arXiv:1407.3530v1 [math.AG] 14 Jul 2014

THE MODULI OF KLEIN COVERS OF CURVESCHARLES SIEGEL
Abstract. We study the moduli space V4 Mg of Klein four covers of genus
g curves and its natural compactification. This requires the construction ofa related space which has a choice of basis for the Klein four group. This
space has the interesting property that the two components intersect along
a component of the boundary. Further, we carry out a detailed analysis of
the boundary, determining components, degrees of the components over theirimages in Mg , and computing the canonical divisor of V4 Mg .

Contents
Introduction
1. Background
2. The space Z22 Mg
3. Geometry of the boundary
4. The moduli of Klein curves
5. Pluricanonical forms
References

1
3
4
7
10
14
18

Introduction
Ever since the nineteenth century, unramified double covers have been an essential tool for studying curves. They correspond to square roots of the trivial line
bundle which form a group. This formulation can be used to study surface groups
and the mapping class group, and also theta characteristics, the square roots of thecanonical line bundle. There are intricate relationships between double covers andtheta characteristics, in particular, and the difference between them only becomecompletely clear after passing to moduli.
The moduli of double covers of curves has two components, one that is isomorphic to Mg , where the double cover is a disconnected union of two copies of thebase curve, and one where the double cover is nontrivial, traditionally denoted by
Rg . While the moduli of theta characteristics also has two components, neither
one is isomorphic to Mg . The components, Sg± , correspond to whether the thetacharacteristic has an even or odd dimensional space of global sections.1
Another, slightly more subtle, connection between the two moduli spaces is thatthe theta characteristics on a curve correspond to quadratic forms on the (F2 vector space of) points of order two on the curve. The quadratic form is given
1The notations come from the French for covering, revêtement and from the fact that curves

with theta characteristics are often called spin curves, due to connections with the quantum
mechanical notion of spin.
1

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CHARLES SIEGEL

by, if L is a theta characteristic, µ 7→ h0 (L ⊗ µ) − h0 (L) mod 2, and induces
a skew-symmetric bilinear form on the points of order two. This bilinear form
is independent of the theta characteristic chosen and is called the Weil pairing.
The Weil pairing, however, is really an invariant of a Klein four subgroup of theJacobian, as hµ, νi = hµ, µ+νi, and in fact, if {0, µ1 , µ2 , µ3 } is a Klein four subgroupof J (C)[2], then the Weil pairing on the group has the value h0 (L ⊗ OC ) + h0 (L ⊗µ1 ) + h0 (L ⊗ µ2 ) + h0 (L ⊗ µ3 ) mod 2, which is manifestly symmetric in the groupelements, suggesting that it could be clarified by studying the moduli of Klein
covers.
The boundaries of these moduli spaces have been studied in detail, and thefibers of the natural projection to Mg have been made very explicit. The approach±originates in [Cor89] for Sg and is pushed through in detail in [CC03], and adaptedto Rg in [BCF04]. More recently, this approach has been adapted to proving that±
pluricanonical forms extend to both Sg [Lud10] and to Rg [FL10], bringing thestudy of the birational geometry of these spaces into reach.

This paper. In this paper, we extend the description of the boundary and pluricanonical forms to the moduli of Klein covers of curves. This, however, is difficult
to do directly, so instead we introduce an intermediate moduli space, Z22 Mg , ofpairs of Prym curves and study it, then use the relationship between it and themoduli of Klein covers V4 Mg to prove the results on this space. In fact, we use thisrelationship to define V4 Mg .
In section 1 of this paper, we recall relevant facts about double covers. In
particular, the classification of points in the fiber over a stable curve from [BCF04],
and the relationship between two competing notations for the components of theboundary of Rg , used in, for instance, [Don87] and [Far12].
In section 2, we initiate the study of Z22 Mg , focusing on the interior. We construct the space and show that there are two connected components, corresponding
to Weil pairing 0, which was studied in [Sie13] under the notation R2 Mg , andWeil pairing 1, and the degree of each component over Mg , reproducing a result in[CLP12], which holds in the degenerate case where the dihedral group is only four
elements.
In section 3, we analyze the boundary of Z22 Mg in detail, describing the fibers
over Mg , then identify the boundary components and determine how many objects
in each fiber lie in each component. Here, we note an interesting fact. Although
Z22 Mg is an unramified covering of Mg and has two components, the natural compactification Z22 Mg is in fact connected, and the boundaries of the two components
intersect nicely along a single component.
In section 4, we proceed to analyzing V4 Mg and its boundary. We do so by
showing that the group action of PSL2 (F2 ) on Z22 Mg extends to the boundary ofeach of the two components separately, identifying several components of ∂Z22 Mg .
This allows us to describe the (much simple) boundary of V4 Mg and to show that thenatural map V4 Mg → Mg is simply ramified along a single boundary component.
In the last section, we follow [FL10], [Lud10] and [HM82], to extend the pluricanonical forms from the smooth locus to an arbitrary resolution of singularities.
The main tool in this is the Reid–Shepherd-Barron–Tai criterion [Rei80, Tai84].

THE MODULI OF KLEIN COVERS OF CURVES

3i

We conclude with a slope criterion for V4 Mg to be of general type analagous tosimilar results for Mg and Rg :i

Theorem 5.8. ForPany g, V4 Mg has general type if there exists a single effective
divisor D ≡ aλ − T b∆T ∆T where T runs over all boundary components, suchaaathat all the ratios baT are less than 13
2 and the ratios bII,III,III , b1,g−1,1:g−1 , b1,1,1 ,
a
a
a
13
a
bg−1,g−1,g−1 , b1,1:g−1,1:g−1 , bg−1,1:g−1,1:g−1 , and b1:g−1,1:g−1,1:g−1 are less than 3 .
Acknowledgments. I would like to thank Gavril Farkas, for suggesting that thebirational geometry of this space might be interesting, as well as for conversations
on the relationship between the Weil pairing, theta characteristics and Klein four
curves. Also I would like to thank Angela Gibney, Joe Harris, Tyler Kelly andAngela Ortega for helpful discussions on the moduli of curves, coverings, the Weilpairing and birational geometry and Amir Aazami and Jesse Wolfson for comments
on an earlier draft. This work was supported by World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan.
1. BackgroundIn this section, we will recall relevant facts about double covers and points oforder two on curves. For C a smooth projective curve over C, we denote by J (C)the group of line bundles of degree 0 on C. It has a natural subgroup J (C)[2]
consisting of the elements whose square is trivial.
Lemma 1.1. The following data are equivalent:(1) C̃ → C an irreducible étale double cover,(2) µ ∈ J (C)[2] nonzero, and(3) C̃ ∈ M2g−1 with ι : C̃ → C̃ a fixed point free involution.
Proof. Given a point of order two, we get an unramified double cover by looking
at Spec(OC ⊕ µ). Conversely, given a double cover, there is a single point of order
two that pulls back to zero.
To get between 1 and 3, we note that C ∼= C̃/ι.
By Lemma 1.1, we have an equivalence between J (C)[2] and the set of étale
double covers, with 0 corresponding to the trivial double cover. This induces agroup structure on double covers, and if C̃µ , C̃ν correspond to µ, ν, then C̃µ+ν isgiven by C̃µ ×C C̃ν /(ιµ , ιν ).
Definition 1.2 (quasistable curve). A genus g ≥ 2 curve X is quasistable if every
smooth rational component has at least two nodes and no two of these components
intersect. We call the stable curve C obtained by removing these rational components and gluing the nodes together the stabilization of X, and the nodes of C
obtained this way are the exceptional nodes and the rational components of X overthem exceptional components.
We define the nonexceptional curve to be the union of the nonexceptional components and denote it by Xne .
Definition 1.3 (Prym curve). A Prym curve is a triple (X, η, β) where X is quasistable, η ∈ J (X) such that for all exceptional components, E, we have ηE ∼=OE (1), and β : η ⊗2 → OX is a homomorphism that is generically nonzero on eachnonexceptional component.

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CHARLES SIEGEL

Remark 1.4. In the notation we will use for other objects, a Prym curve would
be called a Z/2Z curve, but we will continue to refer to them as Prym curves.
Definition 1.5 (Isomorphism of Prym Curves). An isomorphism of Prym curves(X, η, β) and (X ′ , η ′ , β ′ ) is an isomorphism σ : X → X ′ such that there exists an
isomorphism τ : σ ∗ (η ′ ) → η such that the diagram commutes:σ ∗ (η ′ )⊗2τ ⊗2σ∗ (β ′ )σ ∗ (OX ′ )/ η ⊗2
β

∼


/ OX

Note that the definition of isomorphism of Prym curves does not depend on what
τ is chosen, only on σ.
Any automorphism of a Prym curve which induces the identity on the stablemodel of X will be called inessential. The group of automorphisms will be denotedby Aut(X, η, β) and the inessential automorphisms will be Aut0 (X, η, β).
For the rest of this paper, for any curve, denote by ν the normalization morphism
for a curve, and denote by g ν the geometric genus of the normalization:
Proposition 1.6 ([BCF04, Proposition 11]). Let X be a quasistable curve, Z its
stable model, ΓZ the dual graph of Z, and ∆X the set of nodes not lying underan exceptional curve, and assume further that ∆cX is eulerian. Then there areν
22g +b1 (∆X ) Prym curves supported on X and each has multiplicity 2b1 (ΓZ )−b1 (∆X )in the fiber of Rg → Mg .
We will denote the moduli space of these curves by Z2 Mg , and we note that it
has two components. One, isomorphic to Mg where the Prym curve has η ∼= OXover a stable base, and Rg , the nontrivial Prym curves.
Remark 1.7 (Notation). The boundary components of the Prym moduli space haveseveral competing notations in the literature. For the 22g objects, we always have
1 that is the disconnected double cover, and in this setting, it lies over the stable
curve itself.
Additionally, Donagi [Don87] classified the nontrivial points of order two on an
irreducible 1-nodal curve in terms of the vanishing cycle δ. In his notation, ∆I is
the subset with the marked point µ being equal to δ, ∆II when hδ, µi = 0 but µ 6= δand ∆III being when hδ, µi 6= 0, under the Weil pairing, defined below. Alternately,
these three components are denoted by ∆′′0 , ∆′0 and ∆ramby Farkas [FL10] and it0
is noted that ∆III = ∆ramis precisely the set of Prym curves on the quasistable0curve. In the rest of this article, however, we will follow Donagi’s notation.
Over the 1-nodal reducible curves, the notation agrees, and the components are∆i , ∆g−i and ∆i:g−i , for the Prym curves supported on the component of genus i,
g − i or both, respectively.
2. The space Z22 Mg
Definition 2.1 (Weil Pairing). Let µ, ν ∈ J (C)[2], and let κ ∈ Pic(C) such that
κ⊗2 ∼= KC . Then the Weil pairing on the curve C is given byhµ, νi = h0 (κ) + h0 (κ ⊗ µ) + h0 (κ ⊗ ν) + h0 (κ ⊗ µ ⊗ ν)

mod 2.

THE MODULI OF KLEIN COVERS OF CURVES

5The Weil pairing is bilinear, skew-symmetric, and independent of the choice ofκ, and therefore we can see that hµ, νi = hµ, µ + νi = hν, µ + νi, and so it is aninvariant of a rank 2 subgroup of J (C)[2].
Lemma 2.2. The following data are equivalent:(1) A curve C ∈ Mg along with C̃i → C, i = 1, 2 irreducible, nonisomorphic
unramified double covers,(2) A curve C ∈ Mg along with µ1 , µ2 ∈ J (C)[2], with µ1 6= µ2 , and(3) A pair of curves C, D ∈ M2g−1 along with involutions σC , σD that act
freely on C, D respectively and which are such that C/σC and D/σD are
isomorphic, but the pairs (C, σC ) and (D, σD ) are not.
Proof. This is just an application of Lemma 1.1.

A related, but slightly different result is the following, where we do not choosea basis for the Klein four group.
Lemma 2.3. The following data are equivalent:(1) A curve C ∈ Mg , and C̃ → C an étale Klein 4 cover,(2) A curve C ∈ Mg , and φ : V4 → J (C)[2] an injective homomorphism, and(3) A curve C̃ ∈ M4g−3 with a free action of V4 on C̃.
Proof. This is again just an application of Lemma 1.1.

For the rest of this section, we will be working in the case with a basis, and later
will return to the basis-free case.
Definition 2.4 (Z22 curve). A Z22 curve is (X1 , X2 , η1 , η2 , β1 , β2 ) where (Xi , ηi , βi )is a Prym curve for i = 1, 2 and X1 and X2 have isomorphic stabilizations.
An isomorphism of Z22 curves is just a pair of isomorphisms of Prym curves thatinduce the same isomorphism on the stable model. Equivalently, an automorphism
can be seen by looking at the quasistable curve with exceptional nodes the unionof the exceptional nodes of X1 and X2 . From this viewpoint, an isomorphism is
an isomorphism of such quasi-stable curves which induces isomorphisms on the
pullbacks of ηi and ηi′ .
Given this, we can see that the moduli space of Z22 curves is just Z2 Mg ×Mg
Z2 Mg . It is easy to see that it has at least five components, depending on η1 , η2 . If
both are trivial, we have a copy of Mg . If one is trivial but the other is nontrivial,
then we get two copies of Rg . Additionally, when η1 ∼= η2 , we get another copy of
Rg leaving the remnant:Definition 2.5 (Z22 Mg ). We denote by Z22 Mg the closure in Z2 Mg ×Mg Z2 Mg
of the locus of Z22 curves with η1 ∼6 η2 both nontrivial. The locus of smooth curves=
in Z22 Mg will be denoted by Z22 Mg .
Geometrically, the most natural thing to study is the moduli space of Klein
four subgroups, with no choice of basis. This, by Lemma 2.3 is then the moduli
space of Klein four covers of curves. However, it is much easier to construct the
space with a choice of basis, which is Z22 Mg (also note that the geometricity of
the covers is less clear on the boundary). There is a natural PSL2 (F2 ) action on
Z22 Mg , permuting the ordered bases of the Z22 ⊂ J (C)[2]. Below, we will extend it

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CHARLES SIEGEL

to the boundary and construct V4 Mg , and we will deduce many of its properties
from those of Z22 Mg .
Here, we note that Aut(C, η1 , η2 ) is just Aut(X1 , η1 , β1 ) ×Aut(C) Aut(X2 , η2 , β2 )where (Xi , ηi , βi ) are the two Prym structures. Thus, the results of [BCF04] apply
without difficulty, so we have good behavior of the universal deformation, we havea subgroup of inessential automorphisms, etc. We will recall those facts we need in
the last section of the paper.
Lemma 2.6. The map Z22 Mg → Mg has degree (22g − 1)(22g − 2), thus, so does
Z22 Mg → Mg .
Proof. We know that the degree of Z2 Mg ×Mg Z2 Mg → Mg is 24g and that it
breaks down as Mg ∪ Rg ∪ Rg ∪ Rg ∪ Z22 Mg . Each component is dominant andequidimensional, so we can compute the degree on Z22 Mg as 24g − 1 − 3(22g − 1) =(22g − 1)(22g − 2).
The space Z22 Mg is not irreducible. We can see that there must be at least twocomponents because the Weil pairing is deformation invariant; we will write Z22 M0g
and Z22 M1g for the two components, with Weil pairing respectively 0 and 1.
Lemma 2.7. The spaces Z22 M0g and Z22 M1g are both irreducible.
Proof. For any curve C of genus g, the action of Sp(2g, F2 ) on the space J (C)[2] ×
J (C)[2] has two orbits, pairs of points that are orthogonal and pairs that arenonorthogonal, and this is the monodromy of Z22 Mg → Mg .
Remark 2.8. Although slightly more complex, in the case where we look at pointsof order n rather than points of order 2, we get a similar theorem, where the numberof components is indexed by Zn . Similarly, the Klein moduli space, which we will
study below, will have components indexed by Zn /Z×
n . In our case, both of theseare Z2 , and we will identify them with {0, 1}.
Before moving on to a detailed analysis of the boundary, we compute the degreesof the maps Z22 Mig → Mg .
Proposition 2.9. We have natural maps to Mg forgetting the points of order two,and their degrees are:
(1) Z22 M0g → Mg has degree (22g − 1)(22g−1 − 2)(2) Z22 M1g → Mg has degree (22g − 1)22g−1
Proof. Fix a smooth curve C of genus g.
Any element of Z22 Mg lying over C is of the form (C, η, η ′ ) where η, η ′ ∈ J (C)[2],
which is an F2 vector space. To be in Z22 M0g they must be orthogonal under the
Weil pairing, which is a nondegenerate form. Thus, for each η, we must chooseη ′ ∈ η ⊥, a hyperplane in J (C)[2]. However, as they are linearly independent,
we asset that η ′ ∈/ {0, η}. Thus, we have 22g − 1 choices for η, and given one of
2g−1
those, we have 2− 2 choices of η ′ satisfying these conditions, which computes
20
the degree of Z2 Mg → Mg .
To determine the degree of Z22 M1g → Mg , we note that we can again choose η
freely, and now η ′ ∈ J (C)[2] \ η ⊥ , giving us the degree claimed.
THE MODULI OF KLEIN COVERS OF CURVES

7

3. Geometry of the boundary
Proposition 3.1. Let Z be a stable curve with dual graph ΓZ . The fiber over Z
in the Z22 Mg consists of the following objects:νν(1) (22g +b1 (ΓZ ) − 1)(22g +b1 (ΓZ ) − 2) objects of multiplicity 1 where both Prym
curves are supported on Z.
(2) If X is quasistable with stabilization Z and ∆cX is Eulerian, then we getνν
24g +b1 (∆X )+b1 (ΓZ ) − 22g +b1 (∆X ) objects of multiplicity 2b1 (ΓZ )−b1 (∆X ) supported on each of (X, Z) and (Z, X).
(3) If X is quasistable with stabilization Z and ∆cX is Eulerian, then we get
two types of objects supported on (X, X):νν(a) 24g +2b1 (∆X ) −22g +b1 (∆X ) objects of multiplicity 22b1 (ΓZ )−2b1 (∆X ) withtwo distinct Prym structures
ν(b) 22g +b1 (∆X ) objects of multiplicity 22b1 (ΓZ )−2b1 (∆X ) − 2b1 (ΓZ )−b1 (∆X )with the same Prym structure.
(4) If X1 , X2 are quasistable over Z and ∆cX1 and ∆cX2 are Eulerian, then thereν
exist 24g +b1 (∆X )+b1 (∆X ) objects of multiplicity 22b1 (ΓZ )−b1 (∆X1 )−b1 (∆X2 ) on
(X1 , X2 ) and again on (X2 , X1 ).
Proof. The fiber is a subscheme of RZ × RZ which is the complement of Mg ∪ Rg ∪Rg ∪ Rg , components which split off completely over smooth base curves. Away
from the diagonal, we simply remove anything where one of the components is thetrivial line bundle. On the diagonal, the situation is somewhat more intricate. We
look at Z2 Mg ×Mg Z2 Mg \ (Mg ∪ Z2 Mg ∪ Z2 Mg ∪ Z2 Mg ) and then take theclosure. This leaves us with some points on the diagonal, which we can see because
on any degeneration, there will be classes of Prym curves that will degenerate to
the same thing, which is seen by noting the multiplicity greater than 1.
Additionally, a straightforward computation summing these over all of the Eulerνν
paths gives us total degree (22g +2b1 (ΓZ ) − 1)(22g +2b1 (ΓZ ) − 2), which is the degreeof the moduli space, showing that nothing has been missed.
Now, applying the above to the general point on the boundary, we see that fora reducible 1-nodal curve, all Prym curves on it are supported on the stable curve
itself, yielding (22g − 1)(22g − 2) objects of multiplicity 1. The case of an irreducible
curve is a bit more complex:
Corollary 3.2. Let Z be a 1-nodal irreducible stable curve of genus g and ν : Z ν →Z its normalization. There is only one unstable quasistable curve, X = Z ν ∪x,y P1 ,where x, y are the preimages of the node in Z ν . Then the fiber of Z22 Mg → Mg
over Z consists of the following objects:(1) On (Z, Z), we have (22g−1 − 1)(22g−1 − 2) objects of multiplicity 1.
(2) On each (Z, X) and (X, Z), we have a total of 24g−2 − 22g−1 objects ofmultiplicity 2(3) On (X, X) we have two types of objects:(a) 24g−4 − 22g−2 objects of multiplicity 4 with non-isomorphic projections
to Z2 Mg .
(b) 22g−2 objects of multiplicity 2 with the same projections to Z2 Mg .
Now, we must compute the list of boundary divisors. We begin by looking at
the boundary of Z22 Mg ×Mg Z22 Mg . Points on the boundary can be classified into

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CHARLES SIEGEL

products ∆a × ∆b where a, b were in {I, II, III} over an irreducible 1-nodal curveand {i, g − i, i : g − i} over a reducible curve with components of genus i and g − i.
We denote the restrictions of these loci to Z22 Mg by ∆a,b , and note that althoughsome are, many of these are not irreducible.
i

The components with nonirreducible restrictions to Z22 Mg are ∆II,II , ∆III,III ,∆i,i:g−i , ∆i:g−i,i , ∆g−i,i:g−i , ∆i:g−i,g−i , and ∆i:g−i,i:g−i . Specifically, they break
up as∆II,II=′∆±
II,II + ∆II,II∆III,III

=

′
∆diag
III,III + ∆III,III∆i,i:g−i=∆ii,i:g−i + ∆′i,i:g−i∆i:g−i,i=∆ii:g−i,i + ∆′i:g−i,i∆g−i,i:g−i

=

′
∆g−i
g−i,i:g−i + ∆g−i,i:g−i∆i:g−i,g−i

=

′
∆g−i
i:g−i,g−i + ∆i:g−i,g−i∆i:g−i,i:g−i=′∆ii:g−i,i:g−i + ∆g−i
i:g−i,i:g−i + ∆i:g−i,i:g−iwhere (with equality meaning equal to the closure of)
∆±
II,II

=

∆′II,II=∆diag
III,III=∆′III,III=∆ii,i:g−i∆′i,i:g−i∆ii:g−i,i∆′i:g−i,i∆g−ig−i,i:g−i∆′g−i,i:g−i∆g−i
i:g−i,g−i′∆i:g−i,g−i∆ii:g−i,i:g−i∆g−i
i:g−i,i:g−i∆′i:g−i,i:g−i

=
=
=
=
=
=
=
=
=
=
=

{(η1 , η2 )|ν ∗ η1 ∼= ν ∗ η2 }{(η1 , η2 )|ν ∗ η1 6∼= ν ∗ η2 }{(η1 , η2 )|η1 ∼= η2 }{(η1 , η2 )|η1 ∼6 η2 }={(η1 , η2 )|η1 |C1 ∼= η2 |C1 }6 η2 |C }{(η1 , η2 )|η1 |C ∼=1

1{(η1 , η2 )|η1 |C1 ∼= η2 |C1 }6 η2 |C }{(η1 , η2 )|η1 |C ∼=1

1{(η1 , η2 )|η1 |C2 ∼= η2 |C2 }∼ η2 |C }{(η1 , η2 )|η1 |C 6=
2

2{(η1 , η2 )|η1 |C2 ∼= η2 |C2 }∼6 η2 |C2 }{(η1 , η2 )|η1 |C2 ={(η1 , η2 )|η1 |C ∼= η2 |C }1

1{(η1 , η2 )|η1 |C2 ∼= η2 |C2 }∼6 η2 |Ci for i = 1, 2}{(η1 , η2 )|η1 |Ci =Now that we have a list of all of the components, we compute which objects arein which, a straightforward computation:
Proposition 3.3. Over the locus of 1-nodal irreducible curves in Mg of the boundary components of Z22 Mg consist of∆I,II= 22g−1 − 2 objects of multiplicity 1∆II,I= 22g−1 − 2 objects of multiplicity 1

THE MODULI OF KLEIN COVERS OF CURVES∆I,III= 22g−2 objects of multiplicity 2∆III,I= 22g−2 objects of multiplicity 2∆II,III= 22g−2 (22g−1 − 2) objects of multiplicity 2∆III,II= 22g−2 (22g−1 − 2) objects of multiplicity 2

∆±
II,II
′
∆II,II∆diag
III,III∆′III,III= 22g−1 − 2 objects of multiplicity 1

9= (22g−1 − 2)(22g−1 − 4) objects of multiplicity 1= 22g−2 objects of multiplicity 2= 24g−4 − 22g−2 objects of multiplicity 4

Remark 3.4. It is interesting to note that away from ∆′III,III , the Weil pairing
is well-defined by continuity. However, on this divisor, this fails. Let us look at a
simple example. Let C be a smooth genus 2 curve with p1 , . . . , p6 the fixed points of
the hyperelliptic involution. Then any point of order two is pi − pj for i 6= j, andpi − pj ≡ pj − pi . If the vanishing cycle of the degeneration is p1 − p2 , then thepairs (p1 − p3 , p1 − p4 ) and (p1 − p3 , p2 − p4 ) both degenerate to the same point of∆′III,III . We note that, for a genus 2 curve, the Weil pairing can be described as the
cardinality of the intersection of the set of indices appearing in this representation.
Thus, hp1 − p3 , p1 − p4 i = 1 and hp1 − p3 , p2 − p4 i = 0. Thus, the two componentsof this moduli space intersect on the boundary!
In fact, we can say a bit more about the intersection:
0

1

Theorem 3.5. The intersection ∆′III,III = Z22 Mg ∩ Z22 Mg is transverse, in thesense that if (C, η1 , η2 ) ∈ ∆′III,III , then we have01
T(C,η1 ,η2 ) Z22 Mg ∼= T(C,η1 ,η2 ) Z22 Mg ⊕T(C,η1 ,η2 ) ∆′III,III T(C,η1 ,η2 ) Z22 Mg .

Proof. To see that this is precisely the intersection, we look at a degeneration ofPrym curves with vanishing cycle δ. The fiber over the nodal curve only has points
coming together over ∆III , which is the part of the fiber over the quasi-stable
curve. There, every Prym curve structure is the limit of both η and η + δ for η
some Prym curve structure on a smooth curve in the degeneration. The Weil pairing
is well-defined on all components over ∆III other than ∆′III,III , by linearity andthe definitions of ∆I and ∆II . However, on ∆III,III , we have (η1 , η2 ) two Prym
structures giving a Z22 curve. This limit as δ vanishes, is the same as the limitof (η1 , η2 + δ), and also two other loci. However, the Weil pairing is linear, sohη1 , η2 + δi = hη1 , η2 i + hη1 , δi, and because it is in ∆III , hη1 , δi = 1, so this pointis a limit of families of Veil pairing both 0 and 1, and this holds for every point in∆′III,III .
Transversality follows by looking at first order deformations of (C, η1 , η2 ). The
Weil pairing determines which component (C, η1 , η2 ) is on, except along ∆′III,III
where it is indeterminate. So we describe all of the space:
T(C,η1 ,η2 ) Z22 Mg
T(C,η1 ,η2 ) Z22 Mg

0T(C,η1 ,η2 ) ∆′III,III=First order deformations with hη1 , η2 i undefined, 0, or 1,=

First order deformations with hη1 , η2 i undefined, or 0,=

First order deformations with hη1 , η2 i undefined,

10

CHARLES SIEGEL

T(C,η1 ,η2 ) Z22 Mg

1=First order deformations with hη1 , η2 i undefined, or ,From these descriptions, transversality follows immediately.



A similar computation to Proposition 3.3 over reducible 1-nodal curves gives:
Proposition 3.6. Over the locus of 1-nodal reducible curves that are a union of agenus i and a genus g − i curve in Mg , the boundary components of Z22 Mg are of
the following degrees (and all objects are multiplicity 1):∆i,i=(22i − 1)(22i − 2)∆g−i,g−i=(22(g−i) − 1)(22(g−i) − 2)∆i,g−i=(22i − 1)(22(g−i) − 1)

∆g−i,i=(22i − 1)(22(g−i) − 1)

∆ii,i:g−i=(22i − 1)(22(g−i) − 1)∆′i,i:g−i=(22i − 2)(22i − 1)(22(g−i) − 1)

∆ii:g−i,i=(22i − 1)(22(g−i) − 1)∆′i:g−i,i=(22i − 2)(22i − 1)(22(g−i) − 1)∆g−ig−i,i:g−i=(22i − 1)(22(g−i) − 1)∆′g−i,i:g−i=(22(g−i) − 2)(22i − 1)(22(g−i) − 1)∆g−i
i:g−i,g−i=(22i − 1)(22(g−i) − 1)∆′i:g−i,g−i=(22(g−i) − 2)(22i − 1)(22(g−i) − 1)

∆ii:g−i,i:g−i=(22i − 1)(22(g−i) − 1)(22(g−i) − 2)∆g−i
i:g−i,i:g−i=(22i − 1)(22i − 2)(22(g−i) − 1)∆′i:g−i,i:g−i=(22i − 1)(22(g−i) − 1)
×((22i − 1)(22(g−i) − 1) − (22i − 1) − (22(g−i) − 1))
4. The moduli of Klein curves

In this section, we extend the results of the previous section to a compactificationof V4 Mg = Z22 Mg / PSL2 (F2 ), that is, the moduli space where we do not choose abasis of the Deck transformations. We will do so by extending the group action to
the compactification Z22 Mg , and defining the quotient to be V4 Mg .
Proposition 4.1. The group action PSL2 (F2 ) × Z22 Mg → Z22 Mg extends to eachcomponent of Z22 Mg .
′Proof. Let D1 = ∆I,III ∪ ∆III,I ∪ ∆diag
III,III and D2 = ∆III,III ∪ ∆II,III ∪ ∆III,II .

Then the extension is actually straightforward over Z22 Mg \ (D1 ∪ D2 ), as over thislocus, the fibers are reduced, and the action is just by change of basis on a Klein 4group. Only in the cases where multiplicities are no longer 1, namely D1 and D2 ,will these fail to just be Klein four groups.
Now, we take the orbit of a Z22 -curve in the locus where we have the group
action, and degenerate it to D1 . Then, the six objects of multiplicity 1 of this fiber−−−
degenerate to (η, η) ∈ ∆diag
III,III , (η, OX ) ∈ ∆III,I and (OX , η) ∈ ∆I,III , where OX

THE MODULI OF KLEIN COVERS OF CURVES

11

is the Prym curve structure on X lying in ∆I ⊂ Z2 Mg . Each of these appears
with multiplicity 2. Here, the group action can be seen most clearly by notingthat PSL2 (F2 ) ∼= S3 (and in fact, the change of basis on a Klein four group isjust permuting the three nonzero elements) and seeing the action as being that of−
S3 on the ordered set (OX, η, η) followed by forgetting the last element. Deeper
degeneration into the strata D1 \ D2 can be handled in the same way, leaving only
D2 remaining.
To extend the action to D2 , we must first restrict to the individual irreducible
components of Z22 Mg . This is because ∆′III,III is the intersection of the two components, by Theorem 3.5. So, by transversality the fiber multiplicity of elements in
the intersection must be split evenly between the components.
Now, let (ǫ1 , ǫ2 ) ∈ ∆′III,III . Then ǫ2 ⊗ ǫ−1
1 gives a Prym structure on the closureof the complement of the exceptional components. There are two different Prym
structures on the stable curve that pull back to this under the stabilization map,
but one of them lies on each componenet. We will denote by η i the Prym structureisuch that (η i , ǫ1 ), (η i , ǫ2 ) ∈ Z22 Mg . Then, the PSL2 (F2 ) ∼= S3 action is given by
permutation of ǫ1 , ǫ2 , η i .
This extension allows us to take the quotient, which constructs from the moduli
of pairs of Prym structures, which is the same as the moduli of Klein four groupsof Prym structures, with the moduli of Klein four groups of Prym covers withouta basis.
Definition 4.2 (Moduli of Klein four covers). We define the space V4 Mg to be
the quotient of Z22 Mg by the relation described above, and we call it the moduli of
Klein four covers of genus g curves.
Given an orbit {(C, η1i , η2i )} where i runs over the elements of the orbit, we will
denote by (C, {ηji }i,j ) the corresponding point of V4 Mg , with i running over theorbit and j = 1, 2.
The boundary in the Klein moduli space simplifies significantly. Because theaction of PSL2 (F2 ) exchanges some boundary components, we group them together
and give names to their images (fixing the Weil pairing as either 0 or 1 in each case)in the following:∆I,II ∪ ∆II,I ∪ ∆±II,II
∆I,III ∪ ∆III,I ∪ ∆diag
III,III∆II,III ∪ ∆III,II ∪ ∆′III,III∆′II
g−i∆i,g−i ∪ ∆g−i,i ∪ ∆ii,i:g−i ∪ ∆ii:g−i,i ∪ ∆g−i
g−i,i:g−i ∪ ∆i:g−i,g−i∆i,i∆g−i,g−i′′i∆i,i:g−i ∪ ∆i:g−i,i ∪ ∆i:g−i,i:g−i′∆g−i,i:g−i ∪ ∆′i:g−i,g−i ∪ ∆g−i
i:g−i,i:g−i∆′i:g−i,i:g−i→ ∆I,II,II→ ∆I,III,III→ ∆II,III,III→ ∆II,II,II→ ∆i,g−i,i:g−i→ ∆i,i,i→ ∆g−i,g−i,g−i→ ∆i,i:g−i,i:g−i→ ∆g−i,i:g−i,i:g−i→ ∆i:g−i,i:g−i,i:g−i12

CHARLES SIEGEL

Between the degrees computed in the previous section and the maps above all
being PSL2 (F2 ) quotients, we find the following structure on the boundary∆I,II,II=

22g−2 − 1 objects of multiplicity 1∆I,III,III=

22g−2 objects of multiplicity 1∆II,III,III=∆II,II,II=∆i,g−i,i:g−i=22g−2 (22g−2 − 1) objects of multiplicity 2(22g−1 − 2)(22g−1 − 4)objects of multiplicity 1
6(22i − 1)(22(g−i) − 1) objects of multiplicity 1∆i,i,i=∆g−i,g−i,g−i=∆i,i:g−i,i:g−i=∆g−i,i:g−i,i:g−i=∆i:g−i,i:g−i,i:g−i=(22i − 1)(22i − 2)objects of multiplicity 1
6(22(g−i) − 1)(22(g−i) − 2)objects of multiplicity 1
6(22i − 1)(22i − 2)(22(g−i) − 1)
objects of multiplicity 1
2(22(g−i) − 1)(22(g−i) − 2)(22i − 1)
objects of multiplicity 1
2(22i − 1)(22(g−i) − 1)(22i − 2)(22(g−i) − 2)objects of6
multiplicity 1This gives us
2g

Proposition 4.3. The morphsim V4 Mg → Mg has degree (2simply ramified along ∆II,III,III .

−1)(22g −2)
6

and isThus, we have
Corollary 4.4. The canonical divisor of V4 Mg is
KV4 Mg= 13λ − 2∆I,II,II − 2∆I,III,III − 2∆II,II,II − 3∆II,III,III−∆1,g−1,1:g−1 − ∆1,1,1 − ∆g−1,g−1,g−1−∆1,1:g−1,1:g−1 − ∆g−1,1:g−1,1:g−1 − ∆1:g−1,1:g−1,1:g−1
⌊g/2⌋

−2

X

(∆i,g−i,i:g−i + ∆i,i,i + ∆g−i,g−i,g−i + ∆i,i:g−i,i:g−i

i=1+∆g−i,i:g−i,i:g−i + ∆i:g−i,i:g−i,i:g−i ).
Proof. We use the Hurwitz formula, which tells us that KV4 Mg = π ∗ KMg +∆II,III,III . The canonical divisor of Mg is 13λ−2δ0 −3δ1 −2δ2 −. . .−2δ⌊g/2⌋ [HM82].
We note that π ∗ (∆i ) = ∆i,g−i,i:g−i + ∆i,i,i + ∆g−i,g−i,g−i + ∆i,i:g−i,i:g−i +∆g−i,i:g−i,i:g−i + ∆i:g−i,i:g−i,i:g−i and π ∗ ∆0 = ∆I,II,II + ∆I,III,III + ∆II,II,II +2∆II,III,III and π ∗ λ = λ.
Now we’ll work out some numerics of the odd and even components of V4 Mg
Proposition 4.5. The degrees of the natural projection maps to Mg are
2g• for V4 Mg , (2−1)(22g −2).
6

THE MODULI OF KLEIN COVERS OF CURVES
0

13

2g

−1)(22g−1 −2).
6
1 (22g −1)22g−1.
for V4 Mg ,6

• for V4 Mg , (2•This follows directly from Proposition 2.9 and Lemma 2.6.
As a final computation involving the degrees, we compute the content of the0
1boundary divisors when restricted to V4 Mg and V4 Mg .
Theorem 4.6. The fiber over the generic element C of the boundary of Mg inV4 Mg is:0(1) if C is irreducible 1-nodal, the fiber in V4 Mg is(a) 22g−2 − 1 elements of ∆0I,II,II with multiplicity 1,
2g−1

2g−2

−2)(2
−4)
elements of ∆0II,II,II with multiplicity 1,(b) (26(c) (22g−1 − 2)22g−4 elements of ∆0II,III,III with multiplicity 2,
1(2) if C is irreducible 1-nodal, the fiber in V4 Mg is
2g−1

2g−2

−2)2
(a) (2elements of ∆1II,II,II with multiplicity 1,6(b) (22g−1 − 2)22g−4 elements of ∆1II,III,III with multiplicity 2,(c) 22g−2 elements of ∆1I,III,III with multiplicity 1,(3) if C is reducible with components of genus i and g − i, the fiber in V4 Mg
is(a) (22i − 1)(22(g−i) − 1) elements of ∆0i,g−i,i:g−i with multiplicity 1,
2i

2i−1

(b) (2 −1)(26
2(g−i)−2)0elements of ∆0i,i,i with multiplicity 1,2(g−i)−1−1)(2−2)(c) (2elements of ∆0g−i,g−i,g−i with multiplicity 1,6(d) (22i−1 − 1)(22i−2 − 1)(22(g−i) − 1) elements of ∆0i,i:g−i,i:g−i with multiplicity 1,(e) (22(g−i)−1 − 1)(22(g−i)−2 − 1)(22i − 1) elements of ∆0g−i,i:g−i,i:g−i withmultiplicity 1,2i
2(g−i)−1)((22i−1 −2)(22(g−i)−1 −2)+(22i−1 )(22(g−i)−1 ))(f) (2 −1)(2elements of
6∆0i:g−i,i:g−i,i:g−i with multiplicity 1,(4) if C is reducible with components of genus i and g − i, the fiber in V4 Mg
is
2i
2i−1
elements of ∆1i,i,i with multiplicity 1,(a) (2 −1)262(g−i)12(g−i)−1−1)2(b) (2elements of ∆1g−i,g−i,g−i with multiplicity 1,6(c) (22i−1 − 1)(22i−1 )(22(g−i) − 2) elements of ∆1i,i:g−i,i:g−i with multiplicity 1,(d) (22(g−i)−1 − 1)(22(g−i)−2 )(22i − 1) elements of ∆1g−i,i:g−i,i:g−i withmultiplicity 1,2i
2(g−i)−1)((22i−1 −2)(22i−1 )+(22(g−i)−1 )(22(g−i)−1 −2))elements of(e) (2 −1)(26∆1i:g−i,i:g−i,i:g−i with multiplicity 1.

Proof. We will work out parts 1 and 3, parts 2 and 4 being analagous.
With the exception of ∆II,III,III , we can compute the Weil pairing by choosing
any pair of elements in the group. For ∆II,III,III , we note that it must be divided
evenly between the components. The groups consist of two elements from ∆III
and the one element of ∆II , and can be chosen to either be glued by +1 or −1 at14

CHARLES SIEGEL

the node. These will correspond to Weil pairing 0 and 1, thus dividing ∆II,III,III
evenly.
For ∆I,II,II , by definition, we must have ∆0I,II,II = ∆I,II,II . Similarly, we cansee that ∆0I,III,III = ∅. We can finish by computing that ∆0II,II,II must be the
2g

2g−1

−2)
. However, we
correct size to, with the other components, add up to (2 −1)(26
can compute this directly by choosing µ1 , µ2 ∈ ∆II distinct and orthogonal. Then,if δ is the vanishing cycle, we have µ1 ∈ δ ⊥ \ (δ) and µ2 ∈ (δ, µ1 )⊥ \ (δ, µ1 ), which
gives the appropriate number.
Over a reducible curve C = Ci ∪ Cg−i , although the expressions are more complex, the situation is simpler. We begin by noting that everything in ∆i,g−i,i:g−i
must be in ∆0i,g−i,i:g−i , because the generators have no common support curve.
As for ∆i,i,i and ∆g−i,g−i,g−i , they will be precisely the fibers of the lower genus
maps V4 M0i → Mi and V4 M0g−i → Mg−i . On ∆i,i:g−i,i:g−i , we can have any
nonzero square trivial line bundle on the component Ci , and the second generator
can have any restriction to Cg−i , but the restriction to Ci must be orthogonal to thefirst, and here, we only divide by two choices in ∆i:g−i that can be basis elements.
The next component, ∆g−i,i:g−i,i:g−i , can be computed in a similar way. The final component, ∆i:g−i,i:g−i,i:g−i starts with an arbitary element of ∆i:g−i , and thesecond must either have both restrictions orthogonal to those of the first, or else
both nonorthogonal, and then we divide by 6 from choices of basis, completing thecomputation.


5. Pluricanonical forms
In this section, we show that pluricanonical forms on V4 Mg extend to any smoothmodel, allowing us to compute the Kodaira dimension on V4 Mg itself, rather thanhaving to work on the set of smooth models. As such, the goal of this section is to
prove
i

i
2M
2
Theorem 5.1. Fix g ≥ 4 and i ∈ {0, 1}, and let Z\
g → Z2 Mg be any reso2
i,reglution of the singularities. Then every pluricanonical form defined on Z22 Mg,
i
2 M . Specifically, for all integers
the smooth locus, extends holomorphically to Z\g2ℓ ≥ 0, we have isomorphisms
H 0 (Z22 Mgi,reg, K ⊗ℓ2

Z 2 Mgii,reg

2 M , K ⊗ℓ)∼
= H 0 (Z\
g
i)
2
\
2
Z 2 MgAnalogues of this theorem are known for all of the relevant related moduli spaces:
Mg is proved in [HM82, Theorem 1], Rg is proved in [FL10, Theorem 6.1], and themoduli of spin curves in [Lud10, Theorem 4.1]. Our proof will very closely follow
the one in [FL10] for Rg , which can be expected as Z22 Mg is two of the componentsof Rg ×Mg Rg .
Before we can proceed, we need to make a few remarks about the versal deformations of an object X = (X1 , X2 , η1 , η2 , β1 , β2 ) ∈ Z22 Mg . Let Ct3g−3 be the versal
3g−3
deformation space of Z, the stabilization of Xi and Cα
the versal deformation
space of X. There are compatible decompositions
MM
M
3g−3 ∼CαCτ i ⊕Cτ i ⊕Cτ i=pi ∈∆cX ∩∆X2
1

pi ∈∆cX ∩∆X1
2

pi ∈∆cX ∩∆cX
1

2

THE MODULI OF KLEIN COVERS OF CURVES⊕

M

Cτ i ⊕pi ∈∆X1 ∩∆X2

Ct3g−3∼=M

pi ∈Sing(C)

Cti ⊕

M

M

15

H 1 (Cjν , TCjν (−Dj ))Cj ⊂C

H 1 (Cjν , TCjν (−Dj ))Cj ⊂C

where Dj is the sum of the preimages of the nodes under the normalization map.
There is a natural map from the versal deformation space of a Z22 curve to thatof the underlying stable curve, given by ti = α2i if ti = 0 is the locus where theexceptional node pi ∈ ∆cX1 ∪ ∆cX2 persists and ti = αi otherwise. Similarly to thediscussion in Section 1.2 of [BCF04], we can blow up along all of the exceptional
components and extend η1 , η2 using only those in ∆cX1 and ∆cX2 respectively.
This description makes the rest of the work in Section 6 of [FL10] relatively
straightforward to generalize. Set X∆ to be the quasi-stable curve with exceptional
nodes ∆cX1 ∪ ∆cX2 .
Definition 5.2 (Elliptic tail). Let X be a quasi-stable curve, a component Cj isan elliptic tail if it has arithmetic genus 1 and intersects the rest of the curve in asingle point. That point is called an elliptic tail node, and any automorphism of X
that is the identity away from Cj is an elliptic tail automorphism.
Proposition 5.3. Let σ ∈ Aut(X) be an automorphism in genus g ≥ 4. Then σ
3g−3
acts on Cα
as a quasi-reflection if and only if X∆ has an elliptic tail Cj suchthat σ is the elliptic tail involution with respect to Cj .
The proof of this proposition follows from the proof of [FL10, 6.6]. It implies thatthe smooth locus of Z22 Mg is the locus where the automorphism group is generated
by elliptic tail involutions. Now that we have determined the smooth locus, wemust determine the non-canonical locus. If G acts on a vector space V by quasireflections, then V /G ∼= V , so we let H ⊂ Aut(X1 , X2 , η1 , η2 , β1 , β2 ) be generated
by automorphisms acting as quasi-reflections, that is elliptic tail involutions. Then
3g−3
Cα
/H ∼
= Cν3g−3 where νi = α2i if pi is an elliptic tail node and νi = αi else.
3g−3
On Cν , the automorphisms act without quasi-reflections, so the Reid–ShepherdBarron–Tai criterion can be applied.
Theorem 5.4 (Reid–Shepherd-Barron–Tai Criterion [Rei80, Tai84]). Let V be avector space of dimension d, G ⊂ GL(V ) a finite group and V0 ⊂ V the open set
where G acts freely. Fix g ∈ G, and let g be conjugate to a diagonal matrix withζ ai for i = 1, . . . , d on the diagonal for ζ a fixed mth root of unity and 0 ≤ ai < m.
Pd
1
If for all g and ζ, we have m
i=1 ai ≥ 1, then any n-canonical form on V0 /G
extends holomorphically to a resolution Vg/G.

It is straightforward to check that for g ≥ 4, we have a noncanonical singularity
if X∆ has an elliptic tail Cj with j-invariant 0 such that η1 , η2 are both trivial on
Cj . This goes as in [FL10], where the action of σ is determined to be as the squareof a sixth root of unity in two coordinates for an automorphism of order 6 and asa cube root of unity in those two coordinates for an order 3 element. Both of thesefail the Reid–Shepherd-Barron–Tai criterion.
Now, assuming that we have a noncanonical singularity, then we have an automorphism σ of order n failing Reid–Shepherd-Barron–Tai. Our goal is to classify
such things, and eventually show that only the examples above exist. Let pi0 ,
pi1 = σ(pi0 ),. . ., σ m−1 (pi0 ) = pim−1 be distinct nodes of the stabilization, C, which

16

CHARLES SIEGEL

are permuted by σ and not elliptic tail nodes. The action on the subspace corresponding to these nodes is then given by a matrix


0 c1

 ..
..

 .
.


 0
cm−1 
cm 0 . . .
0Qfor some complex numbers cj . We call the pair (X, σ) singularity reduced if m
j=1 cjis not 1.
By [HM82] and [Lud10, Proposition 3.6], we know that there is a deformation′′
X ′ of X such thatevery cycle
Qm σ deforms to σ , an automorphism of X such thatof nodes with j=1 cj = 1 is smoothed and the action of σ and σ ′ on Cν3g−3 andCν3g−3have the same eigenvalues. In particular, one will satisfy Reid–Shepherd′
Barron–Tai if and only if the other does.
Now, we fix a pair (X, σ) that is singularity reduced and fails the Reid–ShepherdBarron–Tai inequality. On C, the stabilization, the induced automorphism σC must
either fix all of the nodes or else exchange a single pair of them. We look at what
the action does on the components. In [FL10, Proposition 6.9] the proof of [Lud10,
Proposition 3.8] is adapted to the situation of Z2 Mg , and this proof goes throughverbatum, telling us that the action fixes each component of the stable model. Now,
we recall that
Theorem 5.5 ([HM82, Page 36]). Assume that (X, σ) is singularity reduced andfails the Reid–Shepherd-Barron–Tai inequality. Denote by ϕj the induced automorphism on the normalization Cjν of the irreducible component Cj of the stabilization
C of X. Then the pair (Cjν , ϕj ) is one of the following:(1) Cjν rational, and the order of ϕj is 2 or 4,(2) Cjν elliptic, and the order of ϕj is 2,4,3 or 6,(3) Cjν hyperelliptic of genus 2, and ϕj is the hyperelliptic involution,(4) Cjν bielliptic of genus 2, and ϕj is the associated involution,(5) Cjν hyperelliptic of genus 3, and ϕj is the hyperelliptic involution, and(6) Cjν arbitary, and ϕj is the identity.
As pointed out in [Lud10, Proposition 3.10], this rules out the possibility of nodes
being exchanged, so the automorphism must fix all nodes and all components on
the stable curve.
Proposition 5.6 ([FL10, Proposition 6.12]). In the same situation as above, set
Dj to be the divisor of the marked points on Cjν that are preimages of nodes. Thenthe triples (Cjν , Dj , ϕj ) are one of the following types, and the contribution to theleft hand side of the Reid–Shepherd-Barron–Tai inequality are at least wj :(1) Cjν arbitary, ϕj is the identity, and wj = 0,
(2) Elliptic tails: Cjν is elliptic, D = p+
1 which is fixed by ϕj , ϕj has order
2,3,4 or 6, and wj is, respectively, 0, 31 , 12 and 31 .
+(3) Elliptic ladder: Cjν is elliptic and D = p+
1 + p2 , with both points fixed, theautomorphism is of order 2, 3, or 4 and wj is, respectively, 21 , 32 , and 34(4) Hyperelliptic tail: Cjν has genus 2, ϕj is the hyperelliptic involution, and1
D j = p+1 fixed by ϕj . Then wj = 2 .

THE MODULI OF KLEIN COVERS OF CURVES

17With a bit of case by case work, essentially [FL10] Propositions 6.13, 6.14, 6.15
and 6.16, we can see that hyperelliptic tails, elliptic ladders, and elliptic tails oforder 4 do not occur, and that there must, in fact, be at least one elliptic tail oforder 3 or 6, giving us our restrictions on the curve. Now, we look to the line
bundles. Because the automorphism must pull back the line bundle to itself on theelliptic curve, it must be trivial on the elliptic tail, and this must hold for bothof the Prym line bundles. Thus, if we start with (X, σ) failing Reid–ShepherdBarron–Tai, then we can deform to a singularity reduced pair (X ′ , σ ′ ) such that
the Reid–Shepherd-Barron–Tai value is constant. The pair (X ′ , σ ′ ) must have an
elliptic tail with j invariant 0, the automorphism must be of order 3 or 6, and η1 , η2
must both be trivial along it. Thus:
Proposition 5.7. Fix g ≥ 4. A point (X1 , X2 , η1 , η2 , β1 , β2 ) ∈ Z22 Mg is a noncanonical singularity if and only if X∆ has an elliptic tail Cj with j-invariant 0
and η1 |Cj ∼= OCj .
= η2 |Cj ∼i,reg. We want
Proof of Theorem 5.1. Let ω be a pluricanonical form on Z22 Mg
to show that it lifts to a desingularization of some neighborhood of any pointi(X1 , X2 , η1 , η2 , β1 , β2 ) ∈ Z22 Mg . Because this can be done for canonical singularities, we assume that (X1 , X2 , η1 , η2 , β1 , β2 ) is a general non-canonical singularity, and thus X∆ = C1 ∪p C2 where (C1 , p) ∈ Mg−1,1 and (C2 , p) ∈ M1,1 withj(C2 ) = 0. We also assume that η1 |C2 ∼= OC2 and ηi |C1 are two arbitrary= η2 |C2 ∼line bundles on C1 , so that we are on a hypersurface in ∆g−1,g−1 . We consider theipencil φ : M1,1 → Z22 Mg given by φ(C, p) = C1 ∪p C and line bundles ηi′ trivial on
C and isomorphic to ηi |Ci on Ci . As φ(M1,1 ) does not intersect the ramification
locus, then just as in [HM82] pages 41-44, we can construct an open neighborhoodiof the pencil, S, such that the restriction of Z22 Mg → Mg to S is an isomorphismand every pluricanonical form on the smooth locus extends to a resolution Ŝ of
S. For the arbitrary case, with more than one node, ω will extend locally to adesingularization, just as in [Lud10, Theorem 4.1].
iThen, Theorem 5.1 in fact implies the same result for V4 Mg . This is becauseii

Z22 Mg → V4 Mg is a quotient by PSL2 (F2 ). The action is free except for along∆I,III ∪ ∆III,I ∪ ∆diag
III,III , where the stabilizer of a point is Z/2Z. Looking at theReid–Shepherd-Barron–Tai criterion for m = 2, we find that either the pluricanonical forms extend or we have a quasi-reflection, in which case the pluricanonicalforms will also extend. So, either way, we can see that what we get are the inii,reg, K ⊗ℓ ) ∼variants: H 0 (V4 Mg= H 0 (Z2 Mg , K ⊗ℓ )PSL2 (F2 ) , and so, because we can
2ialso do this for partial resolutions of Z22 Mg , we can do this for any resolutioniV\
4 Mg .
We conclude with a statement about the birational geometry of these moduli
spaces, justified by the above
i

Theorem 5.8. ForPany g, V4 Mg has general type if there exists a single effective
divisor D ≡ aλ − T b∆T ∆T where T runs over all boundary components, suchaaathat all the ratios baT are less than 13
2 and the ratios bII,III,III , b1,g−1,1:g−1 , b1,1,1 ,
a
a
a
a
13
bg−1,g−1,g−1 , b1,1:g−1,1:g−1 , bg−1,1:g−1,1:g−1 , and b1:g−1,1:g−1,1:g−1 are less than 3 .

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CHARLES SIEGELThis allows us to begin computing the classes of divisors on the Klein moduli
space to determine its Kodaira dimension, and thus begin the study of the birational
geometry of these spaces.
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(Charles Siegel) Kavli Institute for the Physics and Mathematics of the Universe(WPI), Todai Institutes for Advanced Study, the University of Tokyo
E-mail address, Charles Siegel: charles.siegel@ipmu.jp
URL: http://db.ipmu.jp/member/personal/2754en.html