Supermoduli space with Ramond punctures is not projected
arXiv:2308.07957v1 [math.AG] 15 Aug 2023
Ron Donagi, Nadia Ott
Supergeometry is the study of spaces described by a Z2 -graded sheaf of functions, OX = OX,+ ⊕
OX,− whose sections obey the rule of signs,
ab = (−1)|a||b| ba.
Every superspace X determines an ordinary space M which naturally embeds into X, and one can
ask if this embedding M ⊂ X has a projection X → M . If such a projection exists, then X is
said to be projected. If, in addition, the projection X → M makes X into a vector bundle over
M , then X is said to be split. One important class of examples of superspaces are the moduli
spaces of super Riemann surfaces, known as supermoduli spaces and denoted by Mg,nS ,nR . Here g
is the genus, nS is the number of Neveu-Schwarz punctures, and nR the (always even) number of
Ramond punctures. In physics, integrals over supermoduli space calculate important quantities like
the partition function and scattering amplitude. So far, the most successful computations of these
quantities have relied upon the splitness of the associated supermoduli space; notably, D’Hoker and
Phong’s computation of the scattering amplitude in genus g = 2 [3]. However, most supermoduli
spaces are not split or projected, [6]. Specifically, the cited work demonstrates that Mg,nS ,0 is not
projected for all g ≥ 5 when nS = 0, and for all g ≥ nS + 1 ≥ 2 when nS ≥ 1. This paper’s main
result is that most supermoduli spaces with Ramond punctures Mg,0,nr are not projected.
Theorem 1. Let r > 0. The supermoduli space Mg,0,2r is not projected for all g ≥ 5r + 1 ≥ 6.
We start with the finite covering spaces of supermoduli space,
p : M̃d,ρ → M2,1,0 : (π : X̃ → X) → X
(1)
parameterizing branched covers of genus 2 super Riemann surfaces with a fixed degree d, a specified
ramification configuration ρ = (d1 , . . . , ds ), and a single branch point along the one NS puncture
on X. Our main result about M̃d,ρ is that it is not projected (Lemma 14).
Let us now consider those super Riemann surfaces X̃ forming branched covers as in (1). The
genus of X̃ can be calculated using the usual Hurwitz formula,
s
g =1+d+
3
1
1X
(di –1) = 1 + d − s
2
2
2
(2)
i=1
The ramification configuration ρ = (d1 , . . . , ds ) keeps track of the number of sheets di that come
together at the ramification points of π. In our application below an even number 2r of the s
ramification points will turn into Ramond punctures while the remaining s − 2r points will turn
into NS punctures. Furthermore, from [6] we know that X̃ has Ramond punctures corresponding
1
to those ramification points of π with even local degree. Note that by formula (2) there are always
an even number of ramification points of even local degree.
Each M̃d,ρ comes with a natural immersion into supermoduli space,
i : M̃d,ρ → Mg,0,2r : (π : X̃ → X) → X̃.
(3)
[where g is determined by (2) and r is as described above. 1 . In Lemma (15) we use this immersion
to deduce our main result about Mg,0,2r , namely that it is not projected.
The last step of Theorem 1 is the bound g ≥ 5r + 1 ≥ 6. Fix r > 0. For the sake of brevity, we
will say that the supermoduli space Mg,0,2r is realizable if g is a solution to (2) for some d and ρ.
In other words, a supermoduli space Mg,0,2r is realizable if it contains i(M̃d,ρ ) for some d, ρ. From
the above discussion, we know that a realizable supermoduli space is not projected.
Lemma 2. Mg,0,2r is realizable if and only if g ≥ 5r + 1 ≥ 6
Proof. We consider a d-sheeted cover X̃ → X of a genus 2 curve with a single branch
point over
P
which the fiber is specified by the ramification pattern ρ = (d1 , . . . , ds ) with
i di = d. The
expression in (2) has to be a non-negative integer, namely the genus of X̃. Conversely, Theorem 4
of [1] assures us that there are no further constraints: for any d, ρ such that the expression in (2)
is a non-negative integer, there is a cover X̃ → X with the specified behavior.
As long as g ≥ 5r + 1 we can therefore construct a d = g − (1 + r)-sheeted cover of X ∈ M2,1 ,
branched over the single puncture in X with di = 2 for 1 ≤ i ≤ 2r and di = 1 for i > 2r. This gives
a curve X̃ of genus g with 2r Ramond punctures and g − (5r + 1) Neveu-Schwarz punctures. (In
the minimal case, g = 5r + 1, d = 4r, and all the di = 2, so we have 2r Ramond punctures. When
g ≥ 5r + 1, we still have 2r Ramond punctures but now also a positive number of NS punctures.)
The paper is structured as follows: In Section 1, we look at various examples of split and projected supermanifolds. We also state two Corollaries, 3, 4 from [6] that allow one to construct a
large class of non-projected supermanifolds via submanifolds and branched covers. Secondly, in
Section 2, we provide definitions for supercurves and super Riemann surfaces, and describe the two
types of punctures one sees on a super Riemann surface, namely NS punctures and Ramond punctures. Furthermore, we give an account of divisors on supercurves, their connection to points, and
clarify why Ramond punctures are not considered punctures. In Section 3, we consider branched
covers π : X̃ → X of super Riemann surfaces and describe the relationship between ramification
and the types of punctures on X̃. Lastly, in Section 4, we define the morphisms in equations (1)
and (3), and provide proof for Lemmas 14 and 15. Additionally, the stackyness of M̃d,ρ is addressed
in the appendix of this paper.
Acknowledgements. During the preparation of this work, Ron Donagi was supported in part
by NSF grants DMS 2001673 and FRG 2244978, and by Simons HMS Collaboration grant #390287.
Nadia Ott was supported in part by Simons HMS Collaboration grant #390287. We thank Eric
D’Hoker whose question started us thinking about the material presented here.
1
In general, X̃ also has NS punctures, and i is the composition of M̃d,ρ → Mg,n,2r with the forgetful functor
Mg,n,2r → Mg,0,2r .
2
1
Preliminaries: Split and projected supermanifolds
−
.
Let X = (|X|, OX ) be a supermanifold and let J ⊂ OX denote the ideal sheaf generated by OX
Any supermanifold X determines an ordinary manifold Xbos := (|X|, OX /J) called its bosonic
reduction. Henceforth, we set M := Xbos .
Given an ordinary manifold M and a locally free sheaf X on M , we can always construct a
supermanifold, denoted by S(M, X ), by applying the parity reversing-functor Π to X and defining
S(M, ΠX ) := SpecM Sym• ΠX .
Example 1. The supermanifold S(M, J/J 2 ) determined by the pair (M, J/J 2 ) is the normal
bundle, NM/X , to the embedding M ⊂ X. To see this, note that J/J 2 is the conormal sheaf
associated to the embedding M ⊂ X.
We say that X is split if there exists an isomorphism φ : X ∼
= NM/X of supermanifolds such
that the restriction φ|M is equal to the identity on M . A weaker condition than being split is being
projected: The supermanifold X is said to be projected if the embedding M ⊂ X has a projection
X → M . In terms of coordinates and transition functions, a supermanifold is projected if its even
coordinates modulo nilpotents transform like coordinates on M , and it is split if, in addition, its
odd coordinates transform like sections of the normal bundle NM/X .
Example 2. All affine superspaces Am|n and all projective superspaces Pm|n are split. Specifically,
Am|n = SpecAm Sym• ΠOA⊕n
m
Pm|n = SpecPm Sym• ΠO(1)⊕n .
Example 3. Every supermanifold of dimension (m|1) is split. More generally, the first truncation
X 1 := (|X|, OX /J 2 ) of any supermanifold X is split.
In general, a projected supermanifold need not be split. However, there are some special cases
where the two notions are equivalent:
Example 4. A supermanifold of dimension (m|2) is split if and only if it is projected. More
generally, the second truncation X 2 := (|X|, OX /J 3 ) of any supermanifold X is split if and only if
projected. This is (roughly) because the transformation laws for the odd coordinates on X 2 remain
unchanged from those on the split supermanifold X 1 when “adding back” variables in J 2 /J 3 .
The truncations X 1 and X 2 are the first two terms in a finite sequence of truncations, X i :=
(|X|, OX /J i+1 ) which approximate X in the sense of the embeddings
M ⊂ X 1 ⊂ · · · X i ⊂ · · · ⊂ X.
A supermanifold X is projected or split if and only if all of its truncations X i are projected or
split. Since X 1 is always split, the first interesting case is X 2 , the second truncation. X 2 determines
a class ω2 ∈ H 1 (M, TM ⊗ J 2 /J 3 ) called the first obstruction to splitting. Indeed, the surjection
OX /J 3 → OX /J 2 = OM ⊕ J/J 3 making X 2 a square-zero extension of X 1 is an isomorphism on
the odd components, and the conormal sequence associated to (OX /J 3 )+ → OM ,
0 → J 2 /J 3 → Ω1X 2 |M → Ω1M → 0,
3
identifies X 2 with a class ω2 in Ext1 (Ω1M , J 2 /J 3 ) ∼
= H 1 (M, TM ⊗ J 2 /J 3 ) such that ω2 = 0 if and
only if 0 → J 2 /J 3 → (OX /J 3 )+ → OM → 0 is split exact. In this case, X 2 is projected, and thus
also split by example 4.
It is not difficult to find examples of supermanifolds which are not projected.
Example 5. The superconic defined by the equation z12 + z22 + z32 + θ1 θ1 in Proj C[z1 , z2 , z3 |θ1 , θ2 ]
is not projected.
Example 6. Every non-trivial odd first order deformation of a super Riemann surface supermanifold is not projected, see [6].
The following two Corollaries from [6] can be used to construct a large class of non-projected
supermanifolds.
Corollary 3 (Corollary 2.8). Let π : X̃ → X be a finite cover of supermanifolds. If ω2 (X) 6= 0,
then ω2 (X̃) 6= 0. Furthermore, X is split if and only if X̃ is split.
Corollary 4 (Corollary 2.11). Let X be a supermanifold and suppose X ′ ⊂ X is a sub-supermanifold
with bosonic reduction M ′ ⊂ M such that the normal sequence,
0 → T M → T M |M ′ → NM ′ /M → 0
is split exact. If X ′ is not projected, then X is not projected.
Example. Let us apply Corollary 4 to the superconic X ⊂ P2|2 from example 5. Since X is not projected, the normal sequence associated to the embedding Xbos ⊂ P2 is not split exact by Corollary 4.
2
2.1
Preliminaries: Supercurves, divisors on supercurves, super Riemann surfaces and their punctures
Supercurves
A supercurve S is a compact, connected supermanifold of dimension (1|1). The bosonic reduction
of a supercurve is an ordinary projective curve.
Example 7. The superprojective line P1|1 is a supercurve whose bosonic reduction is the ordinary
projective line, P1 .
A family of supercurves over a supermanifold T is a smooth, proper morphism of supermanifolds
π : S → T of relative dimension (1|1). The bosonic reduction of π : S → T is the family of ordinary
curves, πbos : Sbos → Tbos .
Example 8. An example of a family of supercurves is P1|1 × T → T , where T can be any supermanifold. Its bosonic reduction is P1 × Tbos → Tbos .
In general, we write S for a family of supercurves and make no explicit mention of the base
supermanifold. We will sometimes use the phrase “in the split case” to indicate that the base of
a family is an ordinary scheme. If the base is Spec C, then we will refer to S as a “single” supercurve.
Any topological invariant assigned to a supermanifold is determined by the bosonic reduction of
that supermanifold. In particular, the genus of a supercurve is the genus of its bosonic reduction.
4
2.2
Divisors on supercurves
Our treatment of divisors on supermanifolds follows the style of Chapter 6 in [5].
Let X be a supermanifold and let K denote the sheaf of total quotient superrings (see [5] for the
∗ , where K∗ ⊂ K is the subordinary definition). A Cartier divisor on X is a global section of K∗ /OX
sheaf of invertible elements in K. As in the ordinary case, a Cartier divisor on X can be described
∗ ).
by a collection fi ∈ Γ(Ui , K∗ ) on an open cover Ui of X such that for all i, j, fi /fj ∈ Γ(Ui ∩ Uj , OX
Effective and prime divisors A Cartier divisor is said to be effective if fi ∈ Γ(Ui , OUi ,+ ) for
all i. Every effective Cartier divisor on X has an associated closed sub-supermanifold P ⊂ X of
codimension 1|0 defined by the ideal sheaf IP ⊂ OX locally generated by the fi . We say that
P ⊂ X is a prime divisor if its bosonic reduction P = Pbos is a prime divisor on M = Xbos .
Example 9. Let (z, θ) be a choice of local coordinates on a supercurve S over a base with one odd
coordinate η. The divisor defined by the equation z − z0 − θη = 0 is a prime divisor on S.
∗ = O∗
∗
∗
Lemma 5. On any supermanifold X, OX
X,+ and K = K+ . In particular, a one-to-one
correspondence exists between Cartier divisors on X and Cartier divisors on Xev .
Proof. A section f ∈ OX is invertible if and only if its image f in OX /J is invertible. Therefore,
∗ , then f 6= 0 and, by homogeneity, f is even since f 6= 0 is even.
if f ∈ OX
Corollary 6. Let S be a family of supercurves over an ordinary scheme. Then there is a one-to-one
correspondence between Cartier divisors on S and Cartier divisors on the bosonic reduction C of
S. In particular, every Cartier divisor on C extends uniquely to a Cartier divisor on S.
Proof. Over an ordinary scheme, Sbos = Sev since OS,+ = OS /J. Now apply Lemma 5.
2.2.1
Points and divisors
A marked point on a family of supercurves π : S → T is the image of a section s : T → S. When
T = Spec C, marked points are the same as closed points.
Example 10. Consider the (trivial) family π : A1|1 × A0|1 → A0|1 , and choose coordinates
A1|1 ∼
= Spec C[z, θ], A0|1 ∼
= Spec C[η]
where the isomorphism on the right is for both copies of A0|1 . A marked point s : A0|1 → A1|1 × A0|1
is a morphism of superrings
C[η][z, θ] → C[η] : η 7→ η, z 7→ a, θ 7→ bη
for some a, b ∈ C.
Lemma 7. Let S be a family of supercurves over a purely bosonic scheme. Then there is a oneto-one correspondence between prime divisors on S and marked points on S.
5
Proof. Suppose T is purely bosonic. Then any marked point s : T → S on S must factor through
C = Sbos , and so we have a one-to-one correspondence between marked points on C and marked
points on S. On the other hand, marked points on C are the same as prime divisors on C, and
every prime divisor on C extends uniquely to S by Corollary 6.
However, there is no duality between prime divisors and marked points on a supercurve over
a general superscheme. In the next section we will show that the duality is restored, regardless of
the base superscheme, in the presence of a superconformal structure.
2.3
Super Riemann surfaces
A super Riemann surface X = (S, D) is a supercurve S equipped with a rank (0|1) distribution
D ⊂ TS which is maximally non-integrable in the sense of the following short exact sequence,
0 → D → TS/T → D ⊗2 → 0.
(4)
An isomorphism of super Riemann surfaces φ : X → X ′ is an isomorphism φ : S → S ′ of supercurves such that φ∗ D ′ = D.
The distribution D is called a superconformal structure, and there always exists a special choice
of local coordinates (z, θ) on S such that D is locally generated by the odd vector field,
D=
∂
∂
+θ .
∂θ
∂z
(5)
Duality. In the last section we define a duality between points and divisors on a supercurve over
a purely bosonic scheme, and noted that this duality does not hold over general superschemes.
It turns out that on super Riemann surfaces the defined duality holds in general, i.e., with no
condition on the base superscheme. This is because the maximal non-integrability condition on D
ensures that there exists a unique point on each divisor tangent to D.
More precisely, let p = (z0 , θ0 ) be a point on S, and let vp := D|p be the restriction of the local
generator (5) of D to the point p. The pair (p, vp ) determines a prime divisor P on S which is
locally defined by the equation z − z0 − θθ0 = 0. By construction, D is tangent to P at p ∈ P , and
the maximal non-integrability condition on D ensures that p is the unique point on P tangent to D.
Superconformal vector fields. Let X = (S, D) be a super Riemann surface, and let I be an OS module. A superderivation δ : OS → I is a k-linear map, satisfying the super Leibniz rule
δ(ab) = δ(a)b + (−1)|δ||a| aδ(b), a, b ∈ I
(6)
We can organize the set of derivations into a sheaf of super vector spaces on S,
Derk (OS , I) = Derk (OS , I) ⊕ Derk (OS , ΠI).
where Derk (OS , I) is the space of all grading-preserving derivations, and Derk (OS , ΠI) is the space
of grading-reversing derivations. The sheaf Derk (OS , OS ) is the tangent sheaf TS on S, and TS ⊗I =
Derk (OS , I) where
(TS ⊗ I)+ = Derk (OS , I), and (TS ⊗ I)− = Derk (OS , ΠI).
6
On the super Riemann surface X, we can consider the subsheaf Derscf
k (OX , OX ) ⊂ Derk (OX , OX ) =
TX of derivations preserving the superconformal structure D. A derivation V ∈ TX is in Derscf
k (OX , OX )
scf
if and only if [V, D] ∈ D. Henceforth we set AX := Derk (OX , OX ). The maximal non-integrability
condition on D implies that V ∈ TX /D, and
AX ∼
= TX /D ∼
= D2.
(7)
∼
If I is an OX -module, then Derscf
k (OX , I) = AX ⊗ I, where
(AX ⊗ I)+ = Der(OX , I), and (AX ⊗ I)− = Der(OX , ΠI).
Here we are using the fact that AX inherits the structure of a sheaf of OX -modules from D 2 , and
so the tensor product AX ⊗OX − is well-defined.
Lemma 8. I ∼
= OX ⊕ ΠOX , then
Derk (OX , I) ∼
= TX
= Derk (OX , OX ) ∼
scf
scf
∼ AX
∼ Der (OX , OX ) =
Der (OX , I) =
k
k
Proof.
Derk (OX , I) ∼
= (TX ⊗OX (OX ⊕ ΠOX ))+ = (TX ⊕ ΠTX )+
= TX+ ⊕ (ΠTX )+ = TX+ ⊕ TX−
= TX
First order deformations. A first order deformation of a super Riemann surface X is a family
of super Riemann surfaces f : X ′ → Spec D over the Spec of the super dual numbers D =
C[t, η]/(t2 , ηt), |t| = 0, |η| = 1 making the following diagram commute,
X′
X
f0
(8)
f
Spec C
Spec D.
We say that a first order deformation X ′ of X is trivial if there exists an isomorphism X ′ ∼
=
X × Spec D restricting to the identity on X.
Lemma 9. The set of isomorphism classes of first order deformations of X is equal to H 1 (X, AX ).
Proof. The diagram (8) induces the following morphism of short exact sequences on structure
sheaves,
0
I
·t,·η
OX ′
f
0
J = (t, η)
D
7
OX
0
(9)
f0
C
0
where I := ker(OX ′ → OX ), J := ker(D → C). Any first order deformation of is locally on X
∼
isomorphic to the trivial deformation. Let Ui be a trivializing cover for X, and let φi : X ′ |i −→
Ui × Spec D be the respective isomorphism. Then Φij := φij /φji is an automorphism of the trivial
deformation Uij × Spec D. Giving an automorphism of the trivial deformation is equivalent to
giving an even derivation δ : Oij ′ → i∗ I preserving the superconformal structure on X, so δ is
a section of Derscf (i∗ OX ′ , I)(Uij ) = Derscf (OX , I)(Uij ). Here i denotes the top horizontal arrow
X → X ′ in (8). Furthermore, since I ∼
= f0∗ (J) = OX ⊕ ΠOX , we have an isomorphism
Derscf (OX , I) ∼
= AX ,
by Lemma 8, and so δ ∈ AX (Uij ). The set of isomorphism classes of first order deformations
of X is therefore a torsor for H 1 (X, AX ), and there is a canonical isomorphism from the set of
isomorphism classes of first order deformations of X to H 1 (X, AX ) sending the trivial deformation
of X to 0 ∈ H 1 (X, AX ).
2.3.1
Super Riemann surfaces with Neveu-Schwarz (NS) punctures
A Neveu-Schwarz (NS) puncture on a super Riemann surface X is just a choice of marked point,
s : T → X. A super Riemann surface with nS NS punctures is the data (π : X → T, D, s1 , . . . , sn :
T → X). An isomorphism of super Riemann surfaces with NS punctures is an isomorphism
′ )=P
φ : X → X ′ of super Riemann surfaces such that φ−1 (PN
NS.
S
Duality. The duality between points and divisors on super Riemann surfaces identifies each NS
puncture with a prime divisor PN S ⊂ X of degree nS .
Superconformal vector fields. The sheaf of superconformal derivations on a super Riemann surface
with NS punctures is the subsheaf of AX whose sections vanish along PN S . Specifically, if X =
(S, D) is a super Riemann surface and PN S is the NS divisor, then the sheaf of superconformal
vector fields is the subsheaf AX (−PN S ) ⊂ AX , where
and
scf
∼
AX (−PN S ) = Derscf
k (OX , OX ) ⊗ O(−PN S ) = Derk (OX , O(−PN S ))
(10)
AX (−PN S ) ∼
= (TX /D)(−PN S ) ∼
= D 2 (−PN S ).
(11)
First order deformations. A first order deformation of a super Riemann surface X = (S, D) with
′
NS punctures PN S is a first order deformation X ′ of X and divisor PN
S whose pull back to X is
equal to PN S . To prove the next lemma, replace AX in the proof of Lemma 8 with AX (−PN S ).
Lemma 10. The set of isomorphism classes of first order deformations of X = (S, D, PN S ) is
equal to H 1 (X, AX (−PN S )).
2.3.2
Super Riemann surfaces with Ramond punctures.
Let π : S → T be a family of supercurves. A Ramond divisor on S is prime divisor PR ⊂ S for
which there exists a rank 0|1 distribution D ⊂ TS/T fitting into the short exact sequence
0 → D → TS/T → D 2 (PR ) → 0.
8
The prime components Pi ⊂ PR are called Ramond punctures. A super Riemann surface with nR
Ramond punctures is the data (π : S → T, PR , D) where PR is a Ramond divisor of degree nR .
The number of Ramond punctures is always even, and so we set nR = 2r.
We can always find local coordinates (z, θ) on S such that PR is locally defined by the equation
z = 0 and D is locally generated by the odd vector field,
D=
∂
∂
+ zθ .
∂θ
∂z
(12)
Duality. The duality between points and divisors we saw on super Riemann surfaces, and super
Riemann surfaces with NS punctures, fails in the presence of Ramond punctures . Consequently,
there is no canonical way to associate a set of marked points to the Ramond divisor PR . This is in
stark contrast to what we saw with NS punctures, which start out as marked points, but to which
we can canonically associate a divisor PN S using duality. Let us explain this in more detail. First,
note that D is non-integrable away from PR but integrable along PR since
∼
[, , ] : D 2 −→ (TS/T /D)(−PR ) = 0 ∈ D,
where [ , ] is the supercommutator. Recall from ordinary geometry that an integrable distribution
of rank k determines a submanifold of dimension k which is tangent to the distribution at every
point. Since D is integrable along PR , it determines a submanifold of PR of dimension 0|1. Since
PR is a divisor, and thus of dimension 0|1, this submanifold must be equal to PR , and so that PR
is tangent to D at every point. In particular, there does not exist a unique point on PR which we
can use to define a duality. In this sense, Ramond punctures are not marked points!
Superconformal vector fields. The subsheaf Derscf
k (OX , OX ) ⊂ Derk (OX , OX ) = TX of derivations
preserving the superconformal structure D 2 on a super Riemann surface with Ramond punctures
is defined by the following condition: Set AX := Derscf
k (OX , OX ). A derivation V ∈ TX is a section
of AX if and only if [V, D] ∈ D if and only if V ∈ (TX /D))(−PR ), where the last condition follows
from the maximal non-integrability condition on D. Thus,
If I is an OX -module, then
AX ∼
= TX /D(−PR ) ∼
= D2.
(13)
∼
Derscf
k (OX , I) = AX ⊗ I,
(14)
where
(AX ⊗ I)+ = Derscf (OX , I), and (AX ⊗ I)− = Derscf (OX , ΠI).
First order deformations. A first order deformation of X = (S, PR , D) is a super Riemann surface
′ , D ′ ) over Spec D making the diagram in (8)
with Ramond punctures X ′ = (f : S ′ → Spec D, PR
′
commute, and such that the restriction of PR to X is equal to PR .
Lemma 11. The set of isomorphism classes of first order deformations of X = (S, PR , D) is equal
to H 1 (X, AX ).
2
Any derivation that preserves D also preserves the Ramond divisor, PR , since PR is part of the structure of D.
9
3
Branched Covers
In this section, we define branched covers of supercurves and super Riemann surfaces.
3.1
Branched covers of supercurves
Let S be a supercurve, let B ⊂ S be a closed subscheme of codimension (1|0), and let U = S − B.
A branched cover of S with branch locus B is a finite, surjective morphism π : S̃ → S such that the
local morphisms πp# : OS,f (p) → OS̃,p are étale for all p ∈ U . The cover π may be ramified along
a closed subscheme R ⊂ π −1 (B) ⊂ S̃ of codimension 1|0 called the ramification divisor of π. The
bosonic reduction πbos : C̃ → C of π : S̃ → S is ramified along a divisor R = Rbos with branch
locus B = Bbos .
∼
An isomorphism of branched covers is a pair of isomorphisms of supercurves φ : S −→ S ′ and
φ̃ : S̃ → S̃ ′ such that the following diagram commutes,
S̃
φ̃
π
S
S̃ ′
π′
φ
(15)
S ′.
and such that φ(B) = B ′ and φ̃(R) = R′ .
Local description. Let π : S̃ → S be a branched cover, and let P ⊂ R be a connected component
of the ramification divisor of π. Near P, we can find (local) coordinates (w, θ ′ ) and (z, θ) on S̃ and
S, respectively, such that B ⊂ S is defined by the equation z = 0 and π is locally of the form
z = wk , θ = wℓ θ ′
(16)
for some k > 0, ℓ ≥ 0.
Branched covers over bosonic schemes. Let π : S̃ → S be a branched cover over a purely bosonic
scheme. Then S and S̃ are split, and there exist line bundles L and L̃ on C = Sbos and C̃ = S̃bos ,
respectively, such that
OS = OC ⊕ ΠL, OS̃ = OC̃ ⊕ ΠL̃,
and the even and odd components of π # ,
π0# = πbos : OC → (π0 )∗ OC̃ , Ππ1# : L → (π0 )∗ L̃
are, respectively, a branched cover of the curve C, and a morphism of line bundles on C. From
this description, we see that, in the split case, giving a branched cover of S is equivalent to giving
a branched cover π0 : C̃ → C of C, a line bundle L̃ on C̃, and a morphism Ππ1⊥ : π0∗ L → L̃ of
line bundles on C̃. Note that the adjunction map sends Ππ1⊥ to a morphism Ππ1# : L → (π0 )∗ L̃ of
OC -modules.
Semi-étale cover. We continue to work over a purely bosonic scheme. Using the above notation,
we say that the branched cover π : S̃ → S determined by (π0 : C̃ → C, L̃, π1⊥ ) is semi-étale if
∼
π1⊥ : π0∗ L −→ L̃ is an isomorphism. If π is semi-étale then ℓ = 0 in (12).
10
Lemma 12. Let S → T be a family of supercurves over a purely bosonic scheme T , and fix a prime
divisor B ⊂ S. There is a one-to-one correspondence between branched covers of C = Sbos with
branch locus B = Bbos and semi-étale branched covers of S with branch locus B.
Proof. Let π0 : C̃ → C be a branched cover of C = Sbos and let B ⊂ C and R ⊂ C̃ denote its
branch and ramification divisor. First, note that since S is a supercurve over a bosonic scheme, B
extends uniquely to a divisor, B, on S, by Lemma 6. To prove the lemma, we will show that the
morphism π : S ×C C̃ → S in the cartesian diagram
π
S ×C C̃
S
(17)
C̃
π0
C
is the unique extension, up to isomorphism, of π0 to a semi-étale cover of S branched along B. It
suffices to show that π is the branched cover of S determined by the data (π0 , (π0 )∗ L, idπ0∗ L ). That
this is the case, follows from
OS×C C̃ = OC̃ ⊕ Ππ0∗ L.
We define the local degree of π at a connected component of the ramification divisor P ⊂ R to
be the local degree of πbos at P ⊂ R, i.e., the number of sheets of πbos that come together
P at the
ramification point P . These numbers are arranged into a sequence ρ = (d1 , . . . , ds ) where di = d,
assuming the ramification points are ordered. According to Theorem 4 in [1] a degree d branched
cover π : C̃ → C of ordinary curves, with ramification configuration ρ, exists if and only if g(C) ≥ 2
and the sum of the local degrees di minus 1 is congruent to 0 modulo 2, i.e.,
X
di − 1 ≡ 0 (mod 2).
From Lemma 12 we know that, at least in the split case, every branched cover of C = Sbos lifts
uniquely to a branched cover of S. In particular, the cited existence result can also be used for
supercurves.
3.1.1
Branched covers of super Riemann surfaces
Let X = (S, D) be a super Riemann surface. We define a branched cover of X to be a branched
cover π : S̃ → S of the supercurve S such that π ∗ (D) is a superconformal structure on S̃, possibly
with Ramond punctures PR . An isomorphism of branched covers of super Riemann surfaces is an
isomorphism of branched covers of supercurves with the additional condition that the isomorphisms
respect all superconformal structures.
Local description. Let π : X̃ → X be a branched cover, and let P ⊂ R be a connected component
of the ramification divisor of π of even local degree. Then P is a Ramond puncture on X̃, and near
P, we can find (local) coordinates (w, θ ′ ) and (z, θ) on X̃ and X, respectively, such that B ⊂ X is
defined by the equation z = 0 and π is locally of the form
z = w2ℓ , θ = wℓ θ ′
11
(18)
for some ℓ ≥ 1. If P is a connected component of odd local degree, then it is a NS puncture on X̃,
and π is locally of the form
z = w2ℓ+1 , θ = wℓ θ ′
(19)
for some ℓ ≥ 1.
Blow up construction. Given a super Riemann surface X = (S, D), and a branched cover
π : S̃ → S, locally of the form z = wℓ , θ = θ ′ , one may be tempted to think π : X̃ → X, where
X̃ = (S̃, π ∗ (D)), is a branched cover of X. However, this is never the case (unless π is unramified).
The problem is that π ∗ (D) will always have parabolic degenerations along the ramification locus of
π. In [6], it is explained how these parabolic degenerations can be resolved in a way that recovers a
superconformal structure on a certain supercurve canonically associated to S̃. The idea is to blow
up S̃ along the distinguished closed points on each ramification divisor. The result is a supercurve
Bl(S̃) with the same bosonic reduction C̃ as S̃, and with a natural projection q : Bl(S̃) → S̃ such
that q ∗ (π ∗ D) is a superconformal structure on Bl(S̃) with Ramond punctures along the exceptional
divisors on Bl(S̃) corresponding to the pullbacks by q of ramification points of even local degree.
We recall that the blow up of an ordinary (smooth) curve at a point on that curve is just another
copy of the curve. In particular, the bosonic reduction of Bl(S̃) is the ordinary curve C̃ and the
composition q ◦ π reduces to πbos : C̃ → C.
Lemma 12 has the following corollary:
Corollary 13. Let X = (S, D) be a family of super Riemann surfaces over a bosonic scheme T ,
and fix a prime divisor B ⊂ S. Then there is a one-to-one correspondence between branched covers
of C = Sbos with branch locus B = Bbos and branched covers of X with branch locus B.
Proof. A branched cover π : X̃ = (S̃, PR , D̃) → S = (X, D) is semi-étale because (w, θ ′ := θ/wl )
on a supercurve, and one can check in local coordinates that the generator dz − θdθ of D −2 on X
is mapped to dw − wdθ ′ dθ ′ generating D̃ −2 (−PR ). Now apply Lemma 12.
4
Moduli of branched covers
Let M̃d,ρ : sSch → Groupoid denote the functor sending a superscheme T to the groupoid


 Objects: Degree d branched covers π : X̃ → X of all X ∈ M2,1 (T ),
M̃d,ρ (T ) =
with specified ramification configuration ρ and branched along the one NS puncture on X.


Morphisms: Isomorphisms of branched covers of super Riemann surfaces.
The following examples of moduli functors may be familiar:
Moduli of Curves. Mg,n sends an ordinary scheme T to the groupoid of families of genus g curves
with n marked points over T .
12
Moduli of Spin Curves. SMg,nS ,nR sends an ordinary scheme T to the groupoid of families
genus g spin curves with nS marked points (NS punctures) and nR Ramond punctures over T
Supermoduli space. Mg,nS ,nR sends a superscheme T to the groupoid of families of genus g super
Riemann surfaces with nS marked points (NS punctures) and nR Ramond punctures over T .
The bosonic reduction of any functor F : sSch → Groupoid is its restriction to the subcategory
Sch ⊂ sSch of ordinary schemes, i.e., Fbos : Sch → Groupoid.
The bosonic reduction of Mg,nS ,nR is SMg,nS ,nR . This follows immediately from the one-to-one
correspondence between families of super Riemann surfaces over ordinary schemes and families of
spin curves over the same ordinary scheme. To work with the bosonic reduction of M̃d,ρ , we use
the following cartesian diagram, which is the moduli version of Corollary 13,
(M̃d,ρ )bos
pbos
SM2,1
(20)
M̃d,ρ
M2,1 ,
where SM2,1 → M2,1 forgets the spin structure and makes SM2,1 into a finite, étale cover of M2,1 ,
pbos is the bosonic reduction of the morphism p : M̃d,ρ → M2,1 that sends π : X̃ → X to X ∈ M2,1
and (φ̃, φ) to φ, and M̃d,ρ is the moduli of branched covers of M2,1 and M̃d,ρ → M2,1 is the obvious
projection.
The moduli functors Mg,n , SMg,nS ,nR , Mg,nS ,nR , and M̃d,ρ each have a representable finite
étale cover by a manifold, or supermanifold in the case Mg,nS ,nR and M̃d,ρ . The existence of such a
cover means that these functors are Deligne-Mumford stacks, or superstacks in the case of Mg,nS .nR
and M̃d,ρ . We prove that M̃d,ρ is Deligne-Mumford in Theorem 21 in the appendix of this paper.
Henceforth, we will always confuse a Deligne-Mumford stack, or superstack, with its étale cover,
and all morphisms between Deligne-Mumford superstacks with the morphisms induced on their
respective étale covers.
The tangent set to a general functor F : sSch → Groupid at a point f ∈ F (Spec C) is the set
of isomorphism classes in the groupoid F/f (Spec D), i.e., isomorphism classes of first order deformations of f . where F/f is the groupoid of objects in F which pullback to f on Spec C, and where
D denotes the super dual numbers. The set of isomorphism classes of first order deformations
of objects in M̃d,ρ , Mg,n,2r , and M2,1 , are described in Theorem 18, Lemma 11, and Lemma 4,
respectively.
The rest of this section deals with proving the following two lemmas.
Lemma 14. The morphism p : M̃d,ρ → M2,1 makes M̃d,ρ into a finite cover of M2,1 . In particular,
M̃d,ρ is not projected by Corollary 3.
We let i : M̃d,ρ → Mg,n,2r → Mg,0,2r , where Mg,n,2r → Mg,0,2r forgets the NS punctures, denote
the morphism sending π : X̃ → X to X̃ ∈ Mg,0,2r . Note we are using the same notation X̃ for the
super Riemann surface on which the NS punctures have been forgotten.
13
Lemma 15. The morphism i : M̃d,ρ → Mg,0,2r is an immersion of supermanifolds. Furthermore,
the normal sequence associated to the immersion ibos : (M̃d,ρ )bos → SMg,0,2r is split exact, and thus
Mg,0,2r is not projected by Corollary 4.
Assuming the proof of Lemmas 14 and 15, the proof of Theorem 1 goes as follows:
Proof of main result. The possible values of g and 2r for which there exist a branched covering
X̃ ∈ Mg,0,2r for some X ∈ M2,1 are described in Lemma 2. For this range of values, the associated
supermoduli spaces Mg,0,2r are all not projected by Lemma 15.
Proof of Lemma 14. Recall that a morphisms of supermanifolds is finite if and only if its bosonic
reduction is a finite morphism of manifolds. The bosonic reduction of p, denoted as pbos , is the top
horizontal arrow in diagram (20). It suffices to show that pbos is finite. From diagram 20 we can
see that pbos is the base change of the finite map M̃d,ρ → M2,1 , and is therefore itself finite.
Proof of Lemma 15: The morphism
dp|π : (T M̃d,ρ )|π → H 1 (X, AX )
is an isomorphism by Theorem 21. Given this isomorphism , the differential of i at π is
di|π : H 1 (X, AX ) → H 1 (X̃, AX̃ ).
The map di is injective if and only if the pre-image of the trivial deformation X̃ × Spec D in
H 1 (X̃, AX̃ ) is isomorphic to the trivial deformation of X × Spec D. Suppose di(π ′ ) = X̃ × Spec D,
then again by Corollary 3 we must have X ′ ∼
= X × Spec D.
For the second statement of the lemma, we need to show that the normal sequence associated
to ibos ,
0 → T (M̃d,ρ )bos → i∗bos T SMg,0,2r → N → 0
is split exact. To show this, we can repeat the proof of Proposition 5.3 in [6].
5
Appendix
In this section, we prove (Theorem 21) that M̃d,ρ is a Deligne-Mumford superstack. We will first
show that M̃d,ρ satisfies the conditions of the super Artin theorem, Theorem 16, and is, therefore, an
algebraic superstack. Once we have established that M̃d,ρ is algebraic, we can apply Theorem 8.3.3
in [7]: An algebraic stack X is Deligne-Mumford if and only if the diagonal map ∆ : X → X × X
is formally unramified. The condition that ∆ be formally unramified is equivalent to the condition
that the automorphism group Autx for x ∈ X (Spec C) is a reduced finite group scheme over
Spec C. We will show this holds for π ∈ M̃d,ρ (Spec C), concluding that M̃d,ρ is a Deligne-Mumford
superstack.
14
5.1
Preliminaries
In this section, we define the various functors, e.g., Dv (M ), Obv (M ), that appear in the statement
of the super Artin theorem.
Extensions of objects. Let X : sAffSchop → Groupoid be a superstack over the category of affine
superschemes, and let A0 be a superalgebra. An extension of an object v ∈ X (Spec A0 ) is a pair
(f : B → A0 , v ′ ) where f is a morphism in sAffSch and v ′ is an object in X (Spec B) such that
v ′ ×Spec B Spec A0 ∼
= v. We will define two categories, denoted by Def v (M ) and Dv (M ), that keep
track of certain classes of extensions of v.
Extensions of superalgebras. An (infinitesimal) extension of a superalgebra A0 by a coherent A0 module M is a superalgebra A fitting into the short exact sequence
0 → M → A → A0 → 0.
We will henceforth write extension when we mean infinitesimal extensions. Note that M inherits a
multiplication M · M from A, and the A0 -module structure on M forces M · M = 0. We say that
an extension A is trivial if A ∼
= A0 ⊕ M . An important example of a trivial extension is the super
dual numbers, C[t, η]/(t2 , ηt), |t| = 0, |η| = 1 in Ext(C, C ⊕ ΠC).
We let Ext(A0 , M ) denote the set of isomorphism classes of extensions of A0 by M , and let
Coh(A0 ) denote the category of all coherent modules over A0 . Ext(A0 , M ) is functorial in M ;
that is, if φ : M → M ′ is a morphism of A0 -modules, then there exists morphism Ext(A0 , M ) →
Ext(A0 , M ′ ) sending an extension A to its pushout φ∗ A in Ext(A0 , M ′ ). In other words, Ext(A0 , −)
is a category fibered in Sets over Coh(A0 ).
Extension functors. Fix v ∈ X (Spec A0 ), and M ∈ Coh(A0 ). The following functor keeps track of
the extensions of v to the Spec of all extensions of A0 by all coherent A0 -modules:
Def v (M ) : Ext(A0 , M ) → Set
f : A → A0 7→ Def v (M )(A) = {isomorphism classes of extensions of v over Spec A}
Def v (M ) is functorial in M ; that is, if φ : M → M ′ is a morphism of A0 -modules, then there
exist a morphism Devv (M ) → Def v (M ′ ) sending an extension v ′ ∈ Def v (M )(A) to its pushout
φ∗ v ′ ∈ Def v (M ′ )(φ∗ A). We let Defv (M) denote the sheafification of the presheaf Spec A0 ⊃ U 7→
Def v|U (M|U ) on Spec A0 , where M is the coherent sheaf of OSpec A0 -modules associated to M .
The following functor keeps track of all isomorphism classes of extensions of v to the Spec of
all trivial extensions of A0 by all coherent A0 -modules:
Dv : Coh(A0 ) → Set
M 7→ Dv (M ) = {isomorphism classes of extensions of v to Spec(A0 ⊕ M )}.
where A0 ⊕ M is the trivial extension of A0 by M . We we write Dv (M) for the sheafification of
the presheaf Spec A0 ⊃ U 7→ Dv|U (M|U ) on Spec A0 .
Obstructions. There is a natural morphism, functorial in M , φ : Def v (M ) → Ext(A0 , M ) sending
an extension (f : A → A0 , v ′ ) to (f : A → A0 ) ∈ Ext(A0 , M ). The extensions of A0 by M for which
15
there exists an obstruction to lifting v are parameterized by,
Obv (M ) := Ext(A0 , M )/φ(Def v (M )).
This definition was introduced in [2]. Everything we have said in this section holds equally well for
the sheaves Ext(A0 , M), Defv (M) and Obv (M).
Theorem 16 (Super Artin Theorem, [9]). Let X be a limit preserving superstack and assume that
Schlessinger’s conditions hold. (These conditions guarantee that the tangent set to X is a finite
dimensional super vector space). Then X is an algebraic superstack locally of finite type if and only
if
(A1). The diagonal morphism ∆ : X → X × X is represented by an algebraic superspace locally of
finite type.
(A2). Effectivity: If R is a complete, local, Noetherian superalgebra, then the natural map
X (Spec R) → lim X (Spec R/mnR )
←−
is an equivalence of categories.
(A3). Coherence of Ob: For every superscheme T , object v ∈ X (T ) and coherent OTred -module M,
Obv (M) is a coherent OT -module.
(A4). Constructibility: Let v, T be as above and suppose that T ′ ⊆ T is an irreducible reduced
subspace of T . Then there is a Zariski open dense subset V ′ ⊆ T ′ such that for each closed
point t ∈ V ′ , the canonical morphisms
(a) Dv (OT ′ )⊗k(t) → Dv (OT ′ ⊗k(t))t and Dv (ΠOT ′ )⊗k(t) → Dv (ΠOT ′ ⊗k(t))t are bijective,
and
(b) Obv (OT ′ ) ⊗ k(t) → Obv (OT ′ ⊗ k(t))t and Obv (ΠOT ′ ) ⊗ k(t) → Obv (ΠOT ′ ⊗ k(t))t are
injective.
(A5). The sheaf D is compatible with completion: If T0 is a reduced finite type scheme, a0 ∈ X (T0 ),
t ∈ T0 and M is a coherent OT0 -module, then
∼
bT ,t −→
Da0 (M) ⊗OT0 O
lim Da0 (M/mn M)t
0
←−
(A6). The sheaf D is compatible with étale localization: If e : S0 → T0 is an étale morphism of
reduced schemes, with e∗ (a0 ) = b0 , and M is a coherent OT0 -module, then
Db0 (M ⊗OT0 OS0 ) ∼
= Da0 (M) ⊗OT0 OS0
5.2
Deformation theory of branched covers.
In this section, we will describe the deformation theory of the moduli superstack M̃d,ρ of branched
covers of super Riemann surfaces.
By the deformation theory of a (reasonable) superstack X , we mean the following three objects
associated with a fixed, but arbitrary, object v ∈ X (Spec C): The group of infinitesimal automorphisms of v × Spec D, the set of isomorphism classes of first order deformations of v, and the
obstructions to lifting v. The group of infinitesimal automorphisms is the tangent space of automorphism group Aut(v) at idv , and the set of isomorphism classes of first order deformations of v
is the tangent set to X at v. We do not know a tangent description for the obstructions to lifting.
16
Example 11. Let X ∈ Mg,n,2r (Spec C). The group infinitesimal automorphisms Aut/X (X ×
Spec D) of X ′ is equal to H 0 (X, AX ), the set of isomorphism classes of first order deformations of
X is equal to H 1 (X, AX ), and the obstructions to lifting X are parameterized by H 2 (X, AX ).
First order deformations. Let D denote the super dual numbers , C[t, η]/(t2 , ηt), |t| = 0, |η| = 1,
and let (π : X̃ → X) ∈ M̃d,ρ (Spec C). The tangent set to M̃d,ρ at π is the set Def π := Def π (Spec D)
of isomorphism classes of first order deformations of π, that is triples (X̃ ′ , X ′ , π ′ ) where X̃ ′ , X ′ are
first order deformations of X̃, X, respectively, and π ′ : X̃ ′ → X ′ is a morphism making the following
diagram commute
X̃ ′
X̃
π′
π
(21)
X ′.
X
We say that (X̃ ′ , X ′ , π ′ ) is trivial if there exists an isomorphism of branched covers
∼
(X̃ ′ , X ′ , π ′ ) −→ (X̃ × Spec D, X × Spec D, π × idD ).
Infinitesimal automorphisms. The restriction of an automorphism of a branched cover φ : π ×
∼
Spec D −→ π × Spec D to Spec C, denoted by φ|C , is an automorphism of π : X̃ → X, and we call
φ an infinitesimal automorphism if φ|C ∈ Aut(π) is equal to the identity morphism idπ ∈ Aut(π).
The group of infinitesimal automorphisms is, denoted by InfAut(π), is the tangent space of Aut(π)
at idπ ∈ Aut(π).
Lemma 17. InfAut(π) = {id}. In particular, there are no non-trivial infinitesimal automorphisms.
Proof. Aut(π) ⊂ Aut(X) × Aut(X̃) and Aut(X), Aut(X̃) are finite for g > 1.
Obstructions. There are no obstructions to lifting π by Proposition 3.3 in [6], appropriately adapted
to the case that π has a non-zero number of ramification points of even local degree.
The main result of this section is the following theorem which identifies the set of isomorphism
classes of first order deformation of a branched cover π : X̃ → X with the super vector space
H 1 (X, AX ).
Theorem 18. Def π = H 1 (X, AX ). In particular, the tangent set to M̃d,ρ at π naturally admits
the structure of a super vector space of dimension 4|3.
The proof of Theorem 18 occupies the rest of this section.
Variants on Def π . Let Def π (X ′ ) denote the set of all isomorphism classes of first order deformations
of π with X ′ fixed. The elements of Def π (X ′ ) are pairs (X̃ ′ , π ′ ) where π ′ : X̃ ′ → X ′ is a morphism
making diagram (21) commute. Similarly, let Def π (X̃ ′ , X ′ ) denote the set of all isomorphism classes
first order deformations of π with X̃ ′ and X ′ fixed. The elements of Def π (X̃ ′ , X ′ ) are morphisms
π ′ : X̃ ′ → X ′ making the diagram (21) commute.
17
Let (π : X̃ → X) ∈ M̃d,ρ (Spec C), let PN S denote the one NS puncture on X, and recall that
PN S is equal to the branch locus of π. Set F := π −1 (PN S ) ⊂ X̃, and recall that the ramification
divisor R ⊂ F , and recall that R = P̃R + P̃N S , where P̃R is the divisor of ramification divisors of
even local degree and corresponding to Ramond punctures, while P̃N S is the divisor of ramification
divisors (or, by duality, ramification points), of odd local degree and corresponding to NS punctures.
The divisors (or, by duality, points) in the complement F − R also correspond to NS punctures.
In fact, in the examples we use for proving the main theorem, PN S is empty because none of the
local degrees is > 2, so the only NS points we get are in F − R.
Lemma 19. There is a one-to-one correspondence between Def π (X̃ ′ , X ′ ) and H 0 (X̃, AX̃ (−F ),
where π −1 (PN S ) is the pre-image of the one NS puncture PN S on X. In particular, since
H 0 (X̃, AX̃ (−F )) = 0,
there is at most one map π ′ : X̃ ′ → X ′ making the diagram (21) commute for any fixed pair of first
order deformations X̃ ′ , X ′ of X̃, X, respectively.
π
Proof. Let ψ denote the composition X̃ → X → X ′ in diagram (21) and set I = ker(OX̃ ′ → OX̃ ).
By flatness, I ∼
= f0∗ (J) where J = ker(D → C) and f0 : X̃ → Spec C is the structure morphism. In
particular, I is generated by t and η as a OX̃ -module, and therefore I ∼
= OX̃ ⊕ ΠOX̃ .
There is a one-to-one correspondence between the set of arrows π ′ filling in the dotted arrow in
the below diagram, i.e., the set of all lifts of ψ, and the set Derπ (X̃ ′ , X ′ ). :
X̃ ′
X̃
ψ
(22)
π′
X′
To prove the lemma it therefore suffices to show that there is bijection from the set of all lifts of ψ
to H 0 (X̃, AX̃ (−PR ))
Since X ′ is smooth over Spec D, there always exists a lift π ′ of ψ locally on X̃. The difference
′
between any two such lifts is an even superconformal derivation d ∈ Derscf
D (OX ′ , ψ∗ I)(−PN S ) with
′
′
values in ψ∗ I, and preserving the one NS puncture PN S on X , i.e., the a lift of ψ is a section
′
of H 0 (X ′ , Derscf
D (OX ′ , ψ∗ I)(−PN S )). The proof of the lemma now follows from the following
isomorphisms:
scf
′
′
Derscf
D (OX ′ , ψ∗ I)(−PN S ) = DerD (OX ′ , ψ∗ I ⊗ O(−PN S ))
=
=
=
=
′
Derscf
D (OX ′ (PN S ), ψ∗ I)
′
∗
Derscf
k (ψ OX ′ (PN S ), I)
′
∗
Derscf
k (ψ OX ′ (PN S ), OX̃ )
Derscf
k (OX̃ (F ), OX̃ )
= AX̃ (−F )
(see (10))
(Adjunction)
ψ
∗
′
PN
S
(Lemma 8)
= F, ψ OX ′ ∼
= OX̃
∗
(see (14)).
18
We can use the morphisms D → k[t]/t2 and D → k[η], to pullback π to an even and odd
first order deformation of π, respectively. This breaks Def π (Spec D) into a direct sum of its even
−
2
component Def +
π = Def π (Spec k[t]/t ) and odd component Def π = Def π (Spec k[η]) corresponding
to the even and odd components of the tangent space to M̃d,ρ at π.
′ ∼
′ ∼
Lemma 20. If (X̃ ′ , π ′ ) ∈ Def −
π (X × Spec k[η]), then X̃ = X̃ × Spec k[η] and π = π × Spec k[η].
Proof. Suppose π ′ : X̃ ′ → X × Spec[η] is an odd first order deformation of π : X̃ → X, and note
that X × Spec[η] is split since it is a trivial first order odd deformation of X. Furthermore, since
π ′ is a finite covering of a split super Riemann surface, X̃ ′ must be split by Corollary 3, and thus
X̃ ∼
= π × id by Lemma 19.
= X ′ × Spec D, and π ′ ∼
Proof of Theorem 18. We will prove that differential at π of the projection p : M̃d,ρ → M2,1 is an
isomorphism, i.e., that
dp|π : T M̃d,ρ |π → T M2,1 |p(π
is an isomorphism of super vector spaces. The morphism p is surjective by definition, and so we
are left to show that dp|π is injective.
The tangent space to M̃d,ρ at π is the set Def π of isomorphism classes of first order deformations
of π. The tangent space to M2,1 at X = p(π) is the set of isomorphism classes of first order
deformations of X and we know from Lemma 10 that the set of isomorphism classes of first order
deformations of X is canonically isomorphic to H 1 (X, AX (−PN S )), where PN S denotes the one NS
puncture on X. Under these identifications, dp|π : Def π → H 1 (X, AX (−PN S )) sends a isomorphism
class of first order deformations π ′ : X̃ ′ → X ′ to the isomorphism class of the first order deformation
X ′ of X.
The identity object 0 in H 1 (X, AX (−PN S )) is the class of the trivial deformation, X × Spec D,
of X, and the identity object 0 in Def π is the class of the trivial deformation π ×idD : X̃ ×Spec D →
X × Spec D of π. To prove that dp|π is injective we will show that dp|−1
π (0) = 0.
The map dp|π is a direct sum of its even and odd components,
−
1
−
1
+
−
+
dp|+
π : Def π → H (X, AX (−PN S )) , dp|π : Def π → H (X, AX (−PN S )) .
−
Set D + = Spec C[t]/t2 and D − = Spec C[η]. The identity objects in Def +
π and Def π are π×idD + and
π ×idD− , respectively. The identity objects in H 1 (AX (−PN S ))+ and H 0 (AX (−PN S ))− are X ×D +
and X × D − , respectively. The map dp|−
π is injective by Lemma 20: the lemma says that any first
order deformation of π with target the trivial deformation of X, i.e., π ′ : X̃ ′ → X×D + , is isomorphic
+
+
+
to the trivial deformation. For dp|+
π : Since D is purely bosonic, M̃d,ρ (D ) = (M̃d,ρ )bos (D ), and
dp|+
π = dpbos |πbos : T (M̃d,ρ )bos |πbos → T SM2,1 |pbos (πbos ) .
The diagram 20 shows pbos as the pullback of the étale morphism M̃2,1 → M2,1 . Since étale
morphisms are stable under pullback, pbos is étale, and dp|+
π = dpbos |πbos is therefore an isomorphism.
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5.3
M̃d,ρ is a Deligne-Mumford superstack
A superstack X is said to be Deligne-Mumford if there exist a superscheme U and a morphism
U → X which is representable 3 , surjective, and étale.
Theorem 21. M̃d,ρ is a smooth, proper Deligne-Mumford superstack.
Proof. We first prove that M̃d,ρ is an algebraic superstack by showing that M̃d,ρ satisfies the
conditions of the super Artin theorem. That M̃d,ρ is Deligne-Mumford then follows from the fact
that the automorphism group Autπ is finite, see Theorem [7] in [7].
(A1): We will prove that the diagonal morphism ∆ : M̃d,ρ → M̃d,ρ × M̃d,ρ is representable by an
algebraic superspace locally of finite type. This condition is equivalent to the condition that for
every pair (π1 : X̃ → X), (π2 : X̃ ′ → X ′ ) ∈ M̃d,ρ (T ) the functor
I := Isom(π1 , π2 ) : sSch /T → Set
∼
I(f : S → T ) = {isomorphisms f ∗ π1 −→ f ∗ π2 over T }
is representable by a superscheme.
We will first prove that I is representable in the case π1 = π2 , so that I = AutT (π1 ). Let
CovT (X̃, X) denote the groupoid of all branched covers of X by X̃, and note that π1 is an object in
CovT (X̃, X). There is a natural action of AutT (X̃) × AutT (X) on the set of objects in CovT (X̃, X)
by conjugation,
(φ̃, φ) · π = φ ◦ π ◦ φ̃−1 .
The automorphism group AutT (π1 ) is the stabilizer subgroup Stab(π1 ) ⊂ AutT (X̃)×AutT (X), and
is therefore representable by a superscheme. Coming back to the general case, the sheaf Isom(π1 , π2 )
is a AutT (π1 )-torsor, and is therefore representable.
That (S1’), (S2), (A3), (A4), (A5), (A6) hold follows from the identification of Dπ (M ) with
∗
X/A0 ⊗OX f0 (M )) where f0 : X → Spec A0 is the structure map, which is clearly a finite
A0 -module, and Obπ (M ) = 0 by Proposition 3.3 in [6], adapted appropriately to the case where π
has a non-zero number of ramification points with even local degree.
H 1 (X, A
b0 /mn ), where Xn , X̃n
Condition (A2): if {πn : X̃n → Xn } is a compatible sequence in M̃d,ρ (A
are super Riemann surface over Spec A0 /mn , then it can be approximated by some (π : X̃ → X) ∈
c0 ). Since M2,1 and Mg,0,2r are algebraic superstacks, there exists X ∈ M2,1 (Spec A
b0 ) which
M̃d,ρ (A
n
b
b
maps to limn Xn under the equivalence M2,1 (Spec A0 ) = lim M2,1 (Spec A0 /m ), and similarly there
←−
←−
b0 ) approximating lim X̃n . Now apply Corollary 8.4.6 [4] which states that there
exist X̃ ∈ Mg,0,2r (A
←−n
b0 ) to the category
is an equivalence from the category of finite X-superschemes, proper over Spec(A
n
b
b
of finite formal X-superschemes proper over limn Spec A0 /m . The cited corollary is a corollary of
←−
the Grothendieck existence theorem, a super version of which was proved in [8].
That M̃d,ρ is smooth follows from the vanishing of Ob, and it is proper because (M̃d,ρ )bos is
proper.
3
A morphism of superstacks Y → X is said to be representable if for every superscheme T and morphism T → X
the fiber product Y ×X T is representable by a superscheme. Let P be a property of morphisms of superschemes
which is stable under pullback, e.g., smooth, étale, proper. A representable morphism Y → X has property P if
Y ×X T → T has property P .
20
References
[1]
Dale H Husemoller. “Ramified coverings of Riemann surfaces”. In: (1962).
[2]
Hubert Flenner. “Ein Kriterium für die Offenheit der Versalität”. In: Mathematische Zeitschrift
178 (1981), pp. 449–473.
[3]
Eric D’Hoker and DH Phong. “Two-loop superstrings.: I. Main formulas”. In: Physics Letters
B 529.3-4 (2002), pp. 241–255.
[4]
B. Fantechi et al. “Fundamental algebraic geometry: Grothendieck’s FGA explained”. In: Mathematical Surveys and Monographs 123 (2005), p. 339.
[5]
Robin Hartshorne. Algebraic geometry. Vol. 52. Springer Science & Business Media, 2013.
[6]
R. Donagi and E. Witten. “Supermoduli space is not projected”. In: Proc. Symp. Pure Math.
Vol. 90. 2015, p. 19.
[7]
Martin Olsson. Algebraic spaces and stacks. Vol. 62. American Mathematical Soc., 2016.
[8]
Seyed Faroogh Moosavian and Yehao Zhou. “On the existence of heterotic-string and type-IIsuperstring field theory vertices”. In: arXiv preprint arXiv:1911.04343 (2019).
[9]
Nadia Ott. “Artin’s theorems in supergeometry”. In: arXiv preprint arXiv:2110.12816 (2021).
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