Geometry of Riemann surfaces and their moduli spaces

408 Pages·2009·3.146 MB·English

by  Lizhen Ji, Scott A Wolpert, Shing-Tung Yau

SurveysinDifferentialGeometryXIV
Divisors in the moduli spaces of curves
Enrico Arbarello and Maurizio Cornalba
1. Introduction
ThecalculationbyHarer[12]ofthesecondhomologygroupsofthemod-
uli spaces of smooth curves over C can be regarded as a major step towards
the understanding of the enumerative geometry of the moduli spaces of
curves [21, 17]. However, from the point of view of an algebraic geome-
ter, Harer’s approach has the drawback of being entirely transcendental; in
addition, his proof is anything but simple. It would be desirable to provide
a proof of his result which is more elementary, and algebro-geometric in
nature. While this cannot be done at the moment, as we shall explain in
this note it is possible to reduce the transcendental part of the proof, at
least for homology with rational coefficients, to a single result, also due to
Harer[13],assertingthatthehomologyofM ,themodulispaceofsmooth
g,n
n-pointed genus g curves, vanishes above a certain explicit degree. A sketch
oftheproofofHarer’svanishingtheorem,whichisnotatalldifficult,willbe
presented in Section 5 of this survey. It must be observed that Harer’s van-
ishingresultisanimmediateconsequenceofanattractivealgebro-geometric
conjecture of Looijenga (Conjecture 1 in Section 5); an affirmative answer
to the conjecture would thus give a completely algebro-geometric proof of
Harer’stheoremonthesecondrationalhomologyofmodulispacesofcurves.
In this note we describe how one can calculate the first and second
rational (co)homology groups of M , and those of M , the moduli space
g,n g,n
of stable n-pointed curves of genus g, using only relatively simple algebraic
geometryandHarer’svanishingtheorem.ForM ,thisprogramwascarried
g,n
out in [5], where the third and fifth cohomology groups were also calculated
andshowntoalwaysvanish;inSection6,wegiveanoutlineoftheargument,
whichusesinanessentialwayasimpleHodge-theoreticresultduetoDeligne
[10]. In genus zero, we rely on Keel’s calculation of the Chow ring of M ;
0,n
a simple proof of Keel’s result in the case of divisors is presented in Section
4. We finally give a new proof of Harer’s theorem for H2(M ;Q); we also
g,n
Research partially supported by PRIN 2007 Spazi di moduli e teoria di Lie.
(cid:2)c2009InternationalPress
1
2 E.ARBARELLOANDM.CORNALBA
recover Mumford’s result asserting that H1(M ;Q) always vanishes for
g,n
g≥1. The idea is to use Deligne’s Gysin spectral sequence from [9], applied
to the pair consisting ofM and its boundary∂M . This is possible since
g,n g,n
∂M is a divisor with normal crossings in M , if the latter is regarded as
g,n g,n
an orbifold. Roughly speaking, the Gysin spectral sequence calculates the
cohomology of the open variety M =M (cid:2)∂M in terms of the coho-
g,n g,n g,n
mology of the strata of the stratification of M by “multiple intersections”
g,n
of local components of ∂M . Knowing the first and second cohomology
g,n
groups of the completed moduli spaces M makes it possible to explicitly
g,n
compute the low terms of the spectral sequence, and to conclude. Knowing
the first and second homology of the moduli spaces of curves allows one to
also calculate the Picard groups of the latter, as done for instance in [4].
2. Boundary strata in M
g,n
As customary, we denote by M the moduli stack of stable n-pointed
g,n
genus g curves, and by M the corresponding coarse moduli space. It will
g,n
be notationally convenient to allow the marked points to be indexed by
an arbitrary set P with n=|P| elements, rather than by {1,...,n}. The
correspondingstackandspacewillbedenotedbyM andM .Ofcourse,
g,P g,P
we shall write M and M to indicate the open substack and subspace
g,P g,P
parametrizing smooth curves. By abuse of language, we shall usually view
M and M as complex orbifolds.
g,P g,P
As is well known, to any stable P-pointed curve C of genus g one may
attach a graph Γ, the so-called dual graph, as follows. The vertices of Γ are
the components of the normalization N of C, while the half-edges of Γ are
the points of N mapping to a node or to a marked point of C. The edges of
Γ are the pairs consisting of half-edges mapping to the same node, while the
half-edges coming from marked points are called legs. The vertices joined
by an edge {(cid:3),(cid:3)(cid:3)} are those which correspond to the components containing
(cid:3)
(cid:3) and (cid:3). The dual graph comes with two additional decorations; the legs
are labelled by P, and to each vertex v there is attached a non-negative
integer g , equal to the genus of the corresponding component of N. We
v
shall denote by V(Γ), X(Γ), E(Γ) the sets of vertices, half-edges, and edges
of Γ, respectively. The following formula holds:
(cid:2)
g=h1(Γ)+ g .
v
v∈V(Γ)
This implies, in particular, that g depends only on the combinatorial struc-
ture of Γ; we are thus justified in calling it the genus of Γ. The stability
condition for C is 2g−2+|P|>0, and hence can be stated purely in terms
of Γ. We shall say that Γ is a stable P-pointed graph of genus g. Given
(cid:3) (cid:3)
another P-pointed genus g graph Γ, an isomorphism between Γ and Γ con-
sists of bijections V(Γ)→V(Γ(cid:3)) and X(Γ)→X(Γ(cid:3)) respecting the graph
DIVISORS IN THE MODULI SPACES OF CURVES 3
structures, that is, carrying edges to edges, legs labelled by the same letter
into each other, and vertices into vertices of equal genus.
Moduli spaces can be stratified by graph type. By this we mean the
following. Fix a stable P-pointed genus g graph Γ. For each vertex v let P
v
be the subset of P consisting of all elements labeling legs emanating from
v, and denote by H the set of the half-edges originating from v which are
v
not legs. In particular, (cid:3)
P = P .
v
v∈V(Γ)
We denote by D the closure of the locus of points in M representing
Γ g,P
stable curves with dual graph isomorphic to Γ. It easy to see that D is
Γ
a reduced sub-orbifold of M and that, in suitable local coordinates, it
g,P
is locally a union of coordinate linear subspaces of codimension |E(Γ)|. We
(cid:4) (cid:4)
denote by D the normalization of D ; by what we just observed, D is
Γ Γ Γ
smooth. We also set
(cid:5) (cid:5)
MΓ= Mgv,Pv∪Hv, MΓ= Mgv,Pv∪Hv
v∈V(Γ) v∈V(Γ)
We may define clutching morphisms
ξ : M →M
Γ Γ g,P
as follows (cf. [16], page 181, Theorem 3.4). Let x be a point of M ,
g,P
consisting of a P ∪ H -pointed curve C for each vertex v∈V(Γ). Then
v v v
ξ (x) corresponds to the curve obtained from the disjoint union of the C
Γ v
by identifying the points labelled by (cid:3) and (cid:3)(cid:3), for any edge {(cid:3),(cid:3)(cid:3)} of Γ. By
construction, the image of ξ is supported on D ⊂M . The automor-
Γ Γ g,P
phism group Aut(Γ) acts on M in the obvious way. Again by construction,
Γ
ξ induces a morphism
Γ
ξ(cid:4) : M /Aut(Γ)→M ,
Γ Γ g,P
which induces by restriction an isomorphism between M /Aut(Γ) and an
Γ
open dense substack of D . More generally, one can see that M /Aut(Γ) is
Γ Γ
(cid:4)
isomorphic to the normalization of D , that is, to D .
Γ Γ
The graphs giving rise to codimension one strata, that is, the graphs
with only one edge, are easily described. There is a single such graph Γ
irr
withonlyonevertex,anditiscustomarytodenotethecorrespondingdivisor
with D . The other graphs have two vertices, and are all of the form Γ ,
irr a,A
where a is an integer such that 0≤a≤g, and A is a subset of P. A point
of the corresponding divisors, usually denoted by D , consists of an A-
a,A
pointed genus a curve attached at a single point to a (P (cid:2)A)-pointed curve
of genus g−a.
The union of D and of the D is just the boundary ∂M , that is,
irr a,A g,P
the substack of M parametrizing singular stable curves. This is a normal
g,P
crossings divisor in the sense of stacks, which just means that, in suitable
4 E.ARBARELLOANDM.CORNALBA
“local coordinates”, it is locally a union of coordinate hyperplanes. More
generally, for any integer p, the union of all strata D such that |E(Γ)|=p
Γ
is the locus of points of multiplicity at least p in ∂M .
g,P
(cid:3)
Finally, suppose that a stable P-pointed genus g graph Γ is obtained
from another one, call it Γ, by contracting a certain number of edges. In this
(cid:3)
case we write Γ <Γ. There are natural morphisms
ξΓ(cid:2),Γ: MΓ→MΓ(cid:2),
(cid:3)
defined as follows. Suppose for simplicity that Γ is obtained from Γ by
contracting to a point w a set S of edges forming a connected subgraph. We
define a new graph Σ as follows. The edges of Σ are those in S, the vertices
of Σ are the end-vertices of edges in S, and the legs of Σ are the legs of
Γ originating from vertices of Σ, or the half edges in H that are halves of
v
edges in E(Γ)(cid:2)S; the set of the latter will be denoted H(cid:3). We then have
v
(cid:5)
MΣ= Mgv,Pv∪Hv(cid:2)
v∈V(Σ)
⎛ ⎞
(cid:5)
MΓ=⎝ Mgv,Pv∪Hv⎠×MΣ
v∈V(Γ)(cid:2)V(Σ)
⎛ ⎞
(cid:5)
MΓ(cid:2) =⎝ Mgv,Pv∪Hv⎠×Mgw,Qw
v∈V(Γ)(cid:2)V(Σ)
where
(cid:2) (cid:3) (cid:10) (cid:11)
g =h1(Σ)+ g , Q = P ∪H(cid:3)
w v w v v
v∈V(Σ) v∈V(Σ)
We then set
ξΓ(cid:2),Γ=(1,ξΣ), ξΣ: MΣ→Mgw,Pw.
By definition we have
ξΓ(cid:2) ◦ξΓ(cid:2),Γ=ξΓ, if Γ(cid:3)<Γ
An important case is the one in which D is a codimension one stratum in
Γ
DΓ(cid:2). Suppose that there are exactly k edges in Γ having the property that
(cid:3)
contracting one of them produces a graph isomorphic to Γ, and letF be the
set of these edges. Then there are exactly k branches of DΓ(cid:2) passing through
(cid:4) (cid:4)
DΓ. In general, there is no natural morphism from DΓ to DΓ(cid:2). The two are
however connected as follows. Clearly, F is stable under the action of the
automorphism group of Γ. The group Aut(Γ) then acts on M ×F via the
Γ
product action, and we set
D(cid:4)Γ(cid:2),Γ=(MΓ×F)/Aut(Γ)
DIVISORS IN THE MODULI SPACES OF CURVES 5
(cid:4)
TherearetwonaturalmappingsoriginatingfromDΓ(cid:2),Γ:theprojectionπΓ(cid:2),Γ:
D(cid:4)Γ(cid:2),Γ→D(cid:4)Γ=MΓ/Aut(Γ), and the morphism
ξ(cid:4)Γ(cid:2),Γ: D(cid:4)Γ(cid:2),Γ→D(cid:4)Γ(cid:2)
defined as follows. Given a point of M and an edge e∈F, clutching along
Γ
e produces a point in MΓ(cid:2), well defined up to automorphisms of Γ(cid:3). This
defines a morphism MΓ×F →D(cid:4)Γ(cid:2), and it is a simple matter to show that
(cid:4)
in fact this morphism factors through DΓ(cid:2),Γ.
3. Tautological classes and relations
In the Chow ring and in the cohomology ring of the moduli spaces of
curves there are certain natural, or tautological, classes. Here we describe
those of complex codimension one. First of all, we have the classes of the
components of the boundary of M , that is, of the suborbifolds D and
g,P irr
D introduced in the previous section. We denote these classes by δ and
a,A irr
δ , respectively. We write δ to indicate the sum of all the classes δ such
a,A b a,A
that a=b, and δ to indicate the total class of the boundary, that is, the sum
of δ and of all the classes δ . We also set Ac=P(cid:2)A. Next, consider the
irr a,A
projection morphism
(1) πx: Mg,P∪{x}→Mg,P ,
and denote by ω=ω the relative dualizing sheaf. For every p∈P, there
πx
is a section σp: Mg,P →Mg,P∪{x}, whose image is precisely D0,{p,x}. The
remaining tautological classes on M that we will consider are
g,P
ψ =σ∗(ω), p∈P ,
p p
κ1=πx∗(ψx2),
plus the Hodge class λ, which is just the first Chern class of the locally free
sheafπx∗(ω).TheHodgeclassisrelatedtotheothersbyMumford’s relation
(cf. [20], page 102, just before the statement of Lemma 5.14)
κ =12λ−δ+ψ,
1
(cid:12)
where ψ= ψ ; we will not further deal with it in this section.
p
Consider the clutching morphisms ξ and ξ described in the previ-
Γirr Γa,A
ous section. For convenience, we shall denote them by ξ and ξ , respec-
irr a,A
tively. Thus
ξirr: Mg−1,P∪{q,r}→Mg,P ,
(2)
ξa,A: Ma,A∪{q}×Mg−a,(P(cid:2)A)∪{r}→Mg,P .
We would like to describe the pullbacks of the tautological classes under the
morphisms(1)and(2).Itturnsoutthattheformulasforthepullbackunder
6 E.ARBARELLOANDM.CORNALBA
ξ are somewhat messy. On the other hand, for our purposes it will suffice
a,A
to give pullbacks formulas for the simpler map
(3) ϑ: Ma,A∪{q}→Mg,P
which associates to any A∪{q}-pointed genus a curve the P-pointed genus
g curve obtained by glueing to it a fixed Ac∪{r}-pointed genus g−a curve
C via identification of q and r. The following result is proved in [5], Lemmas
3.1, 3.2, 3.3; for simplicity, in the statement we write ξ in place of ξ , and
irr
π in place of π .
x
Lemma 1. The following pullback formulas hold:
i) π∗(κ )=κ −ψ ;
1 1 q
ii) π∗(ψp)=ψp−δ0,{p,q} for any p∈P;
∗
iii) π (δ )=δ ;
irr irr
∗
iv) π (δa,A)=δa,A+δa,A∪{q};
∗
v) ξ (κ )=κ ;
1 1
vi) ξ∗(ψ )=ψ for any p∈P;
p p (cid:2)
vii) ξ∗(δ )=δ −ψ −ψ + δ ;
irr irr q r b,B
(cid:13) q∈B,r(cid:7)∈B
δ if g=2a, A=P =∅,
∗ a,A
viii) ξ (δ )=
a,A
δa,A+δa−1,A∪{q,r} otherwise;
∗
ix) ϑ (κ1)=κ(cid:13)1;
ψ if p∈A,
∗ p
x) ϑ (ψ )=
p 0 if p∈Ac;
∗
xi) ϑ (δ )=δ .
irr irr
Suppose A=P. Then
⎧
⎪⎨δ2a−g,P∪{q}−ψq if (b,B)=(a,P)
xii) ϑ∗(δ )= or (b,B)=(g−a,∅),
b,B ⎪
⎩
δb,B +δb+a−g,B∪{q} otherwise;
Suppose A(cid:10)=P. Then
⎧
⎪⎪⎪⎪−ψq if (b,B)=(a,A)
⎪
⎪⎨ or (b,B)=(g−a,Ac),
xii’) ϑ∗(δ )= δ if B⊂A, (b,B)(cid:10)=(a,A),
b,B ⎪ b,B
⎪
⎪⎪⎪⎪δb+a−g,(B(cid:2)Ac)∪{q} if B⊃Ac, (b,B)(cid:10)=(g−a,Ac),
⎩
0 otherwise.
As shown in [5], it follows from Lemma 1 that in low genus there are linear
relations between κ , the classes ψ , and the boundary classes. For instance,
1 p
in genus zero
(cid:2)
(4) ψ = δ .
z 0,A
z∈A(cid:7)(cid:8)x,y
DIVISORS IN THE MODULI SPACES OF CURVES 7
This formula will be needed in the next section; for the remaining relations
werefertothestatementofTheorem4,wheretheyappearasformulas(13),
(14), (15), and the first formula in (17).
4. Divisor classes in M
0,n
In [15], Keel describes the Chow ring of the moduli space of pointed
curves of genus 0. For brevity, when writing the boundary divisors of M
0,P
we will drop the reference to the genus (g=0) and we will write D instead
S
of D . Similarly, for the divisor classes we will write δ instead of δ .
0,S S 0,S
Keel’s theorem is the following.
Theorem 1 (Keel [15]). The Chow ring A∗(M ) is generated by the
0,P
classes δ , with S ⊂P and |S|≥2, |Sc|≥2. The relations among these gen-
S
erators are generated by the following:
1) δ =δ ,
S Sc
2) For any quadruple of distinct elements i,j,k,l∈P,
(cid:2) (cid:2) (cid:2)
δ = δ = δ .
S S S
i,j∈S;k,l∈/S i,k∈S;j,l∈/S i,l∈S;j,k∈/S
3) δ δ =0, unless S ⊂T, S ⊃T, S ⊂Tc or S ⊃Tc.
S T
Moreover, A∗(M )=H∗(M ;Z).
0,P 0,P
We will not give a proof of this theorem. Instead, after a few general com-
ments, we will give the complete computation of the first Chow group
A1(M ), showing that it coincides with H2(M ;Z).
0,P 0,P
The relations 1), 2) and 3) can be easily proved. Relations 1) are obvi-
ous. Relations 3) follow immediately from the fact that D and D do not
S T
physically meet except in the cases mentioned; alternatively, one can use
part xii’) of Lemma 1. To get the relations in 2) look at the morphism
πi,j,k,l: M0,P →M0,{i,j,k,l}
defined by forgetting all the points in P with the exception of i, j, k, l
and stabilizing the resulting curve. Look at the divisor classes δ{i,j} and
δ{i,k} on M0,{i,j,k,l}. The pull-backs of δ{i,j} and δ{i,k} via πi,j,k,l are given,
respectively, by (cid:2) (cid:2)
δ , δ .
S S
i,j∈S;k,l∈/S i,k∈S;j,l∈/S
The fact that δ{i,j}=δ{i,k}∈A1(M0,{i,j,k,l})=A1(P1) gives, by pull-back,
the first relation in 2). The second is obtained in a similar way.
We now concentrate our attention on the first Chow group A1(M ).
0,P
Set n=|P|. Recall first that M parametrizes ordered n-tuples of distinct
0,P
points of P1, modulo automorphisms. Fixing the first three points to be
0,1,∞ kills all automorphisms; hence M can be identified with the space
0,P
8 E.ARBARELLOANDM.CORNALBA
of ordered (n−3)-tuples of distinct points of P1(cid:2){0,1,∞}, that is, with the
complement of the big diagonal in (C(cid:2){0,1})n−3. In particular, M is the
0,P
complement of a union of hyperplanes in Pn−3. It follows that the Picard
group of M is trivial, and that M is birationally equivalent to Pn−3.
0,P 0,P
As a consequence, h0,1(M )=h0,2(M )=0, and hence
0,P 0,P
A1(M )=Pic(M )=H2(M ;Z).
0,P 0,P 0,P
We now show that Pic(M ) is generated by boundary divisors. Let L be a
0,P
linebundleonM .SincethePicardgroupofM iszero,therestrictionof
0,P 0,P
LtoM istrivial,i.e.,thereisameromorphicsectionsofLwhichisregular
0,P
and nowhere vanis(cid:12)hing when restricted to M0,P. Therefore the divisor of s is
of the(cid:12)form (s)= niDi, where the Di are boundary divisors, so that L=
O(− n D ). We conclude that H2(M ;Z)=A1(M ) is generated by
i i 0,P 0,P
the classes δ , with S ⊂P and |S|≥2, |Sc|≥2, with the following relations
S
1) δ =δ ;
S Sc
2) for any quadruple of distinct elements i,j,k,l∈P,
(cid:2) (cid:2) (cid:2)
δ = δ = δ .
S S S
i,j∈S;k,l∈/S i,k∈S;j,l∈/S i,l∈S;j,k∈/S
As a consequence, Keel’s theorem for H2(M ;Z)=A1(M ) is implied
0,P 0,P
by the following result.
Proposition 1. Let P be a finite set with n=|P|≥4 elements. Fix
distinct elements i,j,k∈P. Then H2(M ;Z) is freely generated by the
0,P
classes
(5) δ , with i∈S and 2≤|S|≤n−3 or S ={j,k}.
S
In particular
(cid:18) (cid:19)
(cid:10) (cid:11) (cid:10) (cid:11)
n
rank H2(M ;Z) =rank A1(M ) =2n−1− −1.
0,P 0,P
2
Proof. Let F be the subgroup of H2(M ;Z) generated by the ele-
0,P
ments in (5). The only boundary divisors not appearing in (5) are the divi-
sors δ{s,t}, where s(cid:10)=i, t(cid:10)=i and {s,t}(cid:10)={j,k}. We write relation 2) for the
quadruple i,k,s,t
(cid:2) (cid:2)
δ{i,k}+δ{s,t}+ δS =δ{i,t}+δ{s,k}+ δS.
3≤|S|≤n−3 3≤|S|≤n−3
i,k∈S;s,t∈/S i,t∈S;k,s∈/S
Thus δ ≡δ modulo F. Now starting from the quadruple i,j,k,s we get
s,t s,k
δ ≡0 modulo F.
s,k
Denote by FP the set of all elements of the form (5). We must prove
i,j,k
thattheelementsofFP arelinearlyindependent.Weproceedbyinduction
i,j,k
DIVISORS IN THE MODULI SPACES OF CURVES 9
on n=|P|. The case n=4 is trivial. We assume n≥5. Suppose there is a
relation
(cid:2)
(6) νδ + ν δ =0
j,k S S
i∈S,2≤|S|≤n−3
Fix a∈P (cid:2){i}. We are going to pull back this relation to M0,P∪{x}(cid:2){i,a}
via the map ϑ defined in (3), where now A=P (cid:2){i,a} and g=0. Let us
first assume that a(cid:10)=j and a(cid:10)=k. Recalling parts xii) and xii’) of Lemma 1,
the pull-back of the left-hand side of (6) via ϑ is given by
(cid:2)
(7) νSδS∪{x}(cid:2){i,a}−ν{i,a}ψx+νδ{j,k}
S⊃{i,a},|S|≤n−3
We now use the expression for ψ given by (4), where we consider ψ as an
x x
element of H2(M0,P∪{x}(cid:2){i,a}). We get
(cid:2)
ψx=δ{j,k}+ δT .
x∈T⊂P∪{x}(cid:2){i,a}
j,k∈Tc,2≤|T|≤n−4
The relation (7) becomes
(cid:2) (cid:2)
νSδS∪{x}(cid:2){i,a}+ (νS −ν{i,a})δS∪{x}(cid:2){i,a}
S⊃{i,a},|S|≤n−3 S⊃{i,a},|S|≤n−3
j∈Sork∈S j,k∈Sc
+(ν −ν{i,a})δ{j,k}.
We may now apply the induction hypothesis to FP∪{x}(cid:2){i,a}. Since a is
x,j,k
arbitrary, as long as a(cid:10)=j and a(cid:10)=k we deduce that
ν{i,a}=ν for a(cid:10)=j, a(cid:10)=k;
ν =0 for j ∈S or k∈S, i∈S, S (cid:10)={i,j}, S (cid:10)={i,k} and |S|≤n−3;
S
ν =ν for j,k∈Sc, i∈S and |S|≤n−3.
S
Using again the general expression for ψ in terms of the boundary divisors
i
given by (4), the original relation (6) can be written as
(cid:20) (cid:2) (cid:21)
ν{i,j}δ{i,j}+ν{i,k}δ{i,k}+ν δS +νδ{j,k}
2≤|S|≤n−3
i∈S;j,k∈Sc
=ν{i,j}δ{i,j}+ν{i,k}δ{i,k}+νψi=0
We pull back this relation to MP∪{x}(cid:2){j,k} and we get ν =0. Let l(cid:10)=k. We
then pull back the resulting relation to M{i,j,l,x} and we get ν{i,j}=0. But
then ν{i,k}=0 as well. (cid:2)
10 E.ARBARELLOANDM.CORNALBA
5. The vanishing of the high degree homology of M
g,n
We now begin our computation of the first and second cohomology
groupsofthemodulispacesofcurves.Itisusefultoobservethattherational
cohomology of the orbifold M coincides with the one of the space M
g,n g,n
(cf. [2], Proposition 2.12), and likewise the rational cohomology of M is
g,n
the same as the one of M , so that on many occasions we will be able
g,n
to switch from the orbifold point of view to the space one, and conversely,
when needed.
One of the basic results about the homology of M is that it vanishes
g,n
in high degree. Set
⎧
⎪⎨n−3 if g=0;
(8) c(g,n)= 4g−5 if g>0, n=0;
⎪
⎩
4g−4+n if g>0, n>0.
The vanishing theorem, due to John Harer (cf. [13], Theorem 4.1), is the
following:
Theorem 2. H (M ;Q)=0 for k>c(g,n).
k g,n
The case g=0 is straightforward. As we observed in the previous section,
M is an affine variety of dimension n−3, so that its homology vanishes
0,n
in degrees strictly greater than n−3. Actually, a similar proof would yield
the general result if one could establish the following conjecture.
Conjecture 1. (Looijenga). Let g and n be non-negative integers such
that 2g−2+n>0. Then M is the union of g affine subsets if n(cid:10)=0, and
g,n
is the union of g−1 affine subsets if n=0.
We are now going to recall Harer’s proof of Theorem 2. From now on we
assumeg>1.Weonlytreatthecasen=1.Thenwewillshowhowtoreduce
the other cases to this one.
We fix a compact oriented surface S and a point p∈S and we denote
by A the set of isotopy classes, relative to p, of loops in S based at p. We
also require that no class in A represents a homotopically trivial loop in S.
Thearc complex A=A(S;p) isthesimplicialcomplexwhosek-simplicesare
given by (k+1)-tuples a=([α ],...,[α ]) of distinct classes in A which are
0 k
representable by a (k+1)-tuple (α ,...,α ) of loops intersecting only in p.
0 k
ThegeometricrealizationofAisdenotedby|A|.Asimplex([α ],...,[α ])∈
0 k
A is said to be proper if S(cid:2)∪k α is a disjoint union of discs. The improper
i=0 i
simplices form a subcomplex of A denoted by A∞. We set
A0=A(cid:2)A∞ and |A0|=|A|(cid:2)|A∞|.
ThemappingclassgroupΓ actsonAintheobviousway.Fora=([α ],...,
g,1 0
[α ])∈A and [γ]∈Γ we define
k g,1
(9) [γ]·a=([γα ],...,[γα ]).
0 k