For $g\geq1$, $\displaystyle\chi_{g,1}=\zeta(1-2g)=-\frac{B_{2g}}{2g}$, where $\zeta$ is the Riemann zeta function [3].
(5)
For $g\geq2$, $\displaystyle\chi_{g,0}=\frac{B_{2g}}{2g(2g-2)}$ [3].
(6)
For $n\geq1$, $\displaystyle\chi_{1,n}=(-1)^n\frac{(n-1)!}{12}$ [1].
(7)
For $n\geq3$, $\displaystyle\chi_{0,n}=(-1)^{n-3}(n-3)!$ [1].
Comments
(8)
The orbifold characteristic retains every automorphism, including a generic automorphism acting trivially on the local deformation space. Equivalently, it is the Euler characteristic of a torsion-free subgroup of finite index in the pure mapping class group, divided by that index [3].
(9)
For $g=1$ and $n=1$, the generic elliptic involution is retained, so $\chi_{1,1}=-1/12$. Removing that involution doubles the value to $-1/6$. The ordinary Euler characteristic of the coarse moduli space is $1$.
(10)
The curves are smooth: these entries are for $\mathcal M_{g,n}$. Adding nodal stable curves gives the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$, whose Euler characteristic includes boundary contributions [2]. The marked points remain labelled, so no factor of $n!$ divides an entry.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.arith.misc import bernoulli
from math import factorial

def orbifold_chi(g, n):
    if g < 0 or n < 0 or 2*g - 2 + n <= 0:
        raise ValueError("A stable pair is required")
    if g == 0:
        return QQ((-1)**(n-3) * factorial(n-3))
    return (QQ((-1)**n * (2*g-1) * factorial(2*g+n-3))
            * QQ(bernoulli(2*g)) / QQ(factorial(2*g)))

orbifold_chi(2, 0)  # -1/240
References
[1]
A. Giacchetto, D. Lewański and P. Norbury, An intersection-theoretic proof of the Harer–Zagier formula, Algebraic Geometry 10 (2023), 130–147. (doi)
[2]
G. Bini and J. Harer, Euler Characteristics of Moduli Spaces of Curves, preprint (2005), Section 3. (arXiv)
Links
Similar tables
Bernoulli numbers —   supply the rational coefficients in the Harer-Zagier formula
Values of the Riemann zeta function at rational numbers —   include the one-marked characteristics as values at negative odd integers
intersection numbers of cotangent line classes —   are intersection invariants on the compactified moduli stacks of curves
Weil-Petersson volumes —   measure the moduli spaces of hyperbolic surfaces with labelled cusps, with automorphism weights included
Data properties
Entries are of type: rational number
How well the digits are known: exact
Table is complete: no (it holds every stable pair with $1\leq g\leq12$ and $0\leq n\leq12$, covering the first twelve positive genera with up to twelve ordered marked points, including the unmarked cases when $g\geq2$; genus-zero values are signed factorials and are given by the genus-zero formula)
How they were obtained:

Every entry is an exact rational number. The generator starts with $\chi_{1,1}=-1/12$ or $\chi_{g,0}=B_{2g}/(2g(2g-2))$ and applies the forgetful-map recurrence using Sage rational arithmetic. No rounding or floating-point division enters the stored values.

more

The independent checker evaluates the closed formula using Python fractions and Bernoulli numbers obtained from a Stirling-number sum. It also compares every entry with a 512-bit Arb computation from positive even zeta values and the zeta functional equation. The checker first tests known values and deliberately incorrect inputs. It checks every adjacent pair in the stored range against the forgetful-map recurrence, the genus-one factorial formula, and every one-marked and unmarked value. The same checks run on values read back from the database. The generator, checks and numerical evidence are attached.



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