Computing top intersections in the tautological ring of $M_g$

0.2 MB·English

by  Kefeng Liu

COMPUTING TOP INTERSECTIONS IN THE
TAUTOLOGICAL RING OF M
g
KEFENG LIU ANDHAOXU
0
1
Abstract. We derive effective recursion formulae of top intersections in the tautological ring
0
2 R∗(Mg)ofthemodulispaceofcurvesofgenusg≥2. Asanapplication,weproveaconvolution-
typetautological relation in Rg−2(Mg).
n
a
J

5
2 1. Introduction

We denote by M the moduli space of stable n-pointed genus g complex algebraic curves.
] g,n
G
We have the morphism that forgets the last marked point,
A
π :M −→ M .
n+1 g,n+1 g,n
.
h
Denote by σ ,...,σ the canonical sections of π, and by D ,...,D the corresponding divisors
t 1 n 1 n
a in M . Let ω be the relative dualizing sheaf. We have the following tautological classes on
m g,n+1 π
moduli spaces of curves.
[
ψ = c (σ∗(ω )),
i 1 i π
1
v i+1
κ = π c ω D ,
8 i ∗ 1 π i
9 (cid:18) (cid:19)
(cid:16) (cid:16)X (cid:17)(cid:17)
4 λ = c (π (ω )), 1≤ l ≤ g.
l l ∗ π
4
1. The definition of κ classes on Mg,n is due to Arbarello-Cornalba [1], generalizing Mumford-
Morita-Miller classes
0
0 κ = π c (ω )i+1 ∈ Ai(M ),
i ∗ 1 π g
1
: where A∗(M ) is the rational Chow rin(cid:16)g of M . (cid:17)
v g g
The tautological ring R∗(M ) is defined to be the subalgebra of A∗(M ) generated by the
i g g
X
tautological classes κ .
i
r We use Witten’s notation to denote intersection numbers:
a
hτ ···τ κ ···κ |λk1···λkgi := ψd1···ψdnκ ···κ λk1···λkg.
d1 dn a1 am 1 g 1 n a1 am 1 g
ZMg,n
These are rational numbers and are called the Hodge integrals, which can be computed by
Faber’s algorithm [7] based on Mumford’s formula for Chern characters of Hodge bundles and
the celebrated Witten-Kontsevich theorem [22, 14].
1.1. Faber’s conjecture. Around 1993, Faber [6] proposed a series of remarkable conjectures
about the structure of R∗(M ):
g
i) For 0 ≤ k ≤ g−2, the natural product
Rk(M )×Rg−2−k(M ) → Rg−2(M )∼= Q
g g g
is a perfect pairing.
2010 Mathematics Subject Classification. 14H10.
1
2 KEFENGLIUANDHAOXU
ii) The [g/3] classes κ ,...,κ generate the ring R∗(M ), with no relations in degrees
1 [g/3] g
≤ [g/3].
iii) Let n d =g−2 and d ≥ 0. Then
j=1 j j
P (2g−3+n)!
(1) κ = κ ,
σ (2g−2)!! n (2d +1)!! g−2
σX∈Sn j=1 j
where κ is defined as follows: write thQe permutation σ as a product of ν(σ) disjoint
σ
cycles σ = β ···β , where we think of the symmetric group S as acting on the
1 ν(σ) n
n-tuple (d ,...,d ). Denote by |β| the sum of the elements of a cycle β. Then κ =
1 n σ
κ κ ...κ .
|β1| |β2| |βν(σ)|
It is a theorem of Looijenga [18] that
dimRk(M ) = 0, k > g−2,
g
dimRg−2(M )≤ 1.
g
Faber proved that actually dimRg−2(M )= 1.
g
Part (i) of Faber’s conjecture is also called Faber’s perfect pairing conjecture, which is still
open. Faber has verified g ≤ 23.
Part (ii) has been proved independently by Morita [19] and Ionel [11] by very different meth-
ods. As pointed out by Faber [6], Harer’s stability result implies that there is no relation in
degrees ≤ [g/3].
Part (iii) is known as the Faber intersection number conjecture and is equivalent to
(2g−3+n)!
(2) hτ ···τ |λ λ i = hκ | λ λ i .
d1+1 dn+1 g g−1 g (2g−2)!! n (2d +1)!! g−2 g g−1 g
j=1 j
A short and direct proof of the Faber intersection nQumber conjecture can be found in [16]. Also
see [8, 10] and the most recently [3] for different approaches to the problem.
Infact,Faber[6]furtherproposedthatthetautologicalringR∗(M )behaveslikethealgebraic
g
cohomology ring of a nonsingular projective variety of dimension g−2, i.e. it satisfies the Hard
Lefschetz and Hodge Positivity properties with respect to κ .
1
Faber’s intersection number conjecture determines the top intersections in Rg−2(M ). If we
g
assumeFaber’sperfectpairingconjecture, thentheringstructureofR∗(M )is alsodetermined.
g
A central theme in Faber’s conjecture is to explicitly describe relations in the tautological
rings. However, it is a highly nontrivial task to identify tautological relations in R∗(M ) when
g
g becomes larger. In this paper, we will consider only tautological relations of top degree in
Rg−2(M ). Already known examples include Faber-Zagier’s formula [6]
g
1
(3) κg−2 = 22g−5((g−2)!)2κ .
1 g−1 g−2
and Pandharipande’s formula [21]
g−2 2g−1
(4) (−1)iλ κ = κ .
i g−2−i g−2
g!
i=0
X
in Rg−2(M ),g ≥ 2.
g
Now we describethemain results of this paper. We will provetwo effective recursive formulae
of different flavors for computing top intersections in Rg−2(M ) (see Theorems 3.6 and 3.8).
g
For example, Theorem 3.8 can be equivalently stated as the following:
COMPUTING TOP INTERSECTIONS IN Mg 3
Theorem 1.1. Let g ≥ 3 and |m| = g−2. Then the following relation
1 m
(5) κ(m) = A κ(L′+δ )
g,L |L|
(||m||−1) L
L+XL′=m (cid:18) (cid:19)
||L||≥2
holds in Rg−2(M ), where A = L!D are some explicitly known constants.
g g,L g,L
Note that in the right-hand side of equation (5), ||L′+δ || < ||m||, so equation (5) is indeed
|L|
an effective recursion relation. Our strategy of proof is to exploit the method used in [15]. From
an algorithmic point of view, recursion formulae are often more effective than closed formulae,
since previously computed values can be reused in recursive computations.
Our recursion formulae will be used to compute Faber’s intersection matrix, whose rank is
equal to the dimension of R∗(M ) by Faber’s perfect pairing conjecture. On the other hand,
g
the cohomological dimension H∗(M ) is also an outstanding open problem [2].
g
One objective of this paper is to gain a better understanding of Faber-Zagier’s formula.
In Section 4, we prove an interesting Bernoulli number identity equivalent to Faber-Zagier’s
formula.
In the final section, we prove the following convolution-type tautological relation:
Theorem 1.2. Let g ≥ 3. We have the following relation in Rg−2(M ),
g
g−2 κiκ
(6) D 1 g−2−i = 0,
g,g−2−i
i!
i=0
X
where D are given by
g,k
3 1 2g−1 k (2j +1)(−1)j+122jB
2j
(7) D = · + .
g,k
2(g−2) k! 2(g−2) j!(k−j)!
j=0
X
where B is the 2j-th Bernoulli number. We have D = −1, D = g+1, D = 17g−4.
2j g,0 g,1 g−2 g,2 6(g−2)
We don’t know whether the expression of D can be simplified.
g,k
Acknowledgements. We would like to thank Professor Jian Zhou for helpful discussions and
for kindly providing an elegant proof of Lemma 4.6.
2. The Faber intersection matrix
First we fix notation. Consider the semigroup N∞ of sequences m= (m(1),m(2),...) where
m(i) are nonnegative integers and m(i) = 0 for sufficiently large i. We sometimes also use
(1m(1)2m(2)...) to denote m.
Let m,a ,...,a ∈ N∞, m = n a .
1 n i=1 i
P m m(i)
|m| := im(i) ||m|| := m(i) := .
a ,...,a a (i),...,a (i)
i≥1 i≥1 (cid:18) 1 n(cid:19) i≥1(cid:18) 1 n (cid:19)
X X Y
Let m∈ N∞, we denote a formal monomial of κ classes by
m(i)
κ(m) := κ .
i
i≥1
Y
4 KEFENGLIUANDHAOXU
If |m| =g−2, then from Faber’s intersection number identity (2) and the formula expressing
ψ classes by κ classes [13], we have the following relation in Rg−2(M ),
g
(8) κ(m) = Fab (m)κ ,
g g−2
where the proportional constant Fab (m) is given by
g
||m||(−1)||m||−r m (2g−3+r)!
Fab (m) = .
g r! m ,...,m (2g−2)!! r (2|m |+1)!!
Xr=1 m=mmX1i+6=·0··+mr(cid:18) 1 r(cid:19) j=1 j
Q
Let g ≥ 2 and 0 ≤ k ≤ g−2. Denote by p(n) the number of partitions of n. Define a matrix
Vk of size p(k)×p(g−2−k) with entries
g
(9) (Vgk)L,L′ = Fabg(L+L′),
where L,L′ ∈N∞ and |L| = k, |L′|= g−2−k.
We call Vk the Faber intersection matrix. If Faber’s perfect pairing conjecture is true, then
g
we have
(10) rankVk = dimRk(M ), 0≤ k ≤ g−2.
g g
Faber has verified his conjecture for all g ≤ 23, so the above relation holds for at least g ≤ 23.
Thus, we may get useful information of R∗(M ) from the Faber intersection matrix.
g
In the next section, we will develop a recursive method for computing entries of Vk. As a
g
result,wehavecomputedVk forallg ≤ 36. Thesedataisusedtocheckaconjecturalrelationship
g
between rankVk and Ramanujan’s mock theta function (see Section 7 of [17]).
g
The recursive algorithm did not reduce computational complexity in theory. But the recur-
sively reusable data rendered the computation more efficient.
If we calculate Vk by equation (9) on a computer, the required CPU time is proportional to
g
the number of vector partitions.
Definition 2.1. For m∈ N∞, denote by P(m) the number of distinct representations of m as
an unordered sum in N∞,
m = m +m +···+m ,
1 2 r
where m 6= 0. We call such P(m) the vector partition number.
i
We make the convention that P(0) = 1. There is a simple recursion formula for computing
P(m), due to Cheema and Motzkin [5].
Lemma 2.2. (Cheema-Motzkin) For m= (m ,m ,...,m ) with m 6= 0, we have
1 2 s 1
a
1
(11) m1P(m) = P(m1−a1,...,ms−as) .
k
aXi≥0 k|gcdX(ka>10,...,as) 
 
Proof. Let x = (x ,x ,...) be a sequence of formal variables and xm = xm(1)xm(2)... for
1 2 1 2
m∈ N∞. We have
P(m)xm = (1−xa)−1.
m∈N∞ a∈N∞
X aY=6 0
COMPUTING TOP INTERSECTIONS IN Mg 5
Hence
∞ xka
log P(m)xm = − log(1−xa)=
k
m∈N∞ a∈N∞ a∈N∞k=1
X aX=6 0 aX=6 0 X
Differentiating with respect to x , we get
1
∞
m P(m)xm = P(m)xm a xka.
1 1
m∈N∞ m∈N∞ a∈N∞k=1
X X aX=6 0 X
Equating coefficients of xm, we get the desired identity (11). (cid:3)
For a given genus g, the complexity of computing Vk for all 0≤ k ≤ g−2 is measured by the
g
following quantity
D(g−2) := P(m),
|m|=g−2
X
Some values of D(n) are listed as follows.
n 0 1 2 3 4 5 10 20 30 40
D(n) 1 1 3 6 14 27 817 318106 71832114 11668071461
Proposition 2.3. D(n) has a simple generating function
∞ ∞
1
D(n)xn = .
(1−xn)p(n)
n=0 n=1
X Y
Proof. We have
1
P(m)xm = .
(1−xm)
m m6=0
X Y
Substitute x by xi, we get the desired result. (cid:3)
i
D(n) can also be regarded as the number of double partitions of n. The following asymptotic
formula is proved by Kaneiwa [12]
π2n
D(n) ∼e6logn,
which gives the computational complexity of the Faber intersection matrix.
Let I(M ) be the ideal of polynomial relations of κ classes in M and let Ik(M ) be the
g g g
group of relations in degree k. We have
dimRk(M )+dimIk(M )= p(k).
g g
Let s ≥ 0 and g = 3k − s. Faber [6] pointed out that when k ≥ s + 2 (i.e. 2k ≤ g −2),
dimIk(M ) depends only on s. Denoting this number by a(s), Faber has [6] computed the first
g
ten values. Our calculation of the rank of Vk for g ≤ 36 extends Faber’s table of the function a.
g
s 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
a(s) 1 1 2 3 5 6 10 13 18 24 33 41 56 71 91
In [6], Faber says “Zagier and I have a favourite guess of this function a, but there are many
functions with ten prescribed values.”
We have also tried to guess the function a, but failed. For example, define
f(s)= p(s+1−3r)−p(s−3r).
0≤r≤[s/3]
X
6 KEFENGLIUANDHAOXU
We have f(s)= a(s), s ≤ 10, but f(11) = 34 6= a(11).
3. Computing top intersections in Rg−2(M )
g
Let d ≥ 1 and d +|m| =g−2+n. We define the following quantities
j j j
P (2g−2)!! n (2d −1)!! h n τ κ(m) | λ λ i
(12) F (m) := j=1 j · j=1 dj g g−1 g.
g,n
(2g+n−3)!m! hκ | λ λ i
Q Q g−2 g g−1 g
Proposition 3.1. The above definition of F (m) is independent of d and F (0) = 1.
g,n j g,n
Proof. From the formula expressing ψ by κ in [13], we have
n
h τ κ(m) | λ λ i
dj g g−1 g
j=1
Y
||m||(−1)||m||−r m n r
= h τ τ | λ λ i
r! m ,...,m dj |mj|+1 g g−1 g
1 r
Xr=0 m=mmX1i+6=·0··+mr(cid:18) (cid:19) jY=1 jY=1
So the proposition follows directly from Faber’s intersection number identity (2). (cid:3)
We will also write F (m) instead of F (m). We are particularly interested in F (m) when
g g,0 g
|m| = g−2, since they determine relations in Rg−2(M )
g
(2g−3)!!m!F (m)
g
(13) κ(m) = κ .
g−2
2g−2
It is important to notice that we may extend F (m) to be defined for all m ∈ N∞ using the
g
following Lemma 3.3 and Theorem 3.6.
We define constants β and γ by
L L
(−1)||L||β
L
= 0, b 6= 0,
L′!(2|L′|+1)!!
L+L′=b
X
(−1)||L||γ
L
= 0, b 6= 0,
L′!(2|L′|−1)!!
L+L′=b
X
with the initial values β = γ = 1. We also denote their reciprocals by
0 0
(−1)||L|| (−1)||L||
γ−1 := , β−1 := .
L L!(2|L|−1)!! L L!(2|L|+1)!!
Lemma 3.2. Let n ≥ 0 and |m| ≤ g−2. Then
(14) (2g+n−1)F (m) = (2g+n−1−2|L′|)γ F (L′).
g,n+2 L g,n+1
L+L′=m
X
COMPUTING TOP INTERSECTIONS IN Mg 7
Proof. We recall the following recursive formula from Theorem 5.3 of [15]
n
hτ τ τ κ(m) | λ λ i
d d0 dj g g−1 g
j=1
Y
n
γ m!(2d+2d +2|L|−1)!!
= L 0 hτ τ κ(L′) |λ λ i
L′! (2d−1)!!(2d −1)!! d0+d+|L|−1 dj g g−1 g
L+L′=m 0 j=1
X Y
n
γ m!(2d+2d +2|L|−3)!!
+ L j hτ τ τ κ(L′) |λ λ i .
L′! (2d−1)!!(2d −3)!! d0 dj+d+|L|−1 di g g−1 g
L+L′=mj=1 j i6=j
X X Y
When m = 0, it was obtained by Getzler and Pandharipande [8] from degree 0 Virasoro con-
straints for P2.
So we have
(2g+n−1)F (m) = (2d+2d +2|L|−1)γ F (L′)
g,n+2 0 L g,n+1
L+L′=m
X
n
+ (2d −1)γ F (L′)
j L g,n+1
L+L′=mj=1
X X
= (2g+n−1−2|L′|)γ F (L′).
L g,n+1
L+L′=m
X
In the last equation, we used d+d + n d = g+n−|m|. (cid:3)
0 j=1 j
P
Lemma 3.2 can also be proved using the following identity instead
n n
b
(−1)||L|| hτ τ κ(L′)| λ λ i = hτ τ κ(b) |λ λ i ,
L |L| dj g g−1 g dj−1 di g g−1 g
L+L′=b (cid:18) (cid:19) j=1 j=1 i6=j
X Y X Y
which can be proved by the same argument of Proposition 3.1 of [15].
Lemma 3.3. Let n ≥ 0 and |m| ≤ g−2. Then
(15) F (m) = β−1F (L′).
g,n L g,n+1
L+L′=m
X
Proof. We use the following identity
n n
m
(2g−2+n)h τ κ(m) | λ λ i = (−1)||L|| hτ τ κ(L′) |λ λ i ,
dj g g−1 g L |L|+1 dj g g−1 g
j=1 L+L′=m (cid:18) (cid:19) j=1
Y X Y
which can be proved by the same argument of Proposition 3.1 of [15]. By a direct calculation
as Lemma 3.2, we get the desired result. (cid:3)
Lemma 3.4. Let n ≥ 0 and |m| ≤ g−2. Then
(16) 2|m|F (m) = β−1γ (2g+n−1−2|L|)F (L).
g,n+1 e f g,n+1
e+f+L=m
X
L6=m
8 KEFENGLIUANDHAOXU
Proof. The result follows by applying Lemma 3.2 to the right hand side of equation (15) in
Lemma 3.3. We have
(2g+n−1)F (m) = β−1(2g+n−1)F (b)
g,n+1 e g,n+2
e+b=m
X
= β−1 γ (2g+n−1−2|L|)F (L)
e f g,n+1
e+b=m f+L=b
X X
= β−1γ (2g+n−1−2|L|)F (L).
e f g,n+1
e+f+L=m
X
So we get the desired identity. (cid:3)
Proposition 3.5. Let n ≥ 0 and |m|≤ g−2. Then
(17) 2|m|F (m) = (2g+n−2) C F (L′),
g,n L g,n
L+L′=m
X
L6=0
where C = −1 and
0
C = 2|e|β β−1 = − γ−1β , L 6= 0.
L e f e f
e+f=L e+f=L
X X
Proof. From Lemma 3.3 and Lemma 3.4, we have
2|m| β F (L′)= 2|m|F (m)
L g,n g,n+1
L+L′=m
X
= βe−1γf(2g+n−1−2|L|) βL′Fg,n(L′′)
e+f+L=m L′+L′′=L
X X
L6=m
= βe−1γf(2g+n−1−2|L|) βL′Fg,n(L′′)−(2g+n−1−2|m|) βLFg,n(L′).
e+f+L=m L′+L′′=L L+L′=m
X X X
Subtract 2|m| β F (L′) from each side, we have
L+L′=m L g,n
P
(18) 0= (2g+n−1) γfFg,n(L′′)−2 |L′′|βe−1γfβL′Fg,n(L′′)
f+L′′=m e+f+L′+L′′=m
X X
−2 |L′|βe−1γfβL′Fg,n(L′′)−(2g+n−1) βLFg,n(L′).
e+f+L′+L′′=m L+L′=m
X X
Now we simplify the third term in (18)
− 2|L′|βe−1γfβL′Fg,n(L′′) = 2|e|βeγfβL′Fg,n(L′′)
e+f+L′+L′′=m e+f+L′+L′′=m
X X
= (2|e|+1)βe−1γfβL′Fg,n(L′′)− βe−1γfβL′Fg,n(L′′)
e+f+L′+L′′=m e+f+L′+L′′=m
X X
= γe−1γfβL′Fg,n(L′′)− γLFg,n(L′)
e+f+L′+L′′=m L+L′=m
X X
= β F (L′)− γ F (L′).
L g,n L g,n
L+L′=m L+L′=m
X X
COMPUTING TOP INTERSECTIONS IN Mg 9
Substitute into (18), we have
0 = (2g+n−1) γ F (L′)−2 |L′|γ F (L′′)
L g,n L g,n
L+L′=m L+L′=m
X X
+ β F (L′)− γ F (L′)−(2g+n−1) β F (L′).
L g,n L g,n L g,n
L+L′=m L+L′=m L+L′=m
X X X
So we get
2 |L′|γ F (L′)= (2g+n−2) γ F (L′)−(2g+n−2) β F (L′).
L g,n L g,n L g,n
L+L′=m L+L′=m L+L′=m
X X X
Convoluting both sides by γ−1, we have
L
2|m|F (m) = (2g+n−2)F (m)−(2g+n−2) ( γ−1β )F (L′)
g,n g,n e f g,n
L+L′=m e+f=L
X X
= −(2g+n−2) ( γ−1β )F (L′)
e f g,n
L+L′=m e+f=L
X X
L6=0
= −(2g+n−2) ( (2|e|+1)β−1β )F (L′)
e f g,n
L+L′=m e+f=L
X X
L6=0
= (2g+n−2) ( 2|e|β β−1)F (L′).
e f g,n
L+L′=m e+f=L
X X
L6=0
(cid:3)
Our first main result follows as an immediately corollary of the above proposition.
Theorem 3.6. Let |m| ≤ g−2. Then
(19) |m|F (m) = (g−1) C F (L′).
g L g
L+XL′=m
L6=0
The constant C is defined in Proposition 3.5. In particular,
L
2g−2
F (δ )= , k ≥ 1.
g k
(2k+1)!!
δ denotes the sequence with 1 at the k-th place and zeros elsewhere.
k
Proof. For the last assertion, just note that
2k
Cδ = 2k·βδ = .
k k (2k+1)!!
(cid:3)
Theorem 3.6 gives an efficient way to compute constants F (m) when |m| = g − 2, hence
g
tautological relations in Rg−2(M ) through equation (13). The identity (19) may be expanded
g
to get a closed formula of F (m).
g
10 KEFENGLIUANDHAOXU
Corollary 3.7. For m 6= 0, we have
||m|| k C
F (m) = (g−1)k j=1 mj .
g k |m +···+m |
Xk=1 m=mX1+···+mk j=1Q1 j
mi6=0
Q
Now we come to our second main result, in contrast with Theorem 3.6, it is a formula that
gives recursive relations only among those F (m) with |m| = g−2.
g
Theorem 3.8. Let g > 2 and |m| = g−2. Then we have
(L+δ )!
(20) (||m||−1)Fg(m) = Dg,L′ g−2−|L| Fg(L+δg−2−|L|),
L!
L|+|LXL′|′|=≥m2
where the constant Dg,L′ is given by
1 2g−1 (1+2|L |)!!
1
Dg,L′ =−L′! + 2(g−2) CL1 L ! .
L1+XL2=L′ 2
||L1||≥1
The constant C is defined in Proposition 3.5.
L
Proof. By the projection formula and κ = 2g−2, we have
0
(21) hτ κ(m) |λ λ i = π (ψ· (π∗κ +ψi)mi)λ λ
1 g g−1 g ∗ i g g−1
ZMg i≥1
Y
m
= κ(L)κ|L′|λgλg−1
L
L+L′=m(cid:18) (cid:19)ZMg
X
m
= (2g−2) κ(m)λgλg−1+ L κ(L)κ|L′|λgλg−1.
ZMg L+L′=m(cid:18) (cid:19)ZMg
LX′6=0
From equation (12), the above equation becomes
1 1
(22) Fg,1(m) = Fg(m)+ 2g−2 L!L′!Fg(L+δ|L′|)(L+δ|L′|)!
L+L′=m
LX′6=0
||m|| 1 1
= (1+ 2g−2)Fg(m)+ 2g−2 L!L′!Fg(L+δ|L′|)(L+δ|L′|)!
L|+|LXL′|′|=≥m2
Take n = 1 in Proposition 3.5, we have
2|m|F (m) = (2g−1) C F (L′)
g,1 L g,1
L+XL′=m
L6=0