On the Picard Group of the Moduli Space of Curves via -Spin Structures
In this paper, we obtain explicit expressions for Pandharipande-Pixton-Zvonkine relations in the second rational cohomology of ℳ¯,ₙ, and, comparing the result with Arbarello-Cornalba's theorem, we prove Pixton's conjecture in this case.
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| description | In this paper, we obtain explicit expressions for Pandharipande-Pixton-Zvonkine relations in the second rational cohomology of ℳ¯,ₙ, and, comparing the result with Arbarello-Cornalba's theorem, we prove Pixton's conjecture in this case.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 20 (2024), 088, 16 pages
On the Picard Group of the Moduli Space of Curves
via r-Spin Structures
Danil GUBAREVICH ab
a) Laboratoire de Mathématiques de Versailles, UFR des Sciences, Université de Versailles
Saint-Quentin en Yvelines, 45 avenue des États-Unis, 78035 Versailles, France
E-mail: danil.gubarevich@uvsq.fr
b) Faculty of Mathematics, National Research University Higher School of Economics,
6 Usacheva Str., 119048 Moscow, Russia
Received January 05, 2022, in final form August 27, 2024; Published online October 06, 2024
https://doi.org/10.3842/SIGMA.2024.088
Abstract. In this paper, we obtain explicit expressions for Pandharipande–Pixton–Zvon-
kine relations in the second rational cohomology of Mg,n and comparing the result with
Arbarello–Cornalba’s theorem we prove Pixton’s conjecture in this case.
Key words: moduli space of curves; tautological relations; cohomological field theories
2020 Mathematics Subject Classification: 14H10; 14N35
1 Introduction
The study of cohomology of the moduli space of curves was initiated by Mumford [3] in the 1980s.
The most progress has been made in understanding the subring R∗(Mg) of tautological classes.
A systematic study by Faber and Zagier of the algebra of κ classes on the moduli space Mg of
nonsingular genus g curves led to a conjecture in 2000 of a concise set FZ of relations among κ
classes in
R∗(Mg) ⊂ A∗(Mg).
There are several proofs of the Faber–Zagier conjecture, one of them is given by Pandharipande
and Pixton in [5], exploiting moduli space of stable quotients on P1.
Recent progress in 2012 has appeared in Pixton’s paper [8] about hypothetically the full
set of relations in the tautological ring of Mg,n. Pixton’s relations recover FZ when restricted
to Mg ⊂ Mg.
Using homogeneity of Witten’s 3-spin class, authors of [6] proved Pixton’s relations hold
in H∗(Mg,n,Q
)
, proving Faber–Zagier conjecture in cohomology. The set of Pixton’s relations
(in Chow or cohomology) is conjectured to be full.
In this paper, we explicitly obtain the Pandharipande–Pixton–Zvonkine (PPZ or also called
r-spin) relations [6, 7] in the group RH2
(
Mg,n,Q
)
(although we omit the easiest case g = 0 for
brevity). We show that the PPZ relations we obtained coincide with relations from Arbarello–
Cornalba’s result [1, Theorem 2.2]. In [1], the authors proved that their set of relations
in H2
(
Mg,n,Q
)
is full. Also using the fact that PPZ (for r = 3) relations coincide with Pixton’s
relations and hold in H∗(Mg,n,Q
)
, proved in [6], we prove Pixton’s conjecture in cohomology
for this codimension.
mailto:danil.gubarevich@uvsq.fr
https://doi.org/10.3842/SIGMA.2024.088
2 D. Gubarevich
2 Preliminaries
2.1 Tautological classes on Mg,n
Throughout the article, we are working with cohomology with coefficients in Q.
Let Mg,n be the Deligne–Mumford compactification by stable curves of the moduli space of
nonsingular complex genus g curves with nmarkings, see [13] for the survey. Let π : Cg,n → Mg,n
be the universal curve. Considering Cg,n and Mg,n as orbifolds is sufficient for our purposes.
The fiber of a cotangent line bundles Li, i = 1, . . . , n over Mg,n over a point x ∈ Mg,n is
a cotangent line to a curve (Cx, x) ∈ Cg,n at i-th marked points. Denote the first Chern class of
the line bundles Li by ψi (ψ-classes)
ψi = c1(Li) ∈ H2
(
Mg,n,Q
)
.
κ-classes are defined via the universal map π : Mg,n+1 → Mg,n,
κi = π∗
(
ψi+1
n+1
)
∈ H2i
(
Mg,n,Q
)
.
Natural maps between moduli spaces of curves include:
(i) Maps forgetting the last m points and further stabilizing the curve pm : Mg,n+m → Mg,n.
(ii) Gluing maps of separating kind r : Mg1,n1+1 ×Mg2,n2+1 → Mg,n which identify the point
with label n1 + 1 on the first curve with point label n2 + 1 on the second curve.
(iii) Gluing maps of nonseparating kind q : Mg−1,n+2 → Mg,n which identifies points with
labels n+ 1 and n+ 2.
Let RH∗(Mg,n
)
⊂ H∗(Mg,n
)
be the subring of tautological classes in the rational cohomol-
ogy ring. It is defined as the smallest Q-subalgebra closed under push-forwards by forgetting
and gluing maps.
First examples of elements in RH∗(Mg,n
)
include the following classes. 1 ∈ H0
(
Mg,n
)
is
tautological since RH∗(Mg,n
)
is a ring with unit by definition. Classes represented by boundary
strata and their intersections also lie in tautological ring since they are images under gluing maps.
It is clear that κ- and ψ-classes lie in RH∗(Mg,n
)
from their definition and from relation ψi =
−π∗
(
D2
i
)
, where Di ∈ H2
(
Mg,n+1,Q
)
is a class of a divisor whose generic point is represented
by a nodal curve with two irreducible components of genera g and 0 and i-th and (n + 1)-th
points lie on a rational component.
The natural question are the rings RH∗(Mg,n
)
⊂ H∗(Mg,n
)
isomorphic is open but there
was a significant progress from the pioneering studies of moduli of curves initiated by Mumford.
2.2 Cohomological field theories
We recall here the basic definitions of a cohomological field theory (CohFT) by Kontsevich and
Manin, see [4] for a survey.
It consists of the following data (V ,η,1,(cg,n)2g−2+n>0), where
(i) V is a finite-dimensional Q-vector space,
(ii) η is a non-degenerate symmetric bilinear 2-form (metric),
(iii) 1 ∈ V is a distinguished element(unit vector).
Choosing a basis {ei} of V , denote by ηjk = η(ej , ek) the value of the metric on the basis
vectors and by ηjk the inverse matrix. Given these data (V, η,1) the (cg,n)2g−2+n>0 is a col-
lection of linear operators cg,n ∈ H∗(Mg,n
)
⊗ (V ∗)⊗n. It means that cg,n associate a (non-
homogeneous)cohomology class cg,n(x1⊗ · · ·⊗xn)∈ H∗(Mg,n
)
to a vector x1⊗ · · ·⊗xn ∈ V ⊗n.
The CohFT axioms imposed on cg,n are the following.
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 3
Definition 2.1. A tuple (V ,η,1,(cg,n)2g−2+n>0) is called a cohomological field theory (with unit)
if it satisfies the following axioms:
(i) For all xi ∈ V ,
cg,n(x1 ⊗ · · · ⊗ xn) = cg,n(xσ(1) ⊗ · · · ⊗ xσ(n))
for each σ in the symmetric group Σn acting on the set of marked points and simultaneously
on copies of V in the tensor product V ⊗n.
(ii) The pullback of cg,n under the gluing map q of nonseparating kind equals the contraction
of cg−1,n+2 with ηjk ⊗ ej ⊗ ek inserted at the two identified points1
q∗cg,n(x1 ⊗ · · · ⊗ xn) =
∑
j,k
ηjkcg−1,n+2(x1 ⊗ · · · ⊗ xn ⊗ ej ⊗ ek)
in H∗(Mg−1,n+2,Q) for all xi ∈ V . The pullback of cg,n under a gluing map r of separating
kind equals the contraction cg1,n1+1 ⊗ cg2,n2+1 again with ηjk ⊗ ej ⊗ ek inserted at the two
identified points
r∗(cg,n(x1 ⊗ · · · ⊗ xn))
=
∑
j,k
ηjkcg1,n1+1(x1 ⊗ · · · ⊗ xn1 ⊗ ej)⊗ cg2,n2+1(xn1+1 ⊗ · · · ⊗ xn ⊗ ek)
in H∗(Mg1,n1+1,Q
)
⊗H∗(Mg2,n2+1,Q
)
for all xi ∈ V .
(iii) The pullback of cg,n under the map p forgetting the last point is required to satisfy
cg,n+1(x1 ⊗ · · · ⊗ xn ⊗ 1) = p∗cg,n(x1 ⊗ · · · ⊗ xn)
for all xi ∈ V . In addition, the equality c0,3(xi ⊗ xj ⊗ 1) = η(xi, xj) is required for
all xi ∈ V .
Remark 2.2. A tuple (V , η, (cg,n)2g−2+n>0) is called a cohomological field theory (without unit)
if it satisfies properties (i) and (ii) above.
The degree zero part ω of a CohFT c = (cg,n)2g−2+n>0 is called the topological part of c
ωg,n = [cg,n]
0 ∈ H0
(
Mg,n,Q
)
⊗ (V ∗)⊗n.
Via property (ii) one sees that ωg,n(x1 ⊗ · · · ⊗ xn) is determined by considering stable curves
with a maximal number of nodes. Such a curve is obtained by identifying several rational curves
with three marked points. The value of ωg,n(x1 ⊗ · · · ⊗ xn) is thus uniquely specified by the
values of ω0,3 and by the quadratic form η. In other words, given V and η, a topological part is
uniquely determined by the associated quantum product, introduced in the next subsection.
Our main example of CohFT will be the CohFT associated with Witten’s r-spin class and
its shifted version.
Example 2.3 (Witten’s r-spin class). Let r ≥ 2 be an integer. For every r, there is a CohFT
obtained from Witten’s r-spin class, introduced by Witten in [12]. See [9, 10] for an algebraic
construction of Witten’s top Chern class where it was proved that the axioms of CohFT (i)–(iii)
hold.
1Here and in the rest of the paper, we use the Einstein summation convention.
4 D. Gubarevich
Witten’s r-spin theory gives a family of classes
W r
g,n(a1, . . . , an) ∈ H∗(Mg,n
)
for a1, . . . , an ∈ {0, . . . , r − 2}.
Now given an (r − 1)-dimensional Q-vector space Vr with basis e0, . . . , er−2, metric
ηab = η(ea, eb) = δa+b,r−2,
and unit vector 1 = e0 one defines a CohFT Wr
g,n by
Wr
g,n : V ⊗n
r → H∗(Mg,n
)
, Wr
g,n(ea1 ⊗ · · · ⊗ ean) =W r
g,n(a1, . . . , an).
Witten’s class W r
g,n(a1, . . . , an) has (complex) degree given by the formula
degCW
r
g,n(a1, . . . , an) = Dr
g,n(a1, . . . , an) =
(r − 2)(g − 1) +
∑n
i=1 ai
r
.
If Dr
g,n(a1, . . . , an) is not an integer, the corresponding Witten’s class vanishes.
To formulate the Givental–Teleman classification result, we need the following notion of
semisimplicity of a CohFT.
Let (V, η,1) be a data associated with a CohFT cg,n. For x1, x2 ∈ V , the quantum prod-
uct x1 • x2 ∈ V is uniquely determined by the condition: for every x3 ∈ V
η(x1 • x2, x3) = c0,3(x1 ⊗ x2 ⊗ x3) ∈ Q.
The quantum product • is commutative by CohFT axiom (i). Let us check that associativ-
ity of • follows from CohFT axiom (ii). Further, we often write just c0,3(x1, x2, x3) instead
of c0,3(x1 ⊗ x2 ⊗ x3) for short. Fix a basis {ei} of V . Then the product in this algebra gives
rise to structure constants ea • eb = ciabei, where the Einstein summation convention is assumed.
Hence
η((ea • eb) • ec, ed) = ciabη(ei • ec, ed) = ciabc
j
icηjd.
Observe that ciab = ηicc0,3(ea, eb, ec), hence we get
ηipc0,3(ea, eb, ep)η
jqc0,3(ei, ec, eq)ηjd,
which is the same as q∗c0,4(ea, eb, ec, ed) by axiom (ii). On the other hand
η(ea • (eb • ec), ed) = cibcη(ea • ei, ed) = cibcc
j
aiηjd = ηipc0,3(eb, ec, ep)η
jqc0,3(ea, ei, eq)ηjd
= c0,3(eb, ec, ep)η
ipc0,3(ea, ei, ed),
which is again q∗c0,4(ea, eb, ec, ed) by axioms (i) and (ii). Then from the nondegeneracy of η
the associativity follows. Moreover, a simple check shows that (V, •,1) is a Frobenius algebra,
see [2]. A CohFT cg,n is called semisimple if the algebra (V, •,1) is semisimple, i.e., the com-
plexification V ⊗Q C has a basis {ei} of idempotents ei • ej = δijei. Since Wr
g,n itself is not
semisimple, we will need its shifted version.
Example 2.4 (shifted Witten’s r-spin class). Given a vector τ ∈ Vr, the shifted Witten’s class
is defined by
Wr,τ
g,n(v1 ⊗ · · · ⊗ vn) =
∑
m≥0
1
m!
pm∗W
r
g,n+m
(
v1 ⊗ · · · ⊗ vn ⊗ τ⊗m
)
,
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 5
where pm : Mg,n+m → Mg,n is the forgetful map. The shifted Witten’s class Wr,τ determines
a CohFT with unit. Namely, applying a map forgetting the last point, we get
p∗1W
r,τ
g,n(v1 ⊗ · · · ⊗ vn) =
∑
m≥0
1
m!
pm∗p
∗
1W
r
g,n+m
(
v1 ⊗ · · · ⊗ vn ⊗ τ⊗m
)
=
∑
m≥0
1
m!
pm∗W
r
g,n+m+1
(
v1 ⊗ · · · ⊗ vn ⊗ τ⊗m ⊗ 1
)
= Wr
g,n+1(v1 ⊗ · · · ⊗ vn ⊗ 1).
Moreover, let us choose a particular point τ̃ = (0, rϕ, 0, . . . , 0) = rϕe1 ∈ Vr and check the
semisimplicity of this CohFT.
Denote by ⟨a1, . . . , an⟩τ :=
∫
M0,n
Wr,τ
0,n(ea1 ⊗ · · · ⊗ ean) the n-point function. Firstly, we use
the known values for Witten’s class
W0,3(ea, eb, ep) = δp,r−2−a−b, W0,4(e1, e1, er−2, er−2) =
1
r
[pt] ∈ H2
(
M0,4,Q
)
.
Using [7, Proposition 4.1], we compute the 3-point function ⟨a, b, p⟩τ̃ =δp,r−2−a−b+δp,2r−3−a−bϕ.
Then we compute
ea •τ̃ eb = ciabei = ηipWr,τ̃
0,3(ea, eb, ep)ei =
{
ηi,r−2−a−bei = ea+b if a+ b ≤ r − 2,
ϕηi,2r−3−a−bei = ϕea+b−r+1 if a+ b ≥ r − 1.
Then changing a basis to ẽa := ϕ−a/(r−1)ea the new product is
ẽa •τ̃ ẽb = ϕ−(a+b)/r−1ea • eb =
{
ẽa+b if a+ b ≤ r − 2,
ẽa+b−r−1 if a+ b ≥ r − 1
= ẽa+b mod r−1.
Finally, after another change of basis ēi :=
∑r−2
a=0 ξ
aiẽa, ξ
r−1 = 1, we derive a basis of idempo-
tents
ēi •τ̃ ēj =
r−2∑
a,b=0
ξai+bj ẽa+b mod r−1 =
r−2∑
a,c=0
ξa(i−j)+cj ẽj =
r−2∑
a=0
ξa(i−j)ēj = (r − 1)δij ēj ,
since if i− j = x ̸= 0 then 1 + ξx + · · ·+ ξ(r−2)x = 0. So, {ēi/(r− 1)} is a basis of idempotents.
Now we recall how to assign to the given CohFT (V, η,1, (cg,n)2g−2+n>0) a Frobenius manifold
structure on V . Choose a basis V = C⟨e1, . . . , eN ⟩ and a distinguished vector 1 = e1. A full
CohFT potential F ∈ C[[t∗,∗, ℏ]] is the following formal power series in tdk,ik , dk ≥ 0, ik ∈
{1, . . . , N}:
F
({
tdk,ik
}
, ℏ
)
:=
∑
g≥0
ℏg−1Fg,
where
Fg
({
tdk,ik
})
:=
∑
n≥x(g)
1
(n− x(g))!
∑
d1,...,dn
i1,...,in
∫
Mg,n
cg,n(ei1 , . . . , ein)ψ
d1
1 · · ·ψdn
n td1,i1 · · · tdn,in .
The stability condition is captured by
x(g) :=
3 if g = 0,
1 if g = 1,
0 if g ≥ 2.
Note that the intersection numbers above are nonzero only in degrees 3g − 3 + n.
6 D. Gubarevich
A Frobenius potential F ∈ C
[[
t1, . . . , tN
]]
is defined as
F
(
t1, . . . , tN
)
:= F0
({
tdk,ik
})
|{tdk,∗=0,dk>0}
=
∑
n≥3
1
(n− 3)!
N∑
i1,...,in=1
∫
M0,n
c0,n(ei1 , . . . , ein)t
i1 · · · tin ,
where we put tµ := t0,µ.
It gives the structure of formal Frobenius manifold with flat metric ηµν = ∂3F
∂t1∂tµ∂tν
and
a Frobenius algebra structure with multiplication eα • eβ := cαβγη
γθeθ, where
cαβγ(t
∗) =
∂3F
∂tα∂tβ∂tγ
,
(
ηαβ
)
:= (ηαβ)
−1.
CohFT axioms force F to satisfy the WDVV equation cαβθη
θδcδγρ = cαγθη
θδcδβρ.
2.3 Reconstruction
Let c = (cg,n)2g−2+n>0 be a CohFT, not necessarily with unit, associated with data (V, η). Let R
be a matrix series
R(z) =
∞∑
k=0
Rkz
k ∈ Id + z · End(V )[[z]],
which satisfies the symplectic condition η(R(z)x,R(−z)y) = η(x, y) for all x, y in V . The group
of such series acts on a space of CohFTs and we now recall the action, following closely [4,
Section 1.2.1].
One can define a new CohFT Rc on the vector space (V, η) is expressed as a sum over stable
graphs Γ, see [6, Section 0.2] for an exposition, with summands given by products of vertex, edge,
and leg contributions. Let Gg,n denote the set of all stable graphs (up to isomorphism) of genus g
with n legs. To each stable graph Γ, one can associate the moduli space MΓ =
∏
v∈V Mg(v),n(v)
with 2g(v)− 2+n(v) > 0. There is a canonical morphism ιΓ : MΓ → Mg,n with image equal to
the boundary stratum associated to the graph Γ.
Then new CohFT Rc is expressed as
(Rc)g,n =
∑
Γ∈Gg,n
1
|Aut(Γ)|
ιΓ⋆
(∏
v∈V
Cont(v)
∏
e∈E
Cont(e)
∏
l∈L
Cont(l)
)
,
where we place
(i) Cont(v) = cg(v),n(v) at each vertex, where g(v) and n(v) denote the genus and number of
half-edges and legs of the vertex,
(ii) the End(V )-valued cohomology class Cont(l) = R−1(ψl) at each leg, ψl∈H2
(
Mg(v),n(v),Q
)
is the cotangent class at the marking corresponding to the leg,
(iii) Cont(e) =
(
η−1 − R−1(ψ′
e)η
−1R−1(ψ′′
e )
⊤)/(ψ′
e + ψ′′
e ) at each edge, where ψ′
e and ψ′′
e are
the cotangent classes2 at the node which represents the edge e.
Then we can state Givental–Teleman classification result for CohFTs, proved initially in [11]
for underlying Frobenius potentials. Let c be a semisimple CohFT with unit on (V, η,1), and
2Note that if ψ′
e or ψ′′
e corresponds to a marking on genus 0 component with less then four markings, then it
is zero.
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 7
let ω be the topological part of c. For a matrix R satisfying the symplectic condition, de-
fine R.ω = R(T (ω)), where T ∈ V [[z]] is a series with no terms of degree 0 or 1
T (z) = z
(
Id−R−1(z)
)
· 1 ∈ z2V [[z]],
acting on a CohFT cg,n by the formula
(Tc)g,n(x1 ⊗ · · · ⊗ xn) =
∞∑
m=0
1
m!
pm∗(cg,n+m(x1 ⊗ · · · ⊗ xn ⊗ T (ψn+1)⊗ · · · ⊗ T (ψn+m))),
where pm : Mg,n+m → Mg,n is the morphism forgetting the last m markings.
By [6, Proposition 2.12], R.ω is a CohFT with unit on (V, η,1). The Givental–Teleman
classification asserts the existence of a unique R-matrix which exactly recovers c.
Theorem (Teleman). There exists a unique symplectic matrix R ∈ Id + z · End(V )[[z]] which
reconstructs c from ω, c = R.ω, as a CohFT with unit.
3 Tautological relations at semisimple point:
τ̃ = (0, rϕ, 0, . . . , 0)
We chose this point since at this point one can derive the relatively simple expression for the
topological field theory of the corresponding shifted Witten’s CohFT, although the closed for-
mulas for R-matrix are not known. Acting on the topological field theory by R-matrix, we
successfully deduce the PPZ relations (also called r-spin relations) for r ≥ 3 in tautological ring,
using degree vanishing argument. We will follow Pandharipande–Pixton–Zvonkine [7].
The shifted along the second basis vector e1 ∈ Vr Witten’s CohFT is semisimple and has
a relatively simple expression for its topological part. As we saw in Example 2.4, the quantum
product at τ̃ is given by
ẽa •τ̃ ẽb =
{
ẽa+b if a+ b ≤ r − 2,
ẽa+b−r+1 if a+ b ≥ r − 1.
By [7, Proposition 4.2], the associated topological field theory takes the form
ωr,τ̃
g,n(ẽa1 ⊗ · · · ⊗ ẽan) = ϕ(g−1) r−2
r−1 (r − 1)g · δ, (3.1)
where δ equals 1 if g−1−
∑n
i=1 ai is divisible by r−1 and 0 otherwise. The idea is the following.
Firstly, from definition of ẽa observe that
ωr,τ̃
0,3(ẽa, ẽb, ẽc) =
{
ϕ−
r−2
r−1 if a+ b+ c = −1 mod r − 1,
0 else,
and hence ηab = ϕ−
r−2
r−1 δa+b,r−2 in a basis {ẽi}. Then ωr,τ̃
g,n for general g and n is the same as
the restriction of Ωr,τ̃
g,n to a maximally degenerate curve [C] ∈ Mg,n with 2g − 2 + n rational
components and 3g − 3 + n nodes. From this, 3-valent dual graph we see that the sum of the
insertions to ωr,τ̃
0,3 from {0, . . . , r − 2} plus one is divisible by r− 1 and the sum of the insertions
on each edge is r−2. From this, it is clear that g−1+
∑
ai must be divisible by r−1. We place
an arbitrary insertion on a single branch of a node at every independent cycle of the dual graph,
then the other insertions are uniquely determined. This gives us a factor (r − 1)g. The factor
ϕ(g−1) r−2
r−1 comes from matching nodes and rational component powers of ϕ.
8 D. Gubarevich
There is an approach for computing R-matrix recursively explained in [7, Proposition 4.4].
The upshot is the explicit formula for the coefficients.
In a basis ẽ0, . . . , ẽr−2 the R-matrix (Rm)ba, m ≥ 0, a, b ∈ {0, . . . , r − 2} has coefficients
(Rm)ba =
[
−r(r − 1)ϕ
r
r−1
]−m
Pm(r, r − 2− b) if b+m = a mod r − 1,
and 0 otherwise. The inverse matrix R−1(z) has coefficients(
R−1
m
)b
a
=
[
r(r − 1)ϕ
r
r−1
]−m
Pm(r, a) if b+m = a mod r − 1,
and 0 otherwise. Here the polynomials Pm(r, a) with initial condition P0(r, a) = 1 are defined
by the following recursive procedure. For m ≥ 1,
Pm(r, a) =
1
2
a∑
b=1
(2mr − r − 2b)Pm−1(r, b− 1)
− 1
4mr(r − 1)
r−2∑
b=1
(r − 1− b)(2mr − b)(2mr − r − 2b)Pm−1(r, b− 1).
In fact, we will need only first two values
P0 = 1, P1 =
1
2
a(r − 1− a)− 1
24
(2r − 1)(r − 2).
Now let Wr,τ̃ be the cohomological field theory given by the shift of Witten’s r-spin class by
the vector τ̃ = (0, rϕ, 0, . . . , 0). Using the Givental–Teleman classification theorem, Wr,τ̃ can
be recovered as an R-matrix action on the topological field theory ωr,τ̃ and by [7, Theorem 9]
Witten’s r-spin class W r
g,n(a1, . . . , an) equals the part of Wr,τ̃
g,n of degree
Dr
g,n(a1, . . . , an) =
(r − 2)(g − 1) +
∑n
i=1 ai
r
in H∗(Mg,n
)
. The parts of Wr,τ̃
g,n of degree higher than Dr
g,n vanish. This theorem is the key
point to find relations.
Now we describe the procedure of finding r-spin relations at the point τ̃ from the shifted Wit-
ten’s class. It will be convenient to specialize the decoration procedure introduced in Section 2.3
to the case of the reconstruction of the shifted Witten’s CohFT Wr,τ̃
g,n from its topological part.
Fix r ≥ 3. Fix n numbers 0 ≤ a1, . . . , an ≤ r − 2. All constructions below depend on an
auxiliary variable ϕ which keeps track the codimension and we fix its exponent d. A tauto-
logical r-spin relation T (g, n, r, a1, . . . , an, d) = 0 depends on these choices, and it is obtained
as T = rg−1
∑∞
k=0 π
(k)
∗ Tk/k!, where Tk is the coefficient of ϕd in the expression in the decorated
stable graphs of Mg,n+k described below, and π(k) : Mg,n+k → Mg,n forgets the last k points.
Fix a stable graph Γ ∈ Gg,n+k, and then equip
(i) each vertex v of Γ with ωr,τ̃
g(v),n(v),
(ii) the first n legs by
∑∞
m=0
(
R−1
m
)b
ai
ψm
i ẽb, i = 1, . . . , n,
(iii) the k extra legs (the dilaton legs) with −
∑∞
m=1
(
R−1
m
)b
0
ψm+1
n+i ẽb, i = 1, . . . , k,
(iv) each edge e, where we denote by ψ′
e and ψ′′
e the ψ-classes on the two branches of the
corresponding node, with
Cont(e) =
ηi
′i′′ −
∑∞
m′,m′′=0
(
R−1
m′
)i′
j′
ηj
′j′′
(
R−1
m′′
)i′′
j′′
(ψ′
e)
m′
(ψ′′
e )
m′′
ψ′
e + ψ′′
e
ẽi′ ⊗ ẽi′′ .
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 9
The summations above by repeating indices is assumed. Then Tk is defined as the sum over
all decorated stable graphs obtained by the contraction of all tensors assigned to their vertices,
legs, and edges, further divided by the order of the automorphism group of the graph.
Observe that the exponent d of ϕ comes from many places, so let us collect the total degree.
� Every time we fed a leg we get a factor ϕ−mr/(r−1) from R−1
m and if we are interested in
relations in the group RH2D
(
Mg,n,Q
)
, then the sum of indices
∑
mi of matrices R−1
m
should be equal to D. Hence, we get a factor ϕ−
rD
r−1 .
� At every vertex v equipped with ωr,τ̃
g(v),n(v), we get a contribution ϕ(g(v)−1)(r−2)/(r−1) and
the total contribution from all the vertices will be ϕ(
∑ν
v g(v)−ν) r−2
r−1 = ϕg−1−δ, where ν is the
number of irreducible components and δ is the number of nodes the curve Γ has.
� Since we wrote each Rm in a basis {ẽi} and fed the first n legs by the primary fields
e1, . . . , en, we should change a basis to {ẽi}. Hence, we earn an additional factor ϕ
1
r−1
∑
i ai
after the change of basis ei = ϕ
i
r−1 ẽi.
� Dilaton legs will not contribute in the exponent of ϕ since they are fed with e0 = ẽ0.
� Since each edge of Γ contains an inverse metric matrix ηij , it gives a factor ϕ
r−2
r−1 and the
total factor will be ϕ
r−2
r−1
δ. Recall that the number of edges of Γ is the number of nodes on
the corresponding curve.
Summing up, the total degree d of variable ϕ for the graph Γ in codimension D satisfies
d(r − 1) =
∑
ai + (g − 1)(r − 2)− rD.
From the topological part expression (3.1), we can say that∑
ai = g − 1 +D + x(r − 1)
for x ≥ 0. Now if Witten’s r-spin class vanishes when D > Dr
g,n it is easy to see that d must be
negative.
Finally, observe that
d(r − 1) = g − 1 +D + x(r − 1) + (g − 1)(r − 2)− rD = (g − 1−D + x)(r − 1),
and hence d < 0 if and only if D ≥ g + x.
This shows that Pixtons’s relations do not allow us to find relations in small degrees directly.
We are interested in relations in complex degree 1 and to get them we need firstly to find
relations in cohomology of higher degrees of the space with larger number of points and then
push them forward by the forgetful morphism. For example, we could find the relations in the
group RH4
(
M2,n+1,Q
)
and then push them forward by forgetting the last point. Thus, we
would get relations in the group RH2
(
M2,n,Q
)
, but this approach is technically difficult and
we proceed in another way in the genus 2 case. But the genus 1 case can be worked out directly.
4 The case of marked genus 1 curves
In this section by applying the PPZ construction to the genus 1 case, we derive tautological
relations and compare them with Arbarello–Cornalba’s result. In this case, the PPZ approach
works in each positive degree since D ≥ g+x = 1+x is valid for each D ≥ 1 if we choose x = 0.
Firstly, we apply described construction to the space M1,2 and then calculate linear relations
in the space M1,n. As a warm-up, we discuss linear relations in the space M1,2. Further, we
10 D. Gubarevich
1
a1 a2
(i) smooth part
1 0
a1
a2
(ii) δsep
0
a1 a2
(iii) δnonsep
Figure 1. The graphs of strata of M1,2 of codimension at most 1.
fix ϕ = 1 for simplicity. The cohomology group RH2
(
M1,2,Q
)
is generated by ψ1, ψ2, κ1, δsep,
δnonsep with relations
ψ1 = ψ2, 2ψ1 = δsep + κ1, ψ1 =
1
12
δnonsep + δsep,
see [1, Theorem 2.2].
Example 4.1. For r ≥ 3, the PPZ relations between the generators of the group RH2
(
M1,2,Q
)
generate the set of relations from above.
Proof. We have D = 1, x = 0, g = 1, a1+a2 = 1; the degree 1 part of stable graph expressions
gives us tautological relations since 1 = D ≥ g + x = 1. Generators of this group are ψ1, ψ2,
κ1, δsep, δnonsep, where δsep and δnonsep are Poincaré dual classes to the generic point of divisor
strata, see Figure 1. The graphs with more then 1 edge will not contribute by dimensional
reasons.
Our stable graphs will contain 0, 1 or 2 dilaton leaves, so we will consider these cases sepa-
rately. The smooth part gives
∞∑
m=0
r−2∑
b1,b2=0
(
R−1
m
)b1
a1
ψm
1
∞∑
m=0
(
R−1
m
)b2
a2
ψm
2 ω
r,τ̃
1,2(ẽb1 , ẽb2)
2
=
r−2∑
b1,b2=0
((
R−1
0
)b1
a1
+
(
R−1
1
)b1
a1
ψ1
)((
R−1
0
)b2
a2
+
(
R−1
1
)b2
a2
ψ2
)
ωr,τ̃
1,2(ẽb1 , ẽb2)
2
= (r − 1)
((
R−1
1
)a1−1
a1
ψ1 +
(
R−1
1
)a2−1
a2
ψ2
)
.
Since the factor
(
r(r − 1)ϕr/(r−1)
)−1
comes from each graph, we will omit it.
Let us develop the edge factor3 in more detail up to linear terms in the ψ-classes.
Cont(e)(ψ′
e + ψ′′
e ) =
ηi′i′′ − ∞∑
m′,m′′=1
(
R−1
m′
)i′
j′
ηj
′j′′
(
R−1
m′′
)i′′
j′′
(ψ′
e)
m′
(ψ′′
e )
m′′
ẽi′ ⊗ ẽi′′
=
[
−
(
R−1
1
)i′′
r−2−i′
(ψ′ + ψ′′) + higher degree terms
]
ẽi′ ⊗ ẽi′′ .
3See Section 3.
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 11
1
a1 a2
(iv)
Figure 2. The remaining contributing graph with a dilaton leg.
The second and third graphs give
−δsep
r−2∑
b1,b2=0
((
R−1
0
)b1
a1
+
(
R−1
1
)b1
a1
ψ1
)((
R−1
0
)b2
a2
+
(
R−1
1
)b2
a2
ψ2
)(
R−1
1
)i′′
r−2−i′
,
ωr,τ̃
0,3(ẽb1 , ẽb2 , ẽi′)ω
r,τ̃
1,1(ẽi′′) = −(r − 1)δsep
(
R−1
1
)0
1
,
−δnonsep
r−2∑
b1,b2=0
(
R−1
0
)b1
a1
(
R−1
0
)b2
a2
r−2∑
i=0
(
R−1
1
)i′
r−2−i′′
ωr,τ̃
0,4(ẽb1 , ẽb2 , ẽi′ , ẽi′′)ϕ
(r−2)/(r−1).
Given a graph Γ with a dilaton leg, the vertex carrying this leg will support a class of positive
degree after push-forward. Thus a codimension 1 class can only be obtained if the graph Γ was
already trivial, see Figure 2.
The fourth graph gives
−
((
R−1
0
)b1
a1
+
(
R−1
1
)b1
a1
ψ1 + (R−1
2 )b1a1ψ
2
1
)((
R−1
0
)b2
a2
+
(
R−1
1
)b2
a2
ψ1 + (R−1
2 )b2a2ψ
2
1
)
,((
R−1
1
)b
0
κ1 + (R−1
2 )b0κ2
)
ωr,τ̃
1,3(ẽb1 , ẽb2 , ẽb) = −(r − 1)
(
R−1
1
)r−2
0
κ1.
So, summing up we get a relation
(r − 1)[P1(r, a2)ψ2 + P1(r, a1)ψ1]−
1
24
(
2− 3r + r2
)
δnonsep,
−(r − 1)P1(r, 1)δsep − (r − 1)P1(r, 0)κ1 = 0.
This equality must hold for every r ≥ 3, so collecting degrees of a variable r, we get following
relations:
ψ1 = ψ2, 2ψ1 = δsep + κ1, ψ1 =
1
12
δnonsep + δsep. ■
Similarly, we derived all PPZ relations in RH2
(
M1,n,Q
)
for all r simultaneously, and prove
that they in fact are the full set of relations in H2
(
M1,n,Q
)
. In fact, we prove a stronger
statement that the PPZ relations for every particular r ≥ 3 generate the full set of relations.
The group H2
(
M1,n,Q
)
is generated by ψ1, . . . , ψn, κ1, δ
0
J , δnonsep with relations
12ψi = δnonsep + 12
∑
J⊂{1,...,n}|i∈J
|J |≥2
δ0J , i = 1, . . . , n, κ1 =
n∑
i=1
ψi −
∑
J⊂{1,...,n}
δ0J , (4.1)
see [1, Theorem 2.2].
12 D. Gubarevich
Proposition 4.2. For every particular r ≥ 3, the PPZ relations between the generators of the
group H2
(
M1,n,Q
)
generate the set of relations from above.
Proof. There are
n+ 1 +
(
n
2
)
+ · · ·+
(
n
n
)
+ 1 = 2n + 1
generators ψ1, . . . , ψn, κ1, δ
0
J , δnonsep. Here δ0J is the class of the divisor representing a curve
with markings from J ⊂ {1, . . . , n}, |J | ≥ 2 on the genus 0 component. Having n markings,
we have n relations coming from n different choices of a vector (a1, . . . , an) with
∑n
i=0 ai = 1.
Further, in the arguments of the topological part, we write just bi instead of ẽbi for short. The
contribution of the smooth part has the form
n∑
i=1
[(
R−1
0
)ai
ai
+
(
R−1
1
)ai−1
ai
ψi
]
ω1,n(0, . . . , 0)
= (r − 1)
[
ψ1
(
R−1
1
)a1−1
a1
+ · · ·+ ψn
(
R−1
1
)an−1
an
]
.
The contribution of a graph δ0J with separating node has the form
−δ0J
n∏
i=1
[(
R−1
0
)ai
ai
+
(
R−1
1
)ai−1
ai
ψi
] r−2∑
p,q=0
ω1,|I|+1(ai1 , . . . , ai|I| , p),
ω0,|J |+1(aj1 , . . . , aj|J| , q)
(
R−1
1
)p
r−2−q
,
where
ω1,|I|+1(ai1 , . . . , ai|I| , p) = (r − 1)
{
1 if aI + p = 0 mod r − 1,
0 otherwise,
ω0,|J |+1(aj1 , . . . , aj|J| , q) =
{
1 if 1 + aJ + q = 0 mod r − 1,
0 otherwise.
From this, we immediately see that each
(
R−1
1
)p
r−2−q
=
(
R−1
1
)−aI
aJ
is nonzero since r − 2 + aI +
1 − 1 = −aJ mod r − 1 is true for every complementary subsets I, J ⊂ {1 . . . n}. The total
contribution over graphs with one separating node is
−
∑
J⊂1,...,n
|J |≥2
δ0J(r − 1)
(
R−1
1
)−aI
aJ
,
and specifying ai = 1, aj = 0, i ̸= j, we would get
−(r − 1)P1(r, 1)
∑
J |i∈J
δ0J − (r − 1)P1(r, 0)
∑
J |i/∈J
δ0J .
The contribution of a graph with a nonseparating node gives −δnonsep 1
24
(
2−3r+r2
)
as before.
The contribution of a graph with one dilaton leg has the form
−
n∏
i=1
[(
R−1
0
)ai
ai
+
(
R−1
1
)ai−1
ai
ψi
] r−2∑
w=0
(R−1
0 )w0 κ1ω1,n+1(a1, . . . , an, w),
which is nonzero if and only if w = r − 2 and equals −
(
R−1
1
)r−2
0
(r − 1)κ1.
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 13
So, for each fixed i = 1, . . . , n, we have the relation
(r − 1)
[
P1(r, 1)ψi + P1(r, 0)
∑
j ̸=i
ψj
]
,
−(r − 1)P1(r, 1)
∑
J |i∈J
δ0J − (r − 1)P1(r, 0)
∑
J |i/∈J
δ0J ,
−P1(r, 0)(r − 1)κ1 − δnonsep
1
24
(
2− 3r + r2
)
= 0,
which is a polynomial in r of degree 3. Taking coefficients and multiplying by 12 and 24,
respectively, to clear denominators, we get[
r3
]
: κ1 =
n∑
i=1
ψi −
∑
J⊂{1,...,n}
δ0J ,
[
r2
]
: 19
∑
J |i∈J
δ0J + 7
∑
J |i/∈J
δ0J + δnonsep + 7κ1 = 19ψi + 7
∑
j ̸=i
ψj ,
and eliminating kappa class, we find
12ψi = δnonsep + 12
∑
J |i∈J
δ0J , i = 1, . . . , n.
The relations coming from [r], [1] are consequences of relations coming from
[
r3
]
,
[
r2
]
. Suppose
now that we have another PPZ relation Q = 0 in degree 1. Put r = 3 for simplicity. As we
noted at the end of Section 3, this relation came from a certain choice of a tuple (a1, . . . , an) such
that
∑
ai = 1 mod 2 and 1 > 1
3
∑
ai. There are only n possibilities corresponding to a choice
of one ai = 1 and 0 the others. So, Q is a linear combination of them.
Moreover, we claim that we get relations (4.1) for every particular r ≥ 3, not just for simul-
taneously all r. For this, it is sufficient to express ψ-classes in terms of boundary divisor classes
and use the linear independence of the latter, which follows from [1, Theorem 2.2].
Namely, we rewrite the system of relations in the form
(r − 1)P1(r, 1)ψ1 + (r − 1)P1(r, 0)ψ2 + · · ·+ (r − 1)P1(r, 0)ψn − (r − 1)P1(r, 0)κ1 = B1,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
(r − 1)P1(r, 0)ψ1 + · · ·+ (r − 1)P1(r, 0)ψn−1+ (r − 1)P1(r, 1)ψn− (r − 1)P1(r, 0)κ1 = Bn,
ψ1 + · · ·+ ψn − κ1 = Bn+1,
where the last equation is the coefficient of r3 from above. Here the terms Bi are linear combi-
nations of boundary divisors with coefficients depending on r.
The matrix M on the left-hand side has determinant
detM = (−1)n(r − 1)n(P1(r, 0)− P1(r, 1))
n−1((n− 1)P1(r, 0) + P1(r, 1)− nP1(r, 0)),
and after further simplification it is seen that for r ≥ 3
detM = −1
[
(r − 1)(r − 2)
2
]n
̸= 0.
This means that for every r ≥ 3 each ψi can be expressed as a sum of divisor classes, without
passing to taking coefficients
[
ri
]
. Expressions of ψi coming from r ≥ 3 and r′ ≥ 3 give a linear
relation on boundary divisor classes, which are linear independent. In other words, we get
the same relations for each particular r ≥ 3. Moreover, these relations are the same as in
Arbarello–Cornalba’s [1, Theorem 2.2], where it is proved that it is the full set of relations
in H2
(
M1,n,Q
)
. ■
14 D. Gubarevich
1
(i) δirr
1 1
(ii) δ1
2
(iii) κ1
Figure 3. Divisors in M2.
5 The case of marked genus 2 curves
In this section, by applying the PPZ construction to the genus 2 case, we derive tautological
relations and compare them with the Arbarello–Cornalba’s result.
The group RH2
(
M2,n,Q
)
is generated by κ1, ψ1, . . . , ψn and divisor classes δI0 , δ
I
1 , I ⊂
{1, . . . , n}, δ′irr with one relation
5
(
κ1 −
∑
ψi +
∑
I⊂1,...,n
δI0
)
= δ′irr + 7
∑
I⊂1,...,n
δI1 ,
see [1, Theorem 2.2].
Proposition 5.1. For every particular r ≥ 3, the PPZ relations between the generators of the
group RH2
(
M2,n,Q
)
generate the set of relations from above.
Proof. We proceed as follows. Firstly, we find relations in M2 using 3-spin structures, then
pull them back by forgetting n points. Get again a PPZ relation and argue why what we get the
full set of relations. We are using the PPZ relations are stable under pulling back via forgetting
morphism π : Mg,n+k → Mg,n
π∗T (g, n, r, a1, . . . , an, d) = T (g, n+ k, r, a1, . . . , an, 0, . . . , 0, d).
Now, the Picard group of M2 is spanned on κ1, δ1 and δirr, see Figure 3.
Witten’s class has degree D3
2,0 = (r−2)
r , which is less than one, so we get a relation. Graphs
in Figure 3 give the following contributions:
−(r − 1)2κ1
(
R−1
1
)r−2
0
= −(r − 1)2κ1P1(r, 0),
−
r−2∑
i,j=0
ω1,1(i)ω1,1(j)
(
R−1
1
)i
r−2−j
δ1 = −(r − 1)2P1(r, 1)δ1, −(r − 1)δirr
2− 3r + r2
24
.
When we put r = 3, after simple algebra, we get 5κ1 = 7δ1 + δirr.
Now we pullback this relation by forgetting n points π : M2,n → M2. Using the well-known
formulas for the pull-backs along the forgetful map
π∗i κ1 = κ1 − ψi, π∗i ψj = ψj − δ0{i,j}, π∗i δ
h
I = δhI + δhI∪{i},
we get, after iterating n times, the relation
5
(
κ1 −
∑
ψi +
∑
I⊂1,...,n
δI0
)
= δ′irr + 7
∑
I⊂1,...,n
δI1 .
Now suppose we have another PPZ relation Q = 0 in degree 1. As we noted at the end of
Section 3, this relation came from a certain choice of a tuple (a1, . . . , an) such that
∑
ai = 0
mod 2 and 1 > 1+
∑
ai
3 . Now we have only one possibility corresponding to the choice when
all ai’s equal to 0. So, Q is proportional to our relation. Moreover, this relation is the same as in
Arbarello–Cornalba’s [1, Theorem 2.2]. Other choices of r ≥ 3 would lead to the same relation
since the set of Arbarello–Cornalba relations is full and PPZ relations hold in cohomology. So,
we are done. ■
On the Picard Group of the Moduli Space of Curves via r-Spin Structures 15
6 Higher genera and conclusion
In genus 3 with no markings, we get zero contributions since the parity conditions for vertices are
not compatible. We can not satisfy the conditions. For example, curves with a node separating
genus 1 and genus 2 components give
−δ1
1∑
p,q=0
ω1,1(p)ω1,1(p)ω2,1(q)
(
R−1
1
)p
1−q
.
So, p and q must be p = 0, q = 1, hence
(
R−1
1
)0
0
is zero. We can not satisfy the conditions∑
ai = 1 mod 2 and 1 >
∑
ai+2
3 at all.
In genus 4 with no markings, we get the expression of the Witten’s class instead of a tau-
tological relation since D3
4,0 = 1. And in higher genera, the degree of Witten’s class is bigger
than 1 for all r. These considerations agree with early computations and do not contradict
the fact that the only relations in RH2
(
Mg,n,Q
)
with g ≥ 3 have the trivial form δh = δg−h,
h = 0, . . . , g − 1.
From above, it is clear that the tautological PPZ relations for every particular r are in fact
the full set relations in H2
(
Mg,n,Q
)
RH2
(
Mg,n,Q
)
= H2
(
Mg,n,Q
)
,
which confirms Pixton’s hypothesis in this codimension.
Acknowledgements
The author is very thankful to P. Dunin-Barkowski for numerous discussions and competent
advisoring during all the project. I am also grateful to D. Zvonkine for teaching me a trick used in
genus 2 case which led to a significant simplification and for helpful remarks. I thank anonymous
referees for numerous tips. The author is partially supported by International Laboratory of
Cluster Geometry NRU HSE, RF Government grant, ag. no. 075-15-2021-608 dated 08.06.2021.
References
[1] Arbarello E., Cornalba M., Calculating cohomology groups of moduli spaces of curves via algebraic geometry,
Inst. Hautes Études Sci. Publ. Math. 88 (1998), 97–127, arXiv:math.AG/9803001.
[2] Kock J., Frobenius algebras and 2D topological quantum field theories, London Math. Soc. Stud. Texts,
Vol. 59, Cambridge University Press, Cambridge, 2004.
[3] Mumford D., Towards an enumerative geometry of the moduli space of curves, in Arithmetic and Geometry,
Vol. II, Progr. Math., Vol. 36, Birkhäuser, Boston, MA, 1983, 271–328.
[4] Pandharipande R., Cohomological field theory calculations, in Proceedings of the International Congress
of Mathematicians – Rio de Janeiro 2018. Vol. I. Plenary lectures,World Scientific Publishing, Hackensack,
NJ, 2018, 869–898, arXiv:1712.02528.
[5] Pandharipande R., Pixton A., Relations in the tautological ring of the moduli space of curves, Pure Appl.
Math. Q. 17 (2021), 717–771, arXiv:1301.4561.
[6] Pandharipande R., Pixton A., Zvonkine D., Relations on Mg,n via 3-spin structures, J. Amer. Math. Soc.
28 (2015), 279–309, arXiv:1303.1043.
[7] Pandharipande R., Pixton A., Zvonkine D., Tautological relations via r-spin structures, J. Algebraic Geom.
28 (2019), 439–496, arXiv:1607.00978.
[8] Pixton A., Conjectural relations in the tautological ring of Mg,n, arXiv:1207.1918.
[9] Polishchuk A., Witten’s top Chern class on the moduli space of higher spin curves, in Frobenius Manifolds,
Aspects Math., Vol. E36, Friedr. Vieweg & Sohn, Wiesbaden, 2004, 253–264, arXiv:math.AG/0208112.
https://doi.org/10.1007/BF02701767
https://arxiv.org/abs/math.AG/9803001
https://doi.org/10.1017/CBO9780511615443
https://doi.org/10.1007/978-1-4757-9286-7_12
https://doi.org/10.1142/9789813272880_0031
https://arxiv.org/abs/1712.02528
https://doi.org/10.4310/PAMQ.2021.v17.n2.a7
https://doi.org/10.4310/PAMQ.2021.v17.n2.a7
https://arxiv.org/abs/1301.4561
https://doi.org/10.1090/S0894-0347-2014-00808-0
https://arxiv.org/abs/1303.1043
https://doi.org/10.1090/jag/736
https://arxiv.org/abs/1607.00978
https://arxiv.org/abs/1207.1918
https://doi.org/10.1007/978-3-322-80236-1_10
https://arxiv.org/abs/math.AG/0208112
16 D. Gubarevich
[10] Polishchuk A., Vaintrob A., Algebraic construction of Witten’s top Chern class, in Advances in Algebraic
Geometry Motivated by Physics (Lowell, MA, 2000), Contemp. Math., Vol. 276, American Mathematical
Society, Providence, RI, 2001, 229–249, arXiv:math.AG/0011032.
[11] Teleman C., The structure of 2D semi-simple field theories, Invent. Math. 188 (2012), 525–588,
arXiv:0712.0160.
[12] Witten E., Algebraic geometry associated with matrix models of two-dimensional gravity, in Topological
Methods in Modern Mathematics (Stony Brook, NY, 1991), Publish or Perish, Houston, TX, 1993, 235–269.
[13] Zvonkine D., An introduction to moduli spaces of curves and their intersection theory, in Handbook of
Teichmüller Theory. Vol. III, IRMA Lect. Math. Theor. Phys., Vol. 17, European Mathematical Society
(EMS), Zürich, 2012, 667–716.
https://doi.org/10.1090/conm/276/04523
https://doi.org/10.1090/conm/276/04523
https://arxiv.org/abs/math.AG/0011032
https://doi.org/10.1007/s00222-011-0352-5
https://arxiv.org/abs/0712.0160
https://doi.org/10.4171/103-1/12
https://doi.org/10.4171/103-1/12
1 Introduction
2 Preliminaries
2.1 Tautological classes on M_{g,n}
2.2 Cohomological field theories
2.3 Reconstruction
3 Tautological relations at semisimple point
4 The case of marked genus 1 curves
5 The case of marked genus 2 curves
6 Higher genera and conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-212607 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T12:57:53Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gubarevich, Danil 2026-02-09T08:05:32Z 2024 On the Picard Group of the Moduli Space of Curves via -Spin Structures. Danil Gubarevich. SIGMA 20 (2024), 088, 16 pages 1815-0659 2020 Mathematics Subject Classification: 14H10; 14N35 arXiv:2112.10182 https://nasplib.isofts.kiev.ua/handle/123456789/212607 https://doi.org/10.3842/SIGMA.2024.088 In this paper, we obtain explicit expressions for Pandharipande-Pixton-Zvonkine relations in the second rational cohomology of ℳ¯,ₙ, and, comparing the result with Arbarello-Cornalba's theorem, we prove Pixton's conjecture in this case. The author is very thankful to P. Dunin-Barkowski for numerous discussions and competent advice throughout the project. I am also grateful to D. Zvonkine for teaching me a trick used in the genus 2 case, which led to a significant simplification, and for helpful remarks. I thank anonymous referees for numerous tips. The author is partially supported by the International Laboratory of Cluster Geometry NRU HSE, RF Government grant, etc., no. 075-15-2021-608 dated 08.06.2021. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Picard Group of the Moduli Space of Curves via -Spin Structures Article published earlier |
| spellingShingle | On the Picard Group of the Moduli Space of Curves via -Spin Structures Gubarevich, Danil |
| title | On the Picard Group of the Moduli Space of Curves via -Spin Structures |
| title_full | On the Picard Group of the Moduli Space of Curves via -Spin Structures |
| title_fullStr | On the Picard Group of the Moduli Space of Curves via -Spin Structures |
| title_full_unstemmed | On the Picard Group of the Moduli Space of Curves via -Spin Structures |
| title_short | On the Picard Group of the Moduli Space of Curves via -Spin Structures |
| title_sort | on the picard group of the moduli space of curves via -spin structures |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212607 |
| work_keys_str_mv | AT gubarevichdanil onthepicardgroupofthemodulispaceofcurvesviaspinstructures |