arXiv:1310.7369v3 [math.AG] 15 Apr 2014
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO
CURVES OF COMPACT TYPE
DAN PETERSEN
Abstract. We prove that the tautological ring of Mct
2,n , the moduli space of n-pointed
genus two curves of compact type, is not Gorenstein for any n ≥ 8. This result is obtained
via a more general study of the cohomology groups of Mct
2,n . We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the
tautological cohomology can be identified within this decomposition. Our results allow the
computation of H k (Mct
2,n ) for any k and n considered both as Sn -representation and as
mixed Hodge structure/ℓ-adic Galois representation considered up to semi-simplification.
1. Introduction
The tautological classes on the moduli spaces Mg,n of n-pointed stable genus g curves can
be defined, following [Faber and Pandharipande 2005] to be the algebraic cycle classes of
‘geometric origin’. Specifically, we declare the fundamental classes of all spaces Mg,n to be
tautological, that the pushforward of tautological classes along the natural maps
Mg,n+1 × Mg′ ,n′ +1 → Mg+g′ ,n+n′ ,
Mg−1,n+2 → Mg,n ,
Mg,n+1 → Mg,n
should be tautological, and that intersections of tautological classes are tautological. In this
way we obtain a collection of subrings R• (Mg,n ) ⊂ A• (Mg,n ), the tautological rings.
We can also consider the images of the tautological rings in cohomology, denoted RH• (Mg,n ).
Finally, if U ⊂ Mg,n is Zariski open, then we let R• (U ) = Im R• (Mg,n → A• (U ) , and we
define similarly RH • (U ).
ct
Particularly interesting (for us) are the Zariski open sets Mrt
g,n ⊂ Mg,n ⊂ Mg,n parametrizing
curves with rational tails and of compact type, respectively. A stable n-pointed genus g curves
is of compact type if its dual graph is a tree; equivalently, if its jacobian is an abelian variety.
It has rational tails if one of its components has geometric genus g, implying that all other
components are trees (or ‘tails’) of rational curves attached to this genus g component.
Faber and Pandharipande [Faber 1999; Faber 2001; Pandharipande 2002] proposed a uniform
•
ct
•
conjectural description of the tautological rings R• (Mrt
g,n ), R (Mg,n ) and R (Mg,n ). These
conjectures naturally split into several smaller pieces. First were the vanishing and socle
2010 Mathematics Subject Classification. 14H10, 14C17, 32S60, 14N35, 14D07, 55R55.
Key words and phrases. tautological ring, Faber conjectures, moduli of curves, Gromov–Witten theory,
cohomology of moduli spaces.
1
2
DAN PETERSEN
conjectures. These assert that
∼
Rg−2+n−δ0,g (Mrt
g,n ) = Q,
∼ Q,
R2g−3+n (Mct ) =
g,n
Rk (Mrt
g,n ) = 0 for k > g − 2 + n − δ0,g ,
Rk (Mct
g,n ) = 0 for k > 2g − 3 + n,
R3g−3+n (Mg,n ) ∼
=Q
(and obviously, Rk (Mg,n ) = 0 for k > 3g − 3 + n). Here δ0,g is the Kronecker delta. Given
these statements, one can (after choosing a generator for the top degree) describe the pairing
into the top degree in terms of proportionalities. On Mg,n these top degree intersection
numbers are all determined by Witten’s conjecture (Kontsevich’s theorem). The second
part of the conjectures were explicit expressions determining the top intersections also in
the rational tails and compact type cases: the λg λg−1 -conjecture and the λg -conjecture,
respectively.
The vanishing, socle and intersection number conjectures are now all theorems. This represents work of a large number of people, and all of the statements now have several different
proofs, enlightening in their own way. See the survey [Faber 2013] and the detailed references
therein.
However, the final part of the conjectures is now known to be false in general. The Gorenstein conjecture proposed that the pairing into the top degree is always perfect, so that the
tautological rings enjoy Poincaré duality. A different way of stating this is in terms of the
relations between the generators of the tautological rings: every potential relation between
tautological classes that is consistent with the top degree pairing is actually a true relation.
In [Petersen and Tommasi 2014] we showed that this conjecture fails on M2,n . Before this,
computer calculations had been used to find examples where it seems likely that the Gorenstein conjecture fails, also in the rational tails and compact type cases, see [Faber 2013; Yin
2012; Pixton 2013].
The main result of this paper is that the conjecture fails also on Mct
2,n . In fact we observe
Gorenstein failure much sooner: the results of [Petersen and Tommasi 2014] imply that
R• (M2,20 ) is not Gorenstein, and we conjecture that this is optimal. But in the compact
type case we find Gorenstein failure already when n = 8. Even though computer assisted
computations of tautological rings for small g and n have not yet gotten as far as Mct
2,8 , it
is not inconceivable that one could actually determine the intersection matrices in this case
with enough computing power and a clever implementation. Thus one could for instance hope
to explicitly write down a nonzero tautological class that pairs trivially with all tautological
classes in opposite degree. (The proof given here seems not explicit enough to produce such
a class). Doing something like this for M2,20 is utterly doomed to fail.
The basic strategy in [Petersen and Tommasi 2014] was to study instead the tautological
cohomology ring RH • (M2,n ), and to prove that its Betti numbers are not symmetric about
the middle degree. This certainly implies that the tautological cohomology ring can not have
a perfect pairing, and then also the tautological ring itself. This is what we do in this paper,
too.
The results will follow from a more general study of the cohomology of Mct
2,n . We approach
ct
ct
the cohomology of Mct
2,n via the Leray spectral sequence for f : M2,n → M2 . Our first
result, Theorem 2.1, is that the Leray spectral sequence degenerates, and therefore that the
ct
cohomology of Mct
2,n can be expressed in terms of the cohomology of local systems on M2
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
3
and local systems supported on the boundary. Moreover, after the results in [Petersen 2013b]
we actually know the cohomology of all these local systems in any degree, together with
their mixed Hodge structure up to semi-simplification. This allows us to obtain very detailed
k
ct
information about the cohomology of Mct
2,n : we can compute H (M2,n ) for any k and n considered both as Sn -representation and as mixed Hodge structure/ℓ-adic Galois representation
(considered up to semi-simplification). This has been implemented on a computer, in a way
described in Section 5.
The results described in the preceding paragraph provide a decomposition of the cohomology
of Mct
2,n into pieces corresponding to different local systems. We study both the pure cohok
k
ct
ct
mology PH k (Mct
2,n ) = Wk H (M2,n ) and its subspace RH (M2,n ). We are able to describe
both these subspaces
•
ct
•
ct
RH • (Mct
2,n ) ⊂ PH (M2,n ) ⊂ H (M2,n )
in terms of the decomposition into local systems (Theorems 3.3 and 3.8). Moreover, Poincaré–
ct
p n+3+i
Verdier duality applied to the morphism f : Mct
(Rf∗ Q) ∼
=
2,n → M2 , in the form H
•
p n+3−i
∨
ct
H
(Rf∗ Q) , implies (Theorem 3.4) that PH (M2,n ) can be written as a part that is
symmetric about the middle degree (corresponding to the trivial local systems on Mct
2 and
on the boundary) and a part that is obviously not symmetric. The latter part is nonzero for
any n ≥ 8.
•
ct
The subspace RH • (Mct
2,n ) ⊂ PH (M2,n ) consists of the whole ‘symmetric’ part of the pure
cohomology, and a subspace of the non-symmetric part defined by the image of a certain Gysin
map. In [Petersen and Tommasi 2014] we conjectured that this Gysin map is never zero, but
we could not prove this. However, Section 4 of this paper presents a direct geometric argument
•
•
ct
ct
showing that RH • (Mct
2,8 ) = PH (M2,8 ), implying that the Betti numbers of RH (M2,8 ) are
not symmetric about the middle degree.
Our results imply in fact that the Betti numbers of RH • (Mct
2,n ) are always bigger above the
middle degree than below it for n ≥ 8, and similarly that the Betti numbers of RH • (M2,20 )
are always bigger above the middle for n ≥ 20 (the latter statement improves on what we
proved in [Petersen and Tommasi 2014]). In particular, all the tautological rings for larger
n fail to be Gorenstein, too (although the fact that Gorenstein failures ‘propagate’ to higher
n can be proven more directly by choosing a non-zero tautological class pairing to zero with
everything of complementary degree; its pullback under a forgetful map will have the same
property by the projection formula, and it will be nonzero because pullback in Chow under
a proper surjection is injective).
It would be interesting if this phenomenon, that the tautological ring is bigger above the
middle degree than below it, always were true. Perhaps this could be explained by a weak
version of the hard Lefschetz theorem, that iterated multiplication by an ample class defines
an injection, as in [Björner and Ekedahl 2009].
1.1. Acknowledgements. This work was carried out in the group of Pandharipande at ETH
Zürich, supported by grant ERC-2012-AdG-320368-MCSK. I am grateful to Rahul Pandharipande for his interest in this work and for suggesting the argument in Section 4. In making
the computer implementation described in Section 5 I benefited greatly from code written
by Carel Faber. I also wish to thank an anonymous referee for their careful reading of this
manuscript, and for noticing a major error in the original version of Theorem 2.1.
4
DAN PETERSEN
2. An application of the decomposition theorem
In the study of the spaces M2,n in [Petersen and Tommasi 2014] we found it useful to compute
rt
the cohomology of the spaces Mrt
2,n by means of the fibration f : M2,n → M2 and the Leray
rt
spectral sequence. The particular choice of M2,n and not any other (partial) compactification
of M2,n is motivated by the fact that f is a smooth projective morphism. Then it is known
that the Leray spectral sequence immediately degenerates [Deligne 1968], and that the sheaves
Rq f∗ Q are the underlying local systems of polarized variations of Hodge structure and that
the Leray spectral sequence is compatible with the natural mixed Hodge structures on the
terms (see [Arapura 2005] for a direct proof).
ct
In this paper we will consider instead the maps f : Mct
2,n → M2 . This morphism is no
longer smooth, but it’s still proper. A similar analysis can be carried out in this case, too,
but now perverse sheaves, the decomposition theorem, and Saito’s theory of mixed Hodge
modules become unavoidable. However, it turns out that the situation is in a sense as nice as
possible: all perverse sheaves occuring in the decomposition of Rf∗ Q are actual sheaves, so
the conclusion of the decomposition theorem holds also for the usual t-structure, the ordinary
Leray spectral sequence degenerates, et cetera.
The reader who is not familiar with perverse sheaves and the decomposition theorem can
feel free to skip directly to Theorem 2.1 and proceed to read the rest of the paper. Nice
introductions to this are [de Cataldo and Migliorini 2009; Dimca 2004]. For mixed Hodge
modules specifically see also [Saito 1989].
ct
Let f : Mct
2,n → M2 be the map that forgets the n markings. Note that f is proper, so that
there exists an isomorphism
M
p i
Rf∗ ICMct2,n ∼
H (Rf∗ ICMct2,n )[−i].
=
i
in Dcb (Mct
2 ) according to the decomposition theorem of [Beı̆linson, Bernstein, and Deligne
1982]. Here we denote by p Hi (K) the perverse cohomology sheaves of a complex K, always
with respect to the middle perversity. We can say more. First of all, since Mct
2,n is smooth we
ct
ct
have ICM2,n = Q[dim M2,n ] = Q[3 + n]. Thus the decomposition theorem implies a similar
decomposition of Rf∗ Q in the derived category, just with a degree shift.
Second, according to the semisimplicity theorem, each p Hi (Rf∗ Q) is a direct sum of simple
2
ct
perverse sheaves. Let Mct
2 = M2 ∪ Sym (M1,1 ) be the stratification of M2 according to
topological type. Then f is a stratified map (even a stratified submersion); the restriction of
f to the inverse image of a stratum is a fiber bundle. Since in addition both strata are smooth
(in the sense of stacks), it is known that each of the simple perverse sheaves in p Hi (Rf∗ Q) is
obtained as an intermediate extension of a local system on one of the two strata.
Let V be a local system on one of these strata, which occurs in the decomposition of Rf∗ Q,
and let us consider its intermediate extension. If the stratum is Sym2 (M1,1 ), which is closed
in Mct
2 , then the intermediate extension is simply the usual pushforward. If the stratum is
ct
M2 , then we claim that the local system extends to a local system on Mct
2 . Since M2 is
smooth, this extension will then be equal to the intermediate extension. We should prove
that the monodromy of V vanishes around Sym2 M1,1 . But the monodromy is given by a
Dehn twist around a vanishing cycle, and since we are in the case of compact type, all nodes
of the curves involved will be separating, which means that we only consider Dehn twists
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
5
around separating curves. But such a Dehn twist lies in the Torelli group, that is, it acts
trivially on the cohomology of the universal curve (and then all its fibered powers, et cetera),
from which the result follows. An equivalent way of thinking about this is that the local
systems occurring on M2 will all be symplectic local systems, which means in general that
they are the pullback of a local system on the moduli space of principally polarized abelian
varieties under the Torelli map. But in genus two, we have A2 ∼
= Mct
2 . (And more generally,
all symplectic local systems on Mg extend to local systems on Mct
g .)
By what we have said, the perverse sheaves occurring in the decomposition of Rf∗ Q will all
have the form j∗ V, where V is a local system on a closed subvariety and j is the inclusion.
In particular, these perverse sheaves are bona fide sheaves (not just complexes of sheaves).
Third, we have Poincaré–Verdier duality in this situation,
p
Hi (Rf∗ ICMct2,n ) ∼
= p H−i (Rf∗ ICMct2,n )∨
for all i. Recall that on a smooth variety, one has that
p
Hi (K) = Hi−d (K)[d]
if the complex K has cohomology sheaves which are local systems supported on a closed
subvariety of dimension d. So let us write
Rf∗ Q = A ⊕ B
2
where A has cohomology sheaves supported on Mct
2 and B is supported on Sym M1,1 (this
is possible, according to what we have said so far). Then
p
Hi (ICMct2,n ) = p Hi (Rf∗ Q[n + 3]) = p Hn+3+i (A ⊕ B)
= Hn+i (A)[3] ⊕ Hn+1+i (B)[2].
Thus by Poincaré–Verdier duality, Hn+i (A) ∼
= Hn−i (A)∨ and Hn+1+i (B) ∼
= Hn+1−i (B)∨ . As
said before, these will all be symplectic local systems, which in particular implies that they
are in fact self-dual (as all representations of Sp(2g) are).
Finally we shall also need Saito’s Hodge-theoretic interpretation of the above [Saito 1990].
In the same way that an admissible variation of mixed Hodge structure has an underlying local system, there is the notion of a mixed Hodge module which has an underlying
b
ct
perverse sheaf. The statements above can be lifted from Rf∗ : Dcb (Mct
2,n ) → Dc (M2 ) to
b
ct
Rf∗ : Db (MHM(Mct
2,n )) → D (MHM(M2 )), the derived category of mixed Hodge modules.
In the context of mixed Hodge modules we can talk about weights and purity. It is known that
Rf∗ preserves purity for a proper morphism and that the intersection complex ICMct2,n is pure
(in fact, this is one way of proving the decomposition theorem). What we get from Saito’s
theory is that each p Hi (Rf∗ ICMct2,n ) — but then also each Ri f∗ Q, by what was said above
— naturally obtains the structure of a mixed Hodge module, and that the Leray spectral
sequence is compatible with mixed Hodge structure.
Concretely, what this means is that each Rq f∗ Q (considered as a mixed Hodge module
on Mct
2 ) is a direct sum of terms, each of which is either a polarized variation of Hodge
structure of weight q or a pushforward of one from Sym2 M1,1 . The cohomology of Rq f∗ Q
thus obtains a mixed Hodge structure, and the claim is that the Leray spectral sequence
q
p+q
H p (Mct
(Mct
2 , R f∗ Q) =⇒ H
2,n ) is compatible with this mixed Hodge structure (and
degenerates immediately).
6
DAN PETERSEN
We summarize what has been said so far in the following theorem.
ct
Theorem 2.1. Let f : Mct
2,n → M2 . Then:
(1) Rf∗ Q is a direct sum of its cohomology sheaves in the derived category, both for the
perverse and the classical t-structure. In particular, the Leray spectral sequence for f
degenerates.
(2) Each Ri f∗ Q is a pure Hodge module of weight i.
(3) Write Ri f∗ Q = Ai ⊕ B i , where A• (resp. B • ) consists of those summands supported
2
n+i ∼
on Mct
= An+1−i (−i),
= An−i (−i) and B n+1+i ∼
2 (resp. Sym M1,1 ). Then A
where the (−i) denotes a Tate twist.
In particular, the proposition tells us that we can compute H • (Mct
2,n ) from the knowledge
of the cohomology of the fibers of f (considered as local systems), and the cohomology of
symplectic local systems on the closures of strata in the topological type stratification of
•
ct
Mct
2 . If we know the fibers of f as Sn -representations then we also get H (M2,n ) as an Sn representation; if we know the cohomology of the symplectic local systems as mixed Hodge
structures then we obtain also H • (Mct
2,n ) as a mixed Hodge structure, and so on.
3. Tautological and non-tautological cohomology in genus two and
Poincaré duality
As we saw in the previous section, we need only to determine
H • (Sym2 M1,1 , V)
and
H • (Mct
2 , V)
for symplectic local systems V in order to determine H • (Mct
2,n ). Recall that symplectic local
systems meant ones associated to rational representations of the symplectic group. Thus on
Mct
2 we have local systems corresponding to representations of Sp(4). These are parametrised
by their highest weights, which are integers l ≥ m ≥ 0. We write Vl,m for the corresponding
local system. On M1,1 × M1,1 this means a local system given by a pair of representations of
SL(2) ∼
= Sp(2). On Sym2 M1,1 we should actually be a bit more precise about what the local
systems are: we are really considering representations of (SL(2) × SL(2)) ⋊ S2 . See [Petersen
2013a].
∼
The cohomology of the local systems Vl,m on Mct
2 = A2 is determined in any degree by the
author in [Petersen 2013b], building off the work of [Harder 2012]. In the results of loc. cit. we
consider the cohomology groups as mixed Hodge structures (or ℓ-adic Galois representations)
up to semi-simplification. The cohomology of local systems on Sym2 M1,1 is determined by
the cohomology of local systems on M1,1 , in a way made precise in [Petersen 2013a]. In this
way one can compute the cohomology of Mct
2,n for any n in any degree. We will return to
this later; let us for now focus on the pure cohomology.
3.1. The pure cohomology does not have Poincaré duality.
k
Definition 3.1. Let X be an algebraic variety. We define PH k (X) = grW
k H (X) and we
call this the pure cohomology. More generally, if V is a Hodge module on X that is pure of
k
weight w, then PH k (X, V) = grW
k+w H (X, V).
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
7
Remark 3.2. If X is a singular variety, this notation is potentially confusing since the
constant sheaf Q is not in general pure of weight zero. For our purposes this will not cause
any problems, since we mostly consider smooth varieties. On a smooth variety U , the pure
cohomology is a subalgebra and one can give the simple intrinsic definition
PH • (U ) = Im (H • (X) → H • (U )) ,
where X is any smooth compactification [Deligne 1971, Corollaire 3.2.17]. In particular, it
follows from this that if U ⊂ V is an open immersion of smooth varieties, then PH • (V ) →
PH • (U ) is surjective, a property we will use repeatedly throughout this paper. For smooth
U , the cycle class map A• (U ) → H 2• (U ) has its image contained in the pure cohomology. In
particular, the pure cohomology contains the tautological cohomology.
Since the category of pure Hodge structures of given weight is semi-simple, there is in fact an
isomorphism
M
q
∼
PH • (Mct
PH p (Mct
2,n ) =
2 , R f∗ Q).
p,q
Consider first M1,1 , where we have the local systems Vk associated to the kth symmetric
power of the standard representation of SL2 . Then Eichler–Shimura theory (or rather its
Hodge-theoretic interpretation [Zucker 1979]) tells us that


i=k=0
Q
i
H (M1,1 , Vk ) = S[k + 2] ⊕ Q(−k − 1) i = 1, k ≥ 2 is even


0
else
where S[k + 2] is the Hodge structure attached to cusp forms for SL(2, Z) of weight k + 2.
(That one has a direct sum decomposition can be proven as in [Elkik 1990].) In particular
only the trivial local system has nontrivial pure even cohomology, and for local systems of
weight less than 10 there is no pure odd cohomology either. On Sym2 M1,1 one finds that for
weights below 20 the only local system with nontrivial pure even cohomology is the trivial
local system, but that for higher weights one finds classes in H 2 (Sym2 M1,1 , V) associated to
tensor products of Hodge structures attached to cusp forms. One even finds classes of Tate
type, since if M is a 2-dimensional Hodge structure attached to a cusp eigenform for SL(2, Z)
of weight w then ∧2 M ∼
= Q(1 − w).
We do not give the complete answer for the cohomology of local systems on Mct
2 from [Petersen
2013b] since it is slightly complicated. However, it turns out that most of the pure cohomology
is concentrated in the middle degree, and in particular it has odd weight since Vl,m has no
cohomology for l + m odd. Outside the middle degree there are not many contributions to
the pure cohomology: there is the trivial local system V0,0 , which has
H 0 (A2 , V0,0 ) = Q
and
H 2 (A2 , V0,0 ) = Q(−1),
and there are the local systems V2a,2a , for which H 2 (A2 , V2a,2a ) is pure Tate and of the same
dimension as the space of cusp forms for SL(2, Z) of weight 4a + 4. There is also a possible
contribution in H 4 which is conjectured to always vanish and which we’ll ignore.1
To summarize, the situation is as follows.
1There is a pure Tate class in H 4 (A , V
2
2a,2a ) for each normalized cusp eigenform for SL(2, Z) of weight
4a + 4 such that the central value of the attached L-function vanishes. Maeda’s conjecture would imply that
there are no such cusp forms [Conrey and Farmer 1999]; this has been verified numerically for weights up to
14000 [Ghitza and McAndrew 2012].
8
DAN PETERSEN
Theorem 3.3. The pure even cohomology of Mct
2,n can be decomposed into contributions
from different local systems:
(1)
(2)
(3)
(4)
the trivial local system on Mct
2,
the trivial local system on Sym2 M1,1 ,
the local systems V2a,2a , a > 0, on Mct
2,
local systems of weight ≥ 20 on Sym2 M1,1 .
Let us now study (the failure of) Poincaré duality for PH even (Mct
2,n ). As in Theorem 2.1, let
us write R• f∗ Q = A• ⊕ B • , where A• is a direct sum of local systems Vl,m and B • a direct
sum of pushforwards of local systems from Sym2 M1,1 .
2
Consider first the contributions from the trivial local systems on Mct
2 and Sym M1,1 . We
ct
saw above that the (pure) cohomology of the trivial local system on M2 is symmetric about
degree 1. On the other hand we see from Theorem 2.1 that the trivial local system has the
same multiplicity in An+q and An−q , so that the contribution to PH • (Mct
2,n ) from this local
system is symmetric about degree n + 1.
The space Sym2 M1,1 has the rational cohomology of a point, so its (pure) cohomology is
symmetric about degree 0. On the other hand we see from Theorem 2.1 that the trivial local
system has the same multiplicity in B n+1+q and B n+1−q , so also this local system has the
n+1−q
same contribution to PH n+1+q (Mct
(Mct
2,n ) and PH
2,n ).
On the other hand, this symmetry fails for every other local system with nonzero pure cohomology, also if we restrict to even cohomology or pure Tate cohomology. The nontrivial
local system on Mct
2 of smallest weight with nonzero pure cohomology is V4,4 , which has
1-dimensional pure Tate cohomology in degree 2 (coming from the one-dimensional space of
weight 12 cusp forms for SL(2, Z).) The local system V4,4 occurs in R8 f∗ Q when n = 8 and
then also in other degrees for any n ≥ 8. Similarly all local systems V2a,2a have nonzero
pure cohomology in degree 2 for a ≥ 2, so their contribution to the even pure cohomology of
2
Mct
2,n is symmetric about degree n + 2. The local systems of weight ≥ 20 on Sym M1,1 with
nonzero pure cohomology in degree 2 give a contribution which is symmetric about degree
n + 3.
To summarize, we have proven the following results.
Theorem 3.4. The four summands in Theorem 3.3 are symmetric about degrees n + 1, n + 1,
n + 2 and n + 3, respectively.
n+1+q
Corollary 3.5. The pure cohomology PH • (Mct
(Mct
2,n ) satisfies the symmetry dim PH
2,n ) =
n+1−q
ct
dim PH
(M2,n ) if and only if n < 8.
Proof. The third and fourth summand in Theorem 3.3 both vanish if and only if n < 8.
In Section 4 we shall see that the pure cohomology of Mct
2,8 is tautological, so that this implies
that the tautological ring of Mct
is
not
Gorenstein.
2,8
3.2. Tautological and non-tautological cohomology. In this subsection we will identify
even
(Mct
the subspace RH • (Mct
2,n ) in terms of local systems. In the process we give a
2,n ) ⊂ PH
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
9
simpler proof of the main result of [Petersen and Tommasi 2014, Section 5], that all boundary
classes on M2,20 are tautological.
We will need to recall some results of [Petersen and Tommasi 2014]. Let g : Mrt
2,n → M2 .
Then the Leray spectral sequence for g degenerates and there is an isomorphism
M
∼
PH • (Mrt
PH p (M2 , Rq g∗ Q),
2,n ) =
p,q
all of which is simpler than in the compact type case. In [Petersen and Tommasi 2014, Section
3] we studied the cohomology of local systems on M2 and A2 . After [Petersen 2013b] those
results can be improved: we now know that the only local systems on M2 with nonzero even
pure cohomology are the trivial one and the restrictions of the local systems V2a,2a for a large
enough (a ≥ 5 will suffice, but we can not rule out that some smaller a may be possible).
In [Petersen and Tommasi 2014, Section 4], we identified RH • (Mrt
2,n ) with the contribution
from the trivial local system. Thus there is a decomposition
•
rt
•
PH even (Mrt
2,n ) = RH (M2,n ) ⊕ Q ,
where Q• denotes the non-tautological even pure cohomology: it is the contributions from
n−q
H 2 (M2 , V2a,2a ). The same reasoning as before shows that dim RH n+q (Mrt
(Mrt
2,n ) = dim RH
2,n )
(and it’s not hard to see that the ring is actually Gorenstein), giving a proof in cohomology
of the result of [Tavakol 2013] that the tautological ring of Mrt
2,n is Gorenstein. Also we see
that dim Qn+2+q = dim Qn+2−q .
We need the following lemma.
Lemma 3.6. Let X be a smooth variety, Z ⊂ X a closed subvariety of complex codimension
e → Z a resolution of singularities. Then there is a short exact sequence
c and Z
e
PH •−2c (Z)(−c)
→ PH • (X) → PH • (X \ Z) → 0.
Proof. Start with the following part of the long exact sequence in Borel–Moore homology:
BM
HkBM (Z) → HkBM (X) → HkBM (X \ Z) → Hk−1
(Z)
and apply grW
−k , killing the final term. By [Lewis 1999, Lemma A.4] there is a surjection
W
BM e
BM
gr−k Hk (Z) → grW
−k Hk (Z) (The projectivity assumption in loc. cit. is never used, only
that the morphism is proper). We thus get
BM
BM
W
BM e
W
grW
−k Hk (Z) → gr−k Hk (X) → gr−k Hk (X \ Z) → 0.
But now all spaces involved are smooth; applying the cap-product isomorphism with ordinary
cohomology we get the desired result. (Exactness on the right was noted already in Remark
3.2.)
rt
ct
Let N be the normalization of Mct
2,n \M2,n . Thus N is a disjoint union of copies of M1,n−k+1 ×
Mct
1,k+1 for various k. By the previous lemma we have a short exact sequence
•
rt
PH •−2 (N )(−1) → PH • (Mct
2,n ) → PH (M2,n ) → 0.
After the author’s proof [Petersen 2012] of Getzler’s claims in [Getzler 1997], we know that
odd
PH even (Mct
(Mct
1,k ) is tautological, and PH
1,k ) consists of classes attached to cusp forms
even
for SL(2, Z). Thus PH
(N ) consists of tautological classes and tensor products of classes
attached to cusp forms. The latter were called PC classes in [Petersen and Tommasi 2014].
10
DAN PETERSEN
Since the cusp form classes live in H 1 (M1,1 , Vk ) and their tensor products live in H 2 (M1,1 ×
ct
M1,1 , Vk ⊗ Vl ), it is now not hard to see that the pushforwards of PC classes into H • (M2,n )
are exactly the cohomology classes that come from H 2 (Sym2 M1,1 , V) and V a local system
of weight ≥ 20. Moreover, it follows from this that these classes are all non-tautological.
Indeed, on M2,n the only tautological classes are the fundamental classes of boundary strata,
possibly with a ψ-class on a genus two component. In particular, if a non-tautological class
α on the boundary (in this case N ) pushes forward to a tautological class, then we can
write α = β + γ where β is tautological (a linear combination of cycle classes of strata) and
odd
γ pushes forward to zero. But H odd (Mct
(Mct
1,n−k+1 ) ⊗ H
1,k+1 ) contains no tautological
class, so the whole image under the pushforward map of the tensor products of odd classes is
always non-tautological.
Let us remark that this observation gives a short proof of the following result, which was
obtained in a more complicated way in [Petersen and Tommasi 2014].
Theorem 3.7. The even cohomology of M2,20 that is pushed forward from the boundary of
the moduli space is tautological.
Proof. The only nontautological even classes on the boundary (or rather its normalization)
are in H odd (M1,11 ) ⊗ H odd (M1,11 ) = H 11 (M1,11 ) ⊗ H 11 (M1,11 ), so it suffices to prove that
these push forward to zero. These classes would land in H 24 . Since they are products of cusp
form classes, Theorem 3.4 says that there is then an equal contribution from such classes in
degree 22, which is impossible.
Finally let us take a closer look at the local systems V2a,2a , a > 0. We have an exact sequence
2
H 0 (Sym2 M1,1 , V2a,2a )(−1) → H 2 (Mct
2 , V2a,2a ) → H (M2 , V2a,2a ).
As said already, H 2 (Mct
2 , V2a,2a ) is pure and of Tate type. One can check using the branching
formula in [Petersen 2013a] that H 0 (Sym2 M1,1 , V2a,2a )(−1) is always one-dimensional, and
pure of Tate type. I claim now that the image of this space inside H 2 (Mct
2 , V2a,2a ) produces
tautological classes on Mct
.
Indeed,
functoriality
of
the
Leray
spectral
sequence implies
2,n
that the resulting classes are pushed forward from the boundary N ; moreover, since they are
not PC classes, they must be tautological. Conversely, we also know that the pure classes in
H 2 (M2 , V2a,2a ) all give rise to non-tautological classes in H • (Mrt
2,n ). We have proven the
following result.
Theorem 3.8. In the decomposition of PH even (Mct
2,n ) described in Theorem 3.3, the subspace
even
of tautological cohomology RH • (Mct
(Mct
2,n ) ⊆ PH
2,n ) is exactly the subspace corresponding
to:
(1) the trivial local system on Mct
2,
(2) the trivial local system on Sym2 M1,1 ,
(3) the subspace of H 2 (Mct
2 , V2a,2a ) given by the image of the pushforward map
H 0 (Sym2 M1,1 , V2a,2a )(−1) → H 2 (Mct
2 , V2a,2a ).
Corollary 3.9. If the map H 0 (Sym2 M1,1 , V2a,2a )(−1) → H 2 (Mct
2 , V2a,2a ) is not zero for
some a > 0, then the tautological ring of Mct
2,n is not Gorenstein for any n ≥ 4a.
ct
Proof. The local system V2a,2a appears in the derived pushforward of Mct
2,n → M2 for
all n ≥ 4a. The assumption shows that this local system contributes to the tautological
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
11
cohomology. After Theorem 3.4 we therefore know that the tautological cohomology does
not have symmetric Betti numbers.
We also obtain a sharper version of the main contradiction with the Gorenstein conjecture of
[Petersen and Tommasi 2014].
Theorem 3.10. The Betti numbers of RH • (M2,n ) are not symmetric about the middle degree
for any n ≥ 20.
Proof. Let N ′ be the normalization of M2,n \ Mct
2,n . Then Lemma 3.6 gives
H •−2 (N ′ )(−1) → H • (M2,n ) → PH • (Mct
2,n ) → 0.
As all even cohomology of N ′ is tautological, we see that the even non-tautological cohomology
of M2,n is equal to the contributions from products of cusp forms (the fourth summand in
Theorem 3.3) and the cohomology H 2 (M2 , V2a,2a ). Now H • (M2,n ) is symmetric about
degree 3 + n by Poincaré duality, and the products of cusp form classes are also symmetric
about this degree by Theorem 3.4. However, the classes from H 2 (M2 , V2a,2a ) are symmetric
about degree 2 + n, so whenever this summand is nonzero, RH • (M2,n ) must have nonsymmetric Betti numbers. This happens for n ≥ 20, as explained in [Petersen and Tommasi
2014].
Perhaps this is the right moment to remark that the map H 0 (Sym2 M1,1 , V2a,2a )(−1) →
H 2 (Mct
2 , V2a,2a ) is a special case of a generalized modular symbol. Let us dualize to compactly
supported cohomology, and replace Sym2 M1,1 by its double cover. Then we are considering
the map
Hc4 (A2 , V2a,2a ) → Hc4 (A1 × A1 , V2a,2a ) ∼
= Q(−2a − 2),
which is essentially defined by taking a compactly supported differential form and pulling
back to A1 × A1 , where it becomes a top degree form that we can integrate. The notion
of a generalized modular symbol is defined more generally on a Shimura variety XG for a
reductive group G, when we have a subgroup H ⊂ G satisfying certain conditions inducing
a map XH → XG . In our case we can take G = Sp(4) and H = SL(2) × SL(2). Note
that the generalized modular symbol can only evaluate non-trivially on pure cohomology. It
is known in many contexts that they always evaluate to zero on the subspace of cuspidal
cohomology [Ash, Ginzburg, and Rallis 1993]; however, in this case we are considering its
complement within the pure cohomology (i.e. the residual Eisenstein cohomology) and here
we can expect nonvanishing. See for instance the similar situation for G = GL(4), H = Sp(4)
in [Rohlfs and Speh 2011]. (But note that the above authors only treat the case of constant
coefficients.) The classes in Hc4 (A2 , V2a,2a ) that we consider are attached to cusp eigenforms
f for SL(2, Z), and they are known to be cuspidal2 if and only if L(f, 1/2) = 0, see e.g. [Kim
2001]. So it seems plausible that the integrals of the corresponding differential forms can be
evaluated in terms of this central value of the L-function. This would hopefully, and modulo
Maeda’s conjecture, prove that H 0 (Sym2 M1,1 , V2a,2a )(−1) → H 2 (Mct
2 , V2a,2a ) is injective
for any a ≥ 2, as conjectured in [Petersen and Tommasi 2014].
2In a sense this is a silly and rather counterfactual thing to say, since as mentioned earlier Maeda’s
conjecture implies that L(f, 1/2) should never vanish if f is a level 1 cusp form of weight 4k + 4. But we can
for instance choose some level d and consider Hc4 (A2 [d], V2a,2a ) instead, where we will in general find both
cuspidal cohomology and residual Eisenstein cohomology, and in this case the given argument suggests vaguely
that the value of the generalized modular symbol should be related to the central value of the L-function, and
then we expect it also for d = 1.
12
DAN PETERSEN
•
ct
In the next section we give a direct geometric argument that PH • (Mct
2,8 ) = RH (M2,8 ), and
2
0
2
ct
hence that the map H (Sym M1,1 , V4,4 )(−1) → H (M2 , V4,4 ) does not vanish. We do this
using a rational parametrization of M2,8 . A similar argument could probably be carried out
for V6,6 since also M2,12 is rational [Casnati and Fontanari 2007]. However M2,n is irrational
for n ≥ 14 [Faber and Pandharipande 2013] so it will not work for the local system V8,8 . Recall
that there was some ambiguity in [Petersen and Tommasi 2014] over the smallest value of n
such that H even (M2,n ) has non-tautological cohomology: this ambiguity came precisely from
our inability to determine whether the pushforward maps H 0 (Sym2 M1,1 , V2a,2a )(−1) →
H 2 (Mct
2 , V2a,2a ) are non-zero in general. In loc. cit. we stated that n ∈ {8, 12, 16, 20} — the
results of the next section narrow it down to n ∈ {12, 16, 20}.
4. The pure cohomology of Mct
2,8 is tautological
We will show in this section that PH • (M2,8 ) = RH • (M2,8 ), using a rational parametrisation
•
ct
of M2,8 . After the preceding section we know that this implies also PH • (Mct
2,8 ) = RH (M2,8 )
ct
and that the tautological ring of M2,n is not Gorenstein for any n ≥ 8.
Let U be the open subset of M2,8 consisting of (C, p1 , . . . , p8 ) such that |pi + pj | is not the
canonical g21 for any 1 ≤ i < j ≤ 8. Let M02,7 be the open subset of M2,7 consisting of
(C, p1 , . . . , p7 ) such that i(p7 ) 6= pi for i = 1, . . . , 7, where i is the hyperelliptic involution.
There is a natural map M02,7 → M2,8 defined by
(C, p1 , . . . , p7 ) 7→ (C, p1 , . . . , p7 , i(p7 )).
In this way, the disjoint union of 82 copies of M02,7 maps surjectively onto M2,8 \ U and is
a resolution of singularities. We therefore have an exact sequence
(82)
M
PH k−2 (M02,7 )(−1) → PH k (M2,8 ) → PH k (U ) → 0
i=1
by Lemma 3.6.
Observe first of all that PH k (M2,7 ) = 0 for any k > 2, since we already know that PH • (M2,n ) ∼
=
RH • (M2,n ) for n < 8, and R• (M2,n ) vanishes above degree 1 in general. Then the same
vanishing holds for PH k (M02,7 ) by Remark 3.2. Since the only possibly non-tautological class
in PH • (M2,8 ) was in degree 10, it will thus suffice to show that PH • (U ) = RH • (U ). We will
prove that PH • (U ) is generated in degree 2, which is enough to conclude.
For this we need to set up some notation. Let PV be the projective space of (3, 2)-curves in
P1 × P1 (so V ∼
= C12 ). For any point of P1 × P1 , the set of (3, 2)-curves passing through it
is a hyperplane in PV .
Let W ⊂ (P1 × P1 )8 be set of all configurations (p1 , . . . , p8 ) such that
(1) no two points have the same second projections,
(2) no collection of four points have the same first projection,
(3) p1 = (0, 0), p2 = (0, 1), p3 = (0, ∞) and p4 = (∞, y) for some y ∈ P1 \ {0, 1, ∞}.
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
13
There is a Gm -action of W by rescaling the first coordinate, and W/Gm is the space parametrizing 8 points in P1 × P1 where no two points have the same second projection and no four
have the same first coordinate, considered up to the action of PGL(2) × PGL(2).
Let Gm act also on PV by rescaling the first coordinate, and let I ⊂ (PV × W )/Gm be the
incidence variety of all pairs (f, p1 , . . . , p8 ) such that f (pi ) = 0 for i = 1, . . . , 8. Finally let
I sm ⊂ I be the Zariski open subset where V (f ) is nonsingular.
Lemma 4.1. There is an isomorphism U ∼
= I sm .
Proof. Take a point (C, p1 , . . . , p8 ) of U and map C to P1 × P1 using the pairs of linear
systems |p1 + p2 + p3 | and |KC |. Since we chose a point of U the former linear system will
not contain the canonical one, so the image is a smooth (3, 2)-curve, and the images of p1 ,
p2 and p3 in the second copy of P1 are distinct. We can therefore choose coordinates so that
they are 0, 1 and ∞. On the first copy of P1 we can choose coordinates so that p1 , p2 and
p3 map to 0 and p4 is mapped to ∞. It is now clear that we have obtained a point of I sm .
Conversely, there is an obvious map I sm → M2,8 , since V (f ) will be a nonsingular genus 2
curve with 8 ordered markings, whose isomorphism type is invariant under the Gm -action.
Condition (1) in the definition of W ensures that we land inside U ⊂ M2,8 .
Lemma 4.2. Suppose given a configuration (p1 , . . . , p8 ) ∈ W . Then the corresponding eight
hyperplanes in PV are in general position with respect to each other.
Proof. We need to find, for k = 1, . . . , 7, a (3, 2)-curve passing through p1 , . . . , pk but none
of pk+1 , . . . , p8 . By condition (2) in the definition of W we can assume (after reordering
the last four points) that p7 and p8 do not pass through the vertical line through p4 . For
k = 1, 2 and 3, use three horizontal lines (and implicitly two vertical lines passing through
none of the points). For k = 4, 5 and 6 take the vertical line through p1 , p2 and p3 , and three
horizontal lines. Finally take the two vertical lines through p1 —p3 and p4 respectively, and
use horizontal lines for the rest.
Proposition 4.3. PH • (U ) is generated by divisors. In particular, RH • (U ) = PH • (U ) (and
thus it actually vanishes above degree 2).
Proof. We do this in smaller steps. First of all, the set of first coordinates of the points in
W form a Zariski open set in A4 \ {(0, 0, 0, 0)} on which Gm acts by rescaling, and the set of
second coordinates form a Zariski open set in A5 . Thus W/Gm is Zariski open in P3 × A5
and therefore its pure cohomology is generated by divisors by Remark 3.2. Lemma 4.2 shows
that I → W/Gm is a P3 -bundle, and the projective bundle formula shows then that PH • (I)
is also generated by divisors. Since I sm is open in I, the same is true for I sm ∼
= U.
After the results of the previous section, we obtain in particular the following result.
Corollary 4.4. The tautological ring of Mct
2,n is not Gorenstein for n ≥ 8.
14
DAN PETERSEN
5. Computing the cohomology of Mct
2,n
Let us now explain how the results of this paper can be used to compute the cohomology of
•
•
•
•
Mct
2,n in arbitrary degree. As in Theorem 2.1 we let R f∗ Q = A ⊕ B , where A consists of
2
ct
•
local systems supported on M2 and B has support on Sym M1,1 .
5.1. The summand A• : review of results of Getzler. Let S be a scheme. The Grothendieck
group of varieties over S, K0 (VarS ), is the free abelian group generated by isomorphism
classes of finite type schemes over S, modulo the following relation: whenever Z ֒→ X is a
closed immersion, we have
[X] = [Z] + [X \ Z].
Taking Z = Xred shows that we may restrict our attention to reduced X. The Grothendieck
group of varieties becomes a ring with the multiplication
[X] · [Y ] = [X ×S Y ].
If S is a complex algebraic variety, let MHM(S) denote the abelian category of mixed Hodge
modules on S. If X is an finite type reduced scheme over S, then we put
eS (X) = [Rf! Q] ∈ K0 (MHM(S)),
where f : X → S is the structure morphism. This is the ‘Euler characteristic’ or ‘Hodge–
Deligne polynomial’ of X in this category. It satisfies
eS (X) = eS (X \ Z) + eS (Z)
for Z ⊂ X a closed subvariety, and eS (X ×S Y ) = eS (X)eS (Y ). We thus have a ring homomorphism K0 (VarS ) → K0 (MHM(S)). There is a natural λ-ring structure on K0 (MHM(S)):
if M is a mixed Hodge module, then λn ([M ]) = [∧m M ].
Remark 5.1. If X is smooth and X → S is smooth and proper, then Saito’s theory implies
that we can recover Ri f∗ Q by applying grW
i to eS (X).
We denote by Λ the completed ring of symmetric functions,
Y
Λ=
Λn .
n≥0
n
Each Λ is isomorphic to the ring of virtual representations of Sn . As in [Getzler 1995, Theorem 4.8] we can identify K0 (MHM(S))⊗Λn with the Grothendieck group of Sn -representations
in MHM(S). If X is a variety over S with Sn -action, we denote by eSSn (X) its class in
K0 (MHM(S)) ⊗ Λn . Elements of Λ are sequences of virtual representations and can therefore
be thought of as
` ‘virtual S-modules’, where by an S-module we mean a representation of the
b (completed tensor product) with
groupoid S = n≥0 Sn . We can identify K0 (MHM(S))⊗Λ
virtual S-modules in MHM(S). This, too, is naturally a λ-ring.
If X is a variety over S, then we denote by F (X/S, n) the relative configuration space of
n distinct ordered points on X, that is, the complement of the ‘big diagonal’ in the n-fold
b by
fibered product of X with itself over S. Define an element FX/S ∈ K0 (MHM(S))⊗Λ
X
FX/S =
eSSn (F (X/S, n)).
n≥0
The element FX/S has the following explicit formula:
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
Proposition 5.2 (Getzler). FX/S = exp
P
n≥1
P
d|n
15
µ(n/d)
log(1
+
p
)ψ
e
(X)
.
n
d S
n
Here pn ∈ Λn is the nth power sum, and ψd denotes the dth Adams operation (that is, the
λ-ring operation corresponding to pd ). This is proven when S = Spec(C) in [Getzler 1995,
Section 5] (see also the treatment in [Getzler and Pandharipande 2006, Theorem 3.2]) and
for X → S any morphism of quasi-projective complex varieties in [Getzler 1999, Theorem
4.5]. The former proof uses only formal properties of the Euler characteristic e : K0 (VarC ) →
K0 (MHSQ ), whereas the latter proof is more involved and involves a ‘by hand’ construction
of a complex in Db (MHM(X n )) which resolves j! j ∗ Q, where j is the open embedding of
F (X/S, n) in X n , and X n is the n-fold fibered power over S. We remark, however, that the
former proof actually works equally well in the relative setting (and this removes the quasiprojectivity assumption), by replacing e by its relative version eS : K0 (VarS ) → K0 (MHM(S)).
Let us now in addition assume that S is smooth and irreducible, and that X → S is a smooth,
quasi-projective morphism. Then we denote by FM(X/S, n) the relative Fulton–MacPherson
compactification of the configuration space of n distinct ordered points on X, as defined in
[Fulton and MacPherson 1994] when S is a point and in [Pandharipande 1995, Section 1] in
the relative case. Under our hypotheses, the fiber of FM(X/S, n) over a point s ∈ S is exactly
the usual Fulton–MacPherson compactification of n points in the fiber Xs . Define a second
element
X
FMX/S =
eSSn (FM(X/S, n)).
n≥0
Finally let
M=
X
n≥3
b
eSn (M0,n ) ∈ K0 (MHSQ )⊗Λ.
The element M was expressed in terms of Tate type Hodge structures in [Getzler 1995]. Note
that K0 (MHM(S)) is in a natural way a K0 (MHSQ )-algebra.
If f is a symmetric function then we denote by f ⊥ : Λ → Λ the operation which is adjoint to
b → R⊗Λ,
b for any ring R. The plethysm
multiplication by f , as well as its extension to R⊗Λ
operation on Λ is denoted by ◦. If R is in addition a λ-ring, then the plethysm extends in a
b
natural way to an operation ◦ on R⊗Λ.
The following result is proven in [Getzler 1995, Proposition 6.9].
Proposition 5.3 (Getzler). FMX/S = FX/S ◦ (s1 + s⊥
1 M).
Proof. The proof uses a certain yoga of symmetric functions and Sn -representations. In this
formalism, the plethysm should be interpreted as ‘attaching’, in a sense which can be made
precise using Joyal’s theory of species [Bergeron, Labelle, and Leroux 1998].
Seen in this framework, the equation simply states that to obtain a point of one of the Fulton–
MacPherson compactifications, we should first choose a configuration of distinct markings in
a fiber of X → S, and at each of these markings we should either attach ‘a point’ (the
summand s1 ) or ‘a stable pointed tree of projective lines with one distinguished point’ (the
summand s⊥
1 M), the distinguished point being the one which is attached to X.
16
DAN PETERSEN
Sn
n−1
If [V ] ∈ Λn is the class of a representation of Sn , then s⊥
, which
1 ([V ]) = [ResSn−1 V ] ∈ Λ
⊥
explains why the operator s1 corresponds to choosing one of the markings as distinguished.
In our case, we take S = M2 , X = M2,1 , and f the natural forgetful map. Then F (X/S, n) =
M2,n and FM(X/S, n) = Mrt
2,n . We have
eS (X) = Q − V + Q(−1)
with V = V1,0 ∼
= R1 f∗ Q. The only other mixed Hodge modules which will appear from
now on will arise by applying Adams operations to this expression. Hence for our purposes
we can replace K0 (MHM(S)) with the representation ring of GSp(4) — the sub-λ-ring of
K0 (MHM(S)) generated by V and Q(−1) is isomorphic to this representation ring, by an
isomorphism sending V to the standard 4-dimensional representation and Q(−1) to the multiplier.
Propositions 5.2 and 5.3, and the calculation of M from [Getzler 1995], allow us to calculate
as many terms as we wish of FMX/S in terms of the local systems Vl,m and their Tate twists.
By Remark 5.1, the same is true for each of the Ri f∗ Q (considered as Sn -equivariant Hodge
modules on M2 ), where f denotes the projection Mrt
2,n → M2 .
2
∼
5.2. The summand B • . To ease notation in this section, we let T = Mct
2 \M2 = Sym M1,1 ,
and Y = Mct
2,1 \ M2,1 , the universal curve over T . As in the previous subsection we have the
relative configuration space F (Y /T, n); it can be identified with the Zariski open subset of
rt
Mct
2,n \ M2,n parametrizing curves with exactly two irreducible components. Although we do
not give a general definition of the Fulton–MacPherson compactification of a singular variety,
we make the definition
rt
FM(Y /T, n) = Mct
2,n \ M2,n ,
by analogy with the preceding subsection.
To determine the summand B • we will compute
X
b
FMY /T =
eSTn (FM(Y /T, n)) ∈ K0 (MHM(T ))⊗Λ.
n≥0
Each eT (FM(Y /T, n)) is equal to the sum of B • and the restriction A• |Sym2 M1,1 . The latter
is determined by the branching formula from Sp(4) to (SL(2) × SL(2)) ⋊ S2 [Petersen 2013a]
since we have already expressed A• in terms of local systems attached to representations of
Sp(4) and their Tate twists, and in this way we find B • .
We can write [Petersen and Tommasi 2014, Lemma 5.2]
eT (Y ) = Q − V + Q(−1) + Q(−1) ⊗ ε,
where the term V denotes the restriction of the local system V = V1,0 we had on Mct
2,
and where ε is defined by the equality µ∗ Q = Q ⊕ ε, with µ : (M1,1 )2 → Sym2 M1,1 the
double cover. As a local system, ε is defined by the sign representation of S2 . The terms
Q(−1)+ Q(−1)⊗ ε simply mean that H 2 (Yt ) is 2-dimensional for any t ∈ T , and it is spanned
by the sum of the fundamental classes of the two components (which is S2 -invariant) and the
difference of the fundamental classes (which transforms under the sign representation).
TAUTOLOGICAL RINGS OF SPACES OF POINTED GENUS TWO CURVES OF COMPACT TYPE
17
Let Y sm ⊂ Y be the locus where the morphism Y → T is smooth; that is, the complement
of the node in each fiber.
⊥
⊥
Proposition 5.4. FMY /T = FY sm /T ◦ (s1 + s⊥
1 M) · (1 + ε ⊗ s1,1 M + s2 M).
Proof. Again we use the yoga of species. As in the preceding proposition, we can interpret
FY sm /T ◦ (s1 + s⊥
1 M) as a choice of distinct markings in the smooth locus of a fiber of Y → T ,
and for each of them we attach a point or a pointed tree of projective lines with one of the
points distinguished.
In order to obtain FMY /T we should also put markings at the node. Here we can either have
no marking at all (the summand ‘1’), or we need to choose a pointed tree of projective lines
with two of the points distinguished. Choosing two marked points is the same as restricting
an Sn -representation V to Sn−2 × S2 . The part of this restriction that transforms according
to the trivial representation of S2 is s⊥
2 [V ], and the part that transforms according to the sign
representation is s⊥
1,1 [V ]. This explains the formula.
Note that even though the map h : FM(Y /T, n) → T fails to be smooth, each Ri h∗ Q is still
pure of weight i. Indeed, this follows from Theorem 2.1. One could also prove this by writing
the fiber of h over t as an iterated blow-up of (Yt )n (which clearly has pure cohomology)
in loci which themselves have pure cohomology. Thus FMY /T determines each of the terms
Ri h∗ Q.
The expressions for FMX/S that were derived earlier will only contain Schur functors applied
to V and Q(−1). Here, we are going to find an expression for FMY /T in terms of Schur
functors applied to V, Q(−1) and ε. Moreover, both FMX/S and FMY /T are going to be
‘effective’, in the sense that terms of odd weight always occur with a negative coefficient and
terms of even weight with a positive coefficient. Thus it makes sense to say that FMX/S is a
‘direct summand’ of FMY /T . (It would perhaps be more accurate to talk about ‘i∗ R0 j∗ FMX/S ’
ct
as such a summand, where j : S → Mct
2 and i : T → M2 are the respective open and closed
immersions.) In making our calculations we have verified in particular that FMY /T − FMX/S
satisfies the Poincaré duality with a degree shift of Theorem 2.1, a nontrivial consistency
check.
In Table 1 we tabulate the cohomology of Mct
2,n for some small values of n. We encode
P
i
i
ct
the cohomology as the polynomial i [H i (Mct
2,n )] · t , where [H (M2,n )] is considered as an
n
element of K0 (MHSQ )⊗Λ . This class is given as a polynomial in L and the Schur polynomials
sλ . By L we mean the Tate Hodge structure Q(−1). (For n ≤ 9 all the cohomology is of
Tate type.) In general we can only determine the cohomology up to semi-simplification (since
the cohomology of local systems on Mct
2 was only computed with this caveat in [Petersen
2013b]), but in this table there are actually no nontrivial extensions since none of the local
systems have cohomology of different weights in the same degree.
In Table 2 we give the tautological cohomology of Mct
2,8 , i.e. the first non-Gorenstein example.
Here we don’t specify the mixed Hodge structure (obviously it’s all pure Tate).
18
DAN PETERSEN
n
1
2
3
4
5
H • (Mct
2,n )
L2 s1 t4 + 2 Ls1 t2 + s1
L3 s2 t6 + (L5 s2 + L4 s1,1 )t5 + (4 L2 s2 + L2 s1,1 )t4 + (4 Ls2 + Ls1,1 )t2 + s2
L4 s3 t8 +(L6 s3 +L6 s2,1 +L5 s2,1 +L5 s1,1,1 )t7 +(5 L3 s3 +3 L3 s2,1 )t6 +(L5 s3 +L5 s2,1 +2 L4 s2,1 +
2 L4 s1,1,1 )t5 + (10 L2 s3 + 7 L2 s2,1 )t4 + (5 Ls3 + 3 Ls2,1 )t2 + s3
L5 s4 t10 + (L7 s4 + L7 s3,1 + L7 s2,2 + L6 s3,1 + L6 s2,1,1 )t9 + (7 L4 s4 + 4 L4 s3,1 + 2 L4 s2,2 )t8 +
(2 L6 s4 + 4 L6 s3,1 + 2 L6 s2,2 + L6 s2,1,1 + L6 s1,1,1,1 + L5 s4 + 5 L5 s3,1 + 2 L5 s2,2 + 6 L5 s2,1,1 +
2 L5 s1,1,1,1 )t7 + (20 L3 s4 + 18 L3 s3,1 + 9 L3 s2,2 + 3 L3 s2,1,1 )t6 + (L5 s4 + L5 s3,1 + L5 s2,2 +
3 L4 s3,1 + L4 s2,2 + 4 L4 s2,1,1 + L4 s1,1,1,1 )t5 + (20 L2 s4 + 18 L2 s3,1 + 9 L2 s2,2 + 3 L2 s2,1,1 )t4 +
(7 Ls4 + 4 Ls3,1 + 2 Ls2,2 )t2 + s4
L6 s5 t12 + (L8 s5 + L8 s4,1 + L8 s3,2 + L7 s4,1 + L7 s3,1,1 )t11 + (8 L5 s5 + 6 L5 s4,1 + 3 L5 s3,2 )t10 +
(4 L7 s5 +7 L7 s4,1 +6 L7 s3,2 +3 L7 s3,1,1 +2 L7 s2,2,1 +L7 s2,1,1,1 +L7 s1,1,1,1,1 +2 L6 s5 +9 L6 s4,1 +
5 L6 s3,2 +11 L6 s3,1,1 +4 L6 s2,2,1 +4 L6 s2,1,1,1 )t9 +(L7 s3,2 +L7 s2,2,1 +L7 s2,1,1,1 +L7 s1,1,1,1,1 +
33 L4 s5 + 37 L4 s4,1 + 25 L4 s3,2 + 8 L4 s3,1,1 + 6 L4 s2,2,1 )t8 + (4 L6 s5 + 7 L6 s4,1 + 6 L6 s3,2 +
3 L6 s3,1,1 +2 L6 s2,2,1 +2 L6 s2,1,1,1 +2 L6 s1,1,1,1,1 +3 L5 s5 +15 L5 s4,1 +10 L5 s3,2 +21 L5 s3,1,1 +
10 L5 s2,2,1 + 10 L5 s2,1,1,1 + L5 s1,1,1,1,1 )t7 + (51 L3 s5 + 68 L3 s4,1 + 48 L3 s3,2 + 22 L3 s3,1,1 +
L3 s2,1,1,1 +14 L3 s2,2,1 )t6 +(L5 s5 +L5 s4,1 +L5 s3,2 +4 L4 s4,1 +2 L4 s3,2 +6 L4 s3,1,1 +2 L4 s2,2,1 +
2 L4 s2,1,1,1 )t5 + (33 L2 s5 + 37 L2 s4,1 + 25 L2 s3,2 + 8 L2 s3,1,1 + 6 L2 s2,2,1 )t4 + (8 Ls5 + 6 Ls4,1 +
3 Ls3,2 )t2 + s5
even
•
ct
Table 1. Cohomology of Mct
(Mct
2,n for n ≤ 5. The equality H
2,n ) = PH (M2,n )
fails for the first time when n = 5, with the impure cohomology arising from classes
in H 2 (Sym2 M1,1 , V) given by products of classes attached to Eisenstein series, and
V is a Tate twist of a weight 4 local system.
k
0
2
4
6
8
10
12
14
16
18
RH k (Mct
2,8 )
s8
13 s8 + 10 s7,1 + 8 s6,2 + 4 s5,3 + 2 s4,4
87 s8 + 122 s7,1 + 127 s6,2 + 40 s6,1,1 + 90 s5,3 + 52 s5,2,1 + 37 s4,4 + 34 s4,3,1 + 16 s4,2,2 + 6 s3,3,2
307 s8 + 565 s7,1 + 695 s6,2 + 319 s6,1,1 + 563 s5,3 + 485 s5,2,1 + 53 s5,1,1,1 + 234 s4,4 + 355 s4,3,1 +
183 s4,2,2 + 80 s4,2,1,1 + 91 s3,3,2 + 38 s3,3,1,1 + 32 s3,2,2,1 + 4 s2,2,2,2
578 s8 + 1194 s7,1 + 1578 s6,2 + 814 s6,1,1 + 1354 s5,3 + 1328 s5,2,1 + 202 s5,1,1,1 + 569 s4,4 +
1018 s4,3,1 + 547 s4,2,2 + 326 s4,2,1,1 + 13 s4,1,1,1,1 + 290 s3,3,2 + 156 s3,3,1,1 + 136 s3,2,2,1 +
16 s3,2,1,1,1 + 16 s2,2,2,2 + 4 s2,2,2,1,1
578 s8 + 1194 s7,1 + 1578 s6,2 + 814 s6,1,1 + 1354 s5,3 + 1328 s5,2,1 + 202 s5,1,1,1 + 569 s4,4 +
1018 s4,3,1 + 547 s4,2,2 + 326 s4,2,1,1 + 13 s4,1,1,1,1 + 290 s3,3,2 + 156 s3,3,1,1 + 136 s3,2,2,1 +
16 s3,2,1,1,1 + 17 s2,2,2,2 + 4 s2,2,2,1,1
307 s8 + 565 s7,1 + 695 s6,2 + 319 s6,1,1 + 563 s5,3 + 485 s5,2,1 + 53 s5,1,1,1 + 234 s4,4 + 355 s4,3,1 +
183 s4,2,2 + 80 s4,2,1,1 + 91 s3,3,2 + 38 s3,3,1,1 + 32 s3,2,2,1 + 4 s2,2,2,2
87 s8 + 122 s7,1 + 127 s6,2 + 40 s6,1,1 + 90 s5,3 + 52 s5,2,1 + 37 s4,4 + 34 s4,3,1 + 16 s4,2,2 + 6 s3,3,2
13 s8 + 10 s7,1 + 8 s6,2 + 4 s5,3 + 2 s4,4
s8
•
ct
Table 2. The S8 -representations RH • (Mct
2,8 ) = PH (M2,8 ), with failure of the
Gorenstein property highlighted.
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