2.1 $\boldsymbol {\mu }_6$ -covers of ${\mathbb P}^1$

Let C be a nonsingular complex-projective curve endowed with a $\boldsymbol {\mu }_6$ -action that has 12 fixed points, is free elsewhere, and is such that its orbit space P is a copy of ${\mathbb P}^1$ . The discriminant of $C\to P$ is then of the form $5D_C$ , with $D_C$ a $12$ -element subset of P, considered as a reduced divisor and Riemann–Hurwitz shows that C will have genus $25$ . Note that the pair $(P,D_C)$ determines C up to a covering transformation (in $\boldsymbol {\mu }_6$ ). We recall how this gives rise to a point in a nine-dimensional ball quotient, which happens to be the one of highest dimension that appears in the Deligne–Mostow list.

Since we find it convenient to make a distinction between what depends on the complex structure and what only depends on the underlying topology, we also fix a closed oriented surface $\Sigma $ for of genus 25 endowed with a $\boldsymbol {\mu }_6$ -action of the type above, so with $12$ fixed points and acting freely elsewhere. We denote by $\pi :\Sigma \to S$ the formation of its $\boldsymbol {\mu }_6$ -orbit space (a smooth $2$ -sphere), and by $D\subset S$ its set of critical values. Then any complex structure on S (which will make it a copy of the Riemann sphere) determines a unique one on $\Sigma $ for which the $\boldsymbol {\mu }_6$ -action and $\pi $ are holomorphic, yielding a $\boldsymbol {\mu }_6$ -curve as above with discriminant $5D$ . We also note that a diffeomorphism of S which preserves D lifts to S and does so almost uniquely: two such lifts will lie in the same $\boldsymbol {\mu }_6$ -orbit.

2.2 Hodge structure of a $\boldsymbol {\mu }_6$ -cover of ${\mathbb P}^1$

We first note that the standard embedding $\chi : \boldsymbol {\mu }_6\subset {\mathbb C}^\times $ and its complex conjugate are the only two primitive characters of $\boldsymbol {\mu }_6$ . The group $\boldsymbol {\mu }_6$ acts on $H^1(\Sigma; {\mathbb C})$ and decomposes the latter into its character spaces: $H^1(\Sigma; {\mathbb C})=\oplus _{i\in {\mathbb Z}/6} H^1(\Sigma; {\mathbb C})_{\chi ^i}$ . The intersection pairing gives rise to a nondegenerate Hermitian form on $H^1(\Sigma; {\mathbb C})$ defined by $(\alpha , \beta )\mapsto \sqrt {-1}\int _\Sigma \alpha \wedge \overline \beta $ . It has signature $(25,25)$ , and for this form, the above decomposition into character spaces is perpendicular. In the presence of a complex structure, it is also compatible with the Hodge structure. For example, the decomposition

(2.1) $$ \begin{align} H^1(C; {\mathbb C})_{\chi}=H^1(C, \Omega_C)_{\chi}\oplus \overline{H^1(C, \Omega_C)_{\overline\chi}} \end{align} $$

is perpendicular with respect to this Hermitian form with the first summand being positive definite of dimension 1 and the second summand negative definite of dimension 9. The group of covering transformations $\boldsymbol {\mu }_6$ acts trivially on ${\mathbb P} (H^1(\Sigma; {\mathbb C})_\chi )$ and so this projective space only depends on the pair $(S,D)$ ; we therefore denote it by ${\mathbb P}(S,D)$ . Its group $\operatorname {PU} (H^1(\Sigma; {\mathbb C})_\chi )$ of projective unitary transformations, which we denote for a similar reason by $\operatorname {PU}(S,D)$ , is isomorphic to $\operatorname {PU} (1,9)$ . The positive definite complex lines in $H^1(\Sigma; {\mathbb C})_\chi $ make up an open complex ball ${\mathbb B}(S,D)\subset {\mathbb P}(S,D)$ on which $\operatorname {PU} (S,D)$ acts transitively with compact stabilizers; indeed, this ball is the symmetric domain associated with $\operatorname {PU} (S,D)$ . If there is no risk of confusion, we simply write ${\mathbb B}$ for ${\mathbb B}(S,D)$ .

As we observed above, a complex structure on S determines one on $\Sigma $ , turning it into a compact Riemann surface C with $\boldsymbol {\mu }_6$ -action. The Hodge decomposition (2.1) then defines an element, the Hodge point, of ${\mathbb B}$ . The main result of Deligne and Mostow [Reference Deligne and Mostow3], [Reference Mostow6] implies that this Hodge point is a complete invariant of the relative isotopy class of this complex structure.

2.3 Action of a mapping class group

To make this last assertion precise, let $\operatorname {Diff}^+(S,D)$ denote the group of orientation-preserving diffeomorphisms of S which preserve D. Since this group $\operatorname {Diff}^+(S,D)$ acts on $\Sigma $ up to covering transformations, it gives rise to a well-defined action on ${\mathbb P}(S,D)$ via $\operatorname {PU} (S,D)$ (and so preserves ${\mathbb B}$ ). This action factors through its connected component group that we shall denote $\operatorname {Mod}(S,D)$ and which we readily recognize as the spherical braid group on 12 strands. That group has as a distinguished conjugacy class the set of simple braids. We recall that a simple braid is given by an (unoriented) arc $\delta $ in S which connects two points of D and has no points of D in its interior: the associated simple braid $T_\delta \in \operatorname {Mod}(S,D)$ has its support in a regular neighborhood of $\delta $ and lets $\delta $ make a half-turn in a counterclockwise direction. These simple braids generate $\operatorname {Mod}(S,D)$ . In fact, if we enumerate the points of D by $\{p_i\}_{i\in {\mathbb Z}/12}$ and let $\delta _i$ connect $p_i$ with $p_{i+1}$ in such a way as that their juxtaposition produces an embedded circle in S, then $T_{\delta _1},\dots , T_{\delta _{10}}$ already generate $\operatorname {Mod}(S,D)$ . It follows from the work of Deligne–Mostow that the image of $\operatorname {Mod}(S,D)$ in $\operatorname {PU} (S,D)$ is an arithmetic group, which therefore acts properly discretely on ${\mathbb B}$ with finite covolume. We denote this image group by $\Gamma \subset \operatorname {PU} (S,D)$ .

2.4 A lattice over the Eisenstein ring

In order to make the action of a simple braid on ${\mathbb B}$ somewhat more explicit, we make some general observations first. Write $\tau $ for the primitive sixth root of unity with a positive imaginary part, $e^{\pi \sqrt {-1}/3}$ , and regard it as a generator of $\boldsymbol {\mu }_6$ . As a complex number, it satisfies $\tau ^2=\tau -1$ , but this identity is of course not valid in the group ring ${\mathbb Z}[\boldsymbol {\mu }_6]$ . Given a ${\mathbb Z}[\boldsymbol {\mu }_6]$ -module V, let us denote by $V^\circ $ (resp. $V_\circ $ ) the maximal submodule (resp. quotient module) on which $\tau $ satisfies the identity $\tau ^2=\tau -1$ . For example, $V_\circ $ is the quotient of V by the subgroup of $v\in V$ with nontrivial $\boldsymbol {\mu }_6$ -stabilizer. Then $V^\circ $ and $V_\circ $ become modules over the ring of Eisenstein integers ${\mathcal E}:={\mathbb Z}+{\mathbb Z}\tau \subset {\mathbb C}$ , the ring of integers the cyclotomic field ${\mathbb Q}(\sqrt [3]{1})$ . Note that ${\mathbb R}\otimes V^\circ \to {\mathbb R}\otimes V_\circ $ is an isomorphism and the latter can be identified with the quotient of ${\mathbb R}\otimes V$ by the sum of the fixed point subspaces of $\tau ^2$ and $\tau ^3$ .

Since ${\mathcal E}$ is a principal ideal domain, any finitely generated torsion-free ${\mathcal E}$ -module is in fact free. So, if V is a finitely generated free ${\mathbb Z}$ -module, then both $V_\circ $ and $V^\circ $ are finitely generated free ${\mathcal E}$ -modules. If we are in addition given a $\boldsymbol {\mu }_6$ -invariant antisymmetric pairing $a: V\times V\to {\mathbb Z}$ , then this gives rise to the $\frac {1}{2}{\mathcal E}$ -valued Hermitian form on $V^\circ $ defined by

$$\begin{align*}h_a(x,y):=\tfrac{1}{2}\tau a(x, y)-\tfrac{1}{2}a(x,\tau y) \end{align*}$$

(the justification for the perhaps somewhat unexpected normalization factor $\tfrac {1}{2}$ will be given below). Note that $h_a(x,x)=-\tfrac {1}{2}a(x,\tau x)$ and $h_a(x,y)-\overline {h_a(x,y)}=\tfrac {1}{2}(\tau -\overline \tau ) a(x, y)=\tfrac {1}{2}\theta a(x,y)$ , where

$$\begin{align*}\theta:=\tau-\overline\tau=-1+2\tau=\sqrt{-3} \end{align*}$$

is the first purely imaginary number in ${\mathcal E}$ . We apply this to the ${\mathbb Z}[\boldsymbol {\mu }_6]$ -module $H_1(\Sigma )$ . We find that $H_1(\Sigma )^\circ $ and $H_1(\Sigma )_\circ $ are free ${\mathcal E}$ -modules of rank 10. The intersection pairing on $H_1(\Sigma )$ is nondegenerate on $H_1(\Sigma )^\circ $ and defines via the above prescription on this lattice the ${\mathcal E}$ -valued Hermitian form

$$\begin{align*}h(x,y):=\tfrac{1}{2}\tau (x\cdot y)-\tfrac{1}{2}(x\cdot\tau y). \end{align*}$$

We will write L for $H_1(\Sigma )^\circ $ endowed with this form.

Let us see what this induces on the preimage $\pi ^{-1}\delta $ of an arc $\delta $ as above. We now assume $\delta $ oriented and let $\tilde \delta $ be a lift of $\delta $ over $\pi $ . Then $c_\delta :=(1-\tau ^3)\tilde \delta $ is an embedded oriented circle (hence a $1$ -cycle) and meets its $ \tau $ -translate in two points, each with multiplicity $1$ . In other words, $c_\delta \cdot \tau c_\delta =2$ . Now, $H_1(\pi ^{-1}\delta )^\circ $ has the ${\mathcal E}$ -generator $a_\delta :=(1+\tau )(1-\tau ^3)\tilde \delta =(1+\tau )c_\delta $ (which is unique up multiplication by an element of $\boldsymbol {\mu }_6$ ), and the above formula shows that $h(a_\delta ,a_\delta )=\tfrac {1}{2}.-6=-3$ . A similar computation shows that $h(a_{\delta _i},a_{\delta _{i+1}})\in \boldsymbol {\mu }_6\theta $ . So $h|L\times L$ takes values in $\theta {\mathcal E}$ , so that $\theta ^{-1}h$ defines a skew-Hermitian form

$$\begin{align*}h':=\theta^{-1}h: L\times L\to {\mathcal E}. \end{align*}$$

Since we have the freedom of multiplying each $a_\delta $ with an element of $\boldsymbol {\mu }_6$ , we can choose $a_{\delta _1},\dots , a_{\delta _{10}}$ in such a manner that it is an ${\mathcal E}$ -basis of L with

$$\begin{align*}h(a_{\delta_i}, a_{\delta_j})= \begin{cases} -3, &\text{ when }j=i,\\ \pm \theta, &\text{ when }j=i\pm 1,\\ 0, &\text{ otherwise.}\\ \end{cases} \end{align*}$$

This also shows that L is isomorphic to the lattice thus denoted in [Reference Allcock2] (this explains the coefficient $\tfrac {1}{2}$ in our definition of h) and if we replace h by $-h$ , it becomes isomorphic to the lattice denoted $\Lambda $ in [Reference Heckman and Looijenga4]. In [Reference Allcock2], L is in fact constructed from the even ${\mathbb Z}$ -lattice $\mathbf {E}_8\perp \mathbf {E}_8\perp \mathbf {U}\perp \mathbf {U}$ . Going in the opposite direction, this amounts to the statement that we recover this lattice if we take the underlying abelian group and the even symmetric pairing on it defined by the real part of h multiplied with $-\frac {2}{3}$ .

2.5 Mirror hyperballs

The simple braid $T_\delta $ acts in L as the unitary transformation $s_\delta $ defined by

$$\begin{align*}s_\delta(x)=x+\tfrac{1}{3}(1-\tau^2)h(x,a_\delta )a_\delta= x+\tau h'(x,a_\delta)a_\delta. \end{align*}$$

It is what is called in [Reference Allcock2] a triflection in ${\mathcal E} a_\delta $ : it multiplies $a_\delta $ with the third root of unity $\tau ^2$ and is the identity on the orthogonal complement of $a_\delta $ . It indeed preserves L, as follows from the fact that $h'$ takes its values in ${\mathcal E}$ .

Poincaré duality identifies $H^1(\Sigma )^\circ $ with $H_1(\Sigma )^\circ =L$ , and we use this to identify ${\mathbb P}(S,D)$ with ${\mathbb P}({\mathbb C}\otimes _{\mathcal E} L)$ . In particular, $T_\delta $ acts on ${\mathbb B}$ as a triflection. So its fixed point set ${\mathbb B}^{T_\delta }$ is a hyperball (to which we shall also refer as a mirror in ${\mathbb B}$ ). Since the $T_\delta $ make up a single conjugacy class in $\operatorname {Mod}(S,D)$ , the group $\operatorname {Mod}(S,D)$ permutes the mirror hyperballs transitively.

The group of unitary transformations of the Hermitian ${\mathcal E}$ -module L acts properly discretely on ${\mathbb B}$ with kernel its center $\boldsymbol {\mu }_6$ . The group $\Gamma $ is a priori contained in the quotient of this group by its center (which is an arithmetic subgroup of $\operatorname {PU}(S,D)$ ), but, as Allcock has shown (see [Reference Allcock2, Th. 5.1]), is in fact equal to it. So $\Gamma $ acts properly discontinuously on ${\mathbb B}$ and the Baily–Borel theory asserts that ${\mathbb B}_\Gamma :=\Gamma \backslash {\mathbb B}$ has the structure of a quasi-projective orbifold that can be completed to a normal projective variety by adding a finite set of cusps (in this case, just one, see below). The image of the Hodge point in this ball quotient ${\mathbb B}_\Gamma $ is a complete invariant of the isomorphism type of the $\boldsymbol {\mu }_6$ -curve C, or equivalently, of the pair $(S,D)$ with the given complex structure.

The $\operatorname {Mod}(S,D)$ -centralizer of $T_\delta $ acts on the mirror ${\mathbb B}^{T_\delta }$ with image an arithmetic subgroup and the resulting ball quotient of ${\mathbb B}^{T_\delta }$ appears in the Deligne–Mostow list as the one defined by the sequence $(\frac {2}{12},\frac {1}{12}, \frac {1}{12},\dots ,\frac {1}{12})$ .

The group $\Gamma $ is as a quotient of $\operatorname {Mod}(S, D)$ obtained by imposing the relation $T_\delta ^3=1$ . The mirrors are locally finite on ${\mathbb B}$ , and since they are transitively permuted by $\Gamma $ , they define an irreducible totally geodesic hypersurface ${\mathcal D}$ in ${\mathbb B}_\Gamma $ that we shall call the confluence locus. We write ${\mathbb B}^\circ $ for the complement of the union of the mirrors (an open subset of ${\mathbb B}$ ). So this is the preimage of ${\mathbb B}_\Gamma ^\circ \smallsetminus {\mathcal D}$ .

2.6 The Deligne–Mostow isomorphism

Let ${\mathcal M}(12)$ denote the $\operatorname {PGL}(2, {\mathbb C})$ -orbit space of the configuration space of $12$ -element subsets of ${\mathbb P}^1$ (this is also equal to the ${\mathfrak S}_{12}$ -orbit space of ${\mathcal M}_{0,12}$ ). The above construction defines a homomorphic map

$$\begin{align*}{\mathcal M}(12)\to {\mathbb B}_\Gamma. \end{align*}$$

If we think of a $12$ -element subset of ${\mathbb P}^1$ as a reduced degree $12$ -divisor, then the above map extends to the locus ${\mathcal M}(12)^{\mathrm {st}}$ of Hilbert–Mumford stable positive degree $12$ -divisors, that is, divisors for which the multiplicities are at most $5$ . The Deligne–Mostow theory asserts that this extension is in fact an isomorphism

$$\begin{align*}{\mathcal M}(12)^{\mathrm{st}}\xrightarrow{\cong} {\mathbb B}_\Gamma \end{align*}$$

that takes ${\mathcal M}(12)$ onto ${\mathbb B}_\Gamma ^\circ $ . It even extends to an isomorphism of their natural compactifications: the GIT compactification on the left and the Baily–Borel compactification on the right. Both are one-point compactifications, with the added point on the left being represented by a $6$ -divisible divisor on ${\mathbb P}^1$ whose support consists of two points. In particular, there is just one cusp.

3.1 $\mathbf {g^1_3}$ ’s on a genus 4 curve

Let X be a smooth connected complex-projective curve of genus $4$ , and let P be a pencil of degree $3$ on X (in other words, a $g^1_3$ ). If P has no base point, then the pencil defines a morphism $X\to P$ of degree $3$ and by Riemann–Hurwitz, the discriminant divisor $D(X, P)$ of this morphism is of degree $12$ . Note that its multiplicities will be $\le 2$ . In case P has a base point, then the residual base-point-free pencil $P'$ will have degree $2$ and this makes X hyperelliptic with $P'$ being the hyperelliptic pencil. We will see that it is then natural to define the discriminant divisor of P to be the degree $12$ -divisor $D(X, P)$ , that is (via the identification of P with $P'$ ), the sum of the discriminant divisor of $X\to P'$ (the image of the Weierstraß points, so of degree 10) plus twice the image in $P'$ of the fixed point P. So its multiplicities are then $\le 3$ .

Recall that if X is nonhyperelliptic, then a canonical image of X in ${\mathbb P}^3$ is the transversal intersection of a cubic surface with a quadric Q surface, the latter being unique and either smooth (so isomorphic to ${\mathbb P}^1\times {\mathbb P}^1$ ) or a quadric cone. In the first case, the projections on the two factors give us two pencils of degree $3$ . In the second case, projection away from the vertex of the cone also yields a pencil of degree $3$ ; the line bundle in question is then an effective (even) theta characteristic. It is well known that this exhausts all the $g^1_3$ ’s in genus 4. For future reference, we state this as a lemma (and only sketch the proof).

3.2 A moduli stack of genus 4 curves with a $\mathbf {g^1_3}$

We interpret this in terms of moduli spaces. Pairs $(X, P)$ as above define a Deligne–Mumford stack that we shall denote by ${\mathcal M}_4(g^1_3)$ . It comes with an involution which assigns to $(X,P)$ the residual pair $(X,P')$ that is characterized by the property that $P+P'$ lies in the canonical system. The forgetful morphism ${\mathcal M}_4(g^1_3)\to {\mathcal M}_4$ evidently factors through the orbit space of this involution. Recall that the moduli space ${\mathcal M}_4$ contains the Theta-null locus ${\mathcal M}_4^\Theta $ (the locus for which the curve admits an even effective theta characteristic) as a substack of codimension one and that this substack ${\mathcal M}_4^\Theta $ in turn contains the hyperelliptic locus ${\mathcal H}_4$ as a substack of codimension one.

It follows from Lemma 3.1 that ${\mathcal M}_4(g^1_3)\to {\mathcal M}_4$ is over ${\mathcal M}_4\smallsetminus {\mathcal H}_4$ a double covering with the covering transformation being induced by the residual involution (and so ramifies over ${\mathcal M}_4^\Theta \smallsetminus {\mathcal H}_4$ ). This lemma also shows that the preimage ${\mathcal H}_4(g^1_3)$ of ${\mathcal H}_4$ in ${\mathcal M}_4(g^1_3)$ is the universal hyperelliptic curve of genus $4$ (so that will be a divisor). The strict transform of ${\mathcal M}_4^\Theta $ in ${\mathcal M}_4(g^1_3)$ is another divisor ${\mathcal M}_4(g^1_3)^\Theta $ on ${\mathcal M}_4(g^1_3)$ which meets ${\mathcal H}_4(g^1_3)$ in the locus parametrizing the Weierstraß points (this locus is geometrically connected, since the monodromy is transitive on the Weierstrass points).

Our goal is to realize ${\mathcal M}_4(g^1_3)$ as an open subset of a quotient of ${\mathbb B}$ relative some finite index subgroup of $\Gamma $ . We determine this subgroup in two ways: first in a purely topological manner and subsequently more in the spirit of algebraic geometry as a level structure on the $\boldsymbol {\mu }_6$ -curve $\Sigma $ .

3.3 The approach via a classification of coverings

This begins with addressing the question of how many topological types of smooth connected degree three coverings of the $2$ -sphere S exist that have the $12$ -element subset $D\subset S$ as reduced discriminant. Such a covering is given by its restriction to $S\smallsetminus D$ (which is then unramified). So if we choose a base point $o\in S\smallsetminus D$ , then it can be given by a group homomorphism $\rho : \pi _1(S\smallsetminus D, o)\to {\mathfrak S}_3$ which assigns to a simple loop around a point of D a transposition. The image of $\rho $ must be a transitive subgroup of ${\mathfrak S}_3$ and hence must be all of ${\mathfrak S}_3$ . Two such epimorphisms define isomorphic coverings if and only if they differ by an inner automorphism of ${\mathfrak S}_3$ . So if we let $\operatorname {Epi}'(\pi _1(S\smallsetminus D, o),{\mathfrak S}_3)$ stand for the set of group of surjective homomorphisms $\pi _1(S\smallsetminus D, o)\to {\mathfrak S}_3$ that take simple loops to transpositions, then the set of topological types is naturally identified with

$$\begin{align*}R(S,D):= {\mathfrak S}_3\backslash \operatorname{Epi}'(\pi_1(S\smallsetminus D, o),{\mathfrak S}_3). \end{align*}$$

We determine its number of elements. We choose smooth arcs $\{\gamma _i\}_{i\in {\mathbb Z}/12}$ from o to the distinct points of D that only meet at o and depart from there along rays in $T_oS$ in a counterclockwise order. This means that the associated simple loop associated with $\gamma _i$ defines a $c_i\in \pi _1(S\smallsetminus D, o)$ so that $c_{11}\cdots c_0=1$ (we read composition of loops from right to left) and $c_1, \dots , c_{11}$ are free generators. Then a $\rho $ as above is given by its values on $c_1, \dots , c_{11}$ . Note that $\rho (c_1), \dots , \rho (c_{11})$ is a sequence in the 3-element set of transpositions $\{(12), (23), (31)\}$ of ${\mathfrak S}_3$ that cannot be nonconstant, but can otherwise be arbitrary. So there are $3^{11}-3$ such $\rho $ . The action of ${\mathfrak S}_3$ on this set by conjugation is free and so

$$\begin{align*}\#R(S,D)=(3^{11}-3)/6=(3^{10}-1)/2=3^9+3^8+\cdots +1. \end{align*}$$

This number is also equal to $\#{\mathbb P}^9({\mathbb F}_3)$ . We shall see that this is not a coincidence.

Evidently, the group of automorphisms of $\pi _1(S\smallsetminus D, o)$ which preserves the simple loops acts on $R(S,D)$ . The group $\operatorname {Mod}(S,D)$ maps to the outer automorphism group of $\pi _1(S\smallsetminus D, o)$ . Since we have divided out by ${\mathfrak S}_3$ -conjugation, this gives us a genuine action of $\operatorname {Mod}(S,D)$ on $R(S,D)$ .

Fix a degree 3 covering $\Sigma '\to S$ as above and denote by $r=r(\Sigma '/S)\in R(S,D)$ the associated element and denote by $\Gamma '\subset \Gamma $ its $\Gamma $ -stabilizer. It follows from Lemma 3.2 that $\Gamma '$ is a subgroup $\Gamma $ of index $(3^{10}-1)/2$ . By construction, the isomorphism type of a connected smooth degree $3$ covering of a copy of ${\mathbb P}^1$ with reduced discriminant of degree $12$ determines a $\Gamma '$ -orbit in ${\mathbb B}^\circ $ . So, if ${\mathcal M}_4(g^1_3)^\circ $ stands for the open subset of ${\mathcal M}_4(g^1_3)$ that parametrizes pairs $(X, P)$ for which P has precisely $12$ nonreduced members (in other words, for which X is nonhyperelliptic and for which $X\to P$ has reduced discrimininant), then we find the following corollary.

In order to get all of ${\mathcal M}_4(g^1_3)$ , we must allow the discriminant divisor D to become nonreduced by letting two of its points coalesce. If we let this happen in the interior of a closed disk $\Delta \subset P$ which contains no other points of D, then the covering over its boundary $\partial \Delta $ is connected (the monodromy is of order 3) or is trivial. In the first case, this confluence creates a point of total ramification and we still have defined a genus $4$ covering of S. In the last case, the confluence creates an ordinary double point of the covering. This we will therefore not allow, except for one particular situation that we should not throw out: if the covering has in addition a section over $P\smallsetminus \Delta $ , then the degeneration will be the union of a smooth double covering $X'\to P$ (whose discriminant is $D\cap (P\smallsetminus \Delta )$ ) and a component $\tilde P$ which maps isomorphically onto P and meets $X'$ transversally over z. So this amounts to specifying a hyperelliptic curve of genus $4$ and a point on that curve; in other words, an element of ${\mathcal H}_4(g^1_3)$ . This happens, for example, when $\rho (c_0)=\rho (c_1)\not =\rho (c_3)=\dots =\rho (c_{11})$ , and we take for $\Delta $ a thin regular neighborhood $\gamma _0\cup \gamma _1$ .

We can state this in terms of the $\Gamma $ -action on $R(S,D)$ . The confluence of two points along an arc $\delta $ as above represents a triflection in $\Gamma $ . It may or may not act trivially on $R(S,D)$ . If the action is nontrivial, then this confluence creates a point of total ramification and we still have defined a genus $4$ covering of S. Otherwise, this creates an ordinary double point. These three cases translate into saying that the stabilizer $\Gamma '$ has three orbits in the collection of mirrors. Equivalently, the preimage of the confluence divisor ${\mathcal D}$ in ${\mathbb B}_\Gamma $ under the natural map

$$\begin{align*}{\mathbb B}_{\Gamma'}\to {\mathbb B}_\Gamma \end{align*}$$

has three irreducible components ${\mathcal D}_{\mathrm {rm}}, {\mathcal D}_{\mathrm {sg}},{\mathcal D}_{{\mathcal H}}$ whose generic points are characterized topologically by the degree 3 covering acquiring, respectively, (rm) a point of total ramification, (sg) an ordinary double point whose complement is connected, and ( ${\mathcal H}$ ) an ordinary double point which disconnects. We can now improve upon Corollary 3.3 as follows.

Theorem 3.4. The above construction gives an identification

$$\begin{align*}{\mathcal M}_4(g^1_3)\cong {\mathbb B}_{\Gamma'}\smallsetminus {\mathcal D}_{\mathrm{sg}}, \end{align*}$$

thus endowing the left-hand side with an (incomplete) complex-hyperbolic metric. This identifies the preimage of ${\mathcal D}_{{\mathcal H}}$ with the universal hyperelliptic curve of genus $4$ .

For a different approach to this theorem (that ultimately yields a more precise statement), we first need to discuss level 3 structures for the Eisenstein lattice L.

3.4 The approach via a level structure

We begin with the observation that ${\mathcal E}/\theta {\mathcal E}$ is an ${\mathbb F}_3$ -vector space of dimension one on which $\tau \in \boldsymbol {\mu }_6$ acts as minus the identity (because $\tau +1=\tau ^{-1}\theta \in \theta {\mathcal E}$ ). It follows that ${\mathbb F}_3\otimes _{\mathcal E} L=L/\theta L$ is an ${\mathbb F}_3$ -vector space of rank 10 on which the skew-Hermitian form $h'$ induces a symplectic form $h^{\prime }_{{\mathbb F}_3}:={\mathbb F}_3\otimes h'$ . The following lemma generalizes an argument of Allcock (appearing in the proof of Theorem 5.2 of [Reference Allcock2]).

The covering group $\boldsymbol {\mu }_6$ acts on the 10-dimensional ${\mathbb F}_3$ -vector space ${\mathbb F}_3\otimes _{\mathcal E} L$ by scalars and so the nine-dimensional projective space ${\mathbb P}({\mathbb F}_3\otimes _{\mathcal E} L)$ only depends on the pair $(S,D)$ .

We regard L as a primitive subgroup of $H^1(\Sigma )$ , so that a one-dimensional subspace $\ell \subset {\mathbb F}_3\otimes _{\mathcal E} L$ defines (by restriction) a surjection $H_1(\Sigma )\cong \operatorname {Hom} (H^1(\Sigma ), {\mathbb Z}) \to \operatorname {Hom}(\ell , {\mathbb F}_3)=\ell ^\vee $ . This surjection is $\boldsymbol {\mu }_6$ -equivariant if we let $\boldsymbol {\mu }_6$ act on $\ell ^\vee $ via the obvious character $\boldsymbol {\mu }_6\twoheadrightarrow \boldsymbol {\mu }_2=\{\pm 1\}$ . This also yields an unramified $\ell ^\vee $ -covering $\Sigma _\ell \to \Sigma $ .

Theorem 3.7. The map which assigns to $[\ell ]\in {\mathbb P} ({\mathbb F}_3\otimes _{\mathcal E} L)$ the cover $\Sigma ^{\prime }_\ell \to S$ defines a $\operatorname {Mod}(S,D)$ -equivariant bijection

$$\begin{align*}[\ell]\in {{\mathbb P}}({\mathbb F}_3\otimes_{\mathcal E} L)\mapsto r(\Sigma^{\prime}_\ell/S) \in R(S,D). \end{align*}$$

In particular, $R(S,D)$ thus acquires the structure of the projective space of a 10-dimensional symplectic space over ${\mathbb F}_3$ on which $\Gamma $ acts through a copy of $\operatorname {Sp}_{10}({\mathbb F}_3)/\{\pm 1\}$ .

3.5 The confluence divisors

Theorem 3.7 also explains why the preimage of ${\mathcal D}\subset {\mathbb B}_\Gamma $ in ${\mathbb B}_{\Gamma '}$ has three irreducible components, namely the confluence divisors ${\mathcal D}_{\mathcal H}$ , ${\mathcal D}_{\mathrm {rm}}$ , and ${\mathcal D}_{\mathrm {sg}}$ . Let us first observe that a mirror is given by the ${\mathcal E}$ -span of $(-3)$ -vector of L and that the mod $\theta $ -reduction of the latter determines a line in ${\mathbb F}_3\otimes _{\mathcal E} L$ and hence a point in ${\mathbb P}({\mathbb F}_3\otimes _{\mathcal E} L)$ .

3.6 The strict transform of the Theta-null locus

The goal of this section is to show that the strict transform ${\mathcal M}_4(g^1_3)^\Theta $ of ${\mathcal M}_4^\Theta $ in ${\mathcal M}_4(g^1_3)$ is also locally a ball quotient in ${\mathbb B}_{\Gamma '}$ . This was actually established in the paper with Heckman [Reference Heckman and Looijenga4], and the goal of this section is to show how. The generic point of that locus parametrizes the canonical genus $4$ curves X that lie on a quadric cone $Q^0$ in a three-dimensional complex projective space and are transversally cut out on $Q^0$ by a cubic hypersurface (so not containing the vertex). The base of the cone $Q^0$ (in other words, the set of rays in $Q^0$ , lines on $Q^0$ that pass through its vertex) is a conic. We denote it by P, because the projection $X\to P$ gives us the unique $g^1_3$ on X. We assume for now that the discriminant divisor D of this projection is reduced.

The vertex of $Q^0$ is the unique singular point of $Q^0$ and is resolved by a single blowup: $\hat Q^0\to Q^0$ . Its exceptional set, which we can identify with P, has self-intersection $-2$ . It defines a section of the evident morphism $\hat Q^0\to P$ whose fibers are the rays of $Q^0$ . In other words, $\hat Q^0$ is a Hirzebruch surface whose exceptional section is P. Its Picard group is freely generated by P and the class f of fiber. The curve X on $\hat Q^0$ has class $3(P+2f)$ and hence the class of $X+P$ is $3(P+2f)+P=4P+6f$ . Since this is $2$ -divisible, it determines a double covering $E_X\to \hat Q^0$ with ramification divisor $X+P$ (Footnote ¹ ). The composite with the projection $\hat Q^0\to P$ defines an elliptic fibration for which the preimage of $P\subset \hat Q^0$ in $E_X$ defines the zero section (with self-intersection number $-1$ ). The covering transformation of the double cover $E_X\to \hat Q^0$ is the fiberwise natural involution with respect to the zero section. Note that D now also appears as the discriminant of the fibration $E_X\to P$ . Indeed, all fibers are smooth, except those over D: over each point of D, we find a nodal curve (a Kodaira fiber of type $I_1$ ) whose double point lies over the singular point of the projection $X\to P$ .

By taking the j-invariant of the fibers, we obtain a morphism

$$\begin{align*}j: P\to \operatorname{PSL}_2({\mathbb Z})\backslash {\mathbb H}^*=\big(\operatorname{PSL}_2({\mathbb Z})\backslash {\mathbb H}\big)\cup\{\infty\}=\overline{\mathcal M}_{1,1}, \end{align*}$$

where ${\mathbb H}^*:={\mathbb H}\cup {\mathbb P}^1({\mathbb Q})$ is endowed with the horocyclic topology. This morphism has degree $12$ and $D=j^*(\infty )$ . In [Reference Heckman and Looijenga4], it was observed that the D that thus appear can be characterized in ${\mathbb B}_\Gamma $ as hyperball quotient as follows. The abelianization of $\operatorname {PSL}_2({\mathbb Z})$ is naturally isomorphic with $\boldsymbol {\mu }_6$ and hence defines a ramified $\boldsymbol {\mu }_6$ -cover $\operatorname {PSL}_2({\mathbb Z})'\backslash {\mathbb H}\to \operatorname {PSL}_2({\mathbb Z})\backslash {\mathbb H}$ . This morphism extends across the cusp $\infty $ with total ramification, so that we get a $\boldsymbol {\mu }_6$ -cover

$$\begin{align*}\overline{\mathcal M}{}^{\prime}_{1,1}:=\operatorname{PSL}_2({\mathbb Z})'\backslash{\mathbb H}^*\to \operatorname{PSL}_2({\mathbb Z})\backslash {\mathbb H}^*=\overline{\mathcal M}_{1,1}. \end{align*}$$

The left-hand side can be identified with the (Eisenstein) elliptic curve ${\mathbb C}/{\mathcal E}$ whose origin is the unique point over $\infty $ and on which $\boldsymbol {\mu }_6$ acts in the standard manner. The differential on ${\mathbb C}/{\mathcal E}$ defined by $dz$ transforms under the $\boldsymbol {\mu }_6$ -action via the character $\chi ^{-1}$ and hence $d\overline z$ transforms with the character $\chi $ . So $H^1(\overline {\mathcal M}{}^{\prime }_{1,1}; {\mathbb C})_\chi $ is of Hodge type $(0,1)$ .

The $\boldsymbol {\mu }_6$ -cover $C\to P$ with discriminant D is then the pullback of the above $\boldsymbol {\mu }_6$ -cover under the j-morphism:

Since j is of degree $12$ , so is $j'$ .

Proof. This is proved in [Reference Heckman and Looijenga4], but let us give here a simpler argument. First, note that since $H^1(\overline {\mathcal M}{}^{\prime }_{1,1})=H^1(\overline {\mathcal M}{}^{\prime }_{1,1})^\circ $ , $j'{}^*$ takes values in $H^1(C)^\circ $ . If $a\in H^1(\overline {\mathcal M}{}^{\prime }_{1,1})$ is primitive, then $a \cdot \tau a=1$ and hence if we put $\varepsilon :=j'{}^*a$ , then

$$\begin{align*}h(\varepsilon,\varepsilon)=-\tfrac{1}{2} (\varepsilon, \tau \varepsilon)=-\tfrac{1}{2}.12 (a\cdot \tau a)=-6, \end{align*}$$

where we used that $j'$ is of degree $12$ . This also implies that $\varepsilon $ is primitive: if it is divisible by $\lambda \in {\mathcal E}$ : $\varepsilon =\lambda \varepsilon '$ with $\varepsilon '\in H^1(C)^\circ $ , then $\lambda \overline \lambda h(\varepsilon ' ,\varepsilon ')=-6$ . But $\lambda \overline \lambda $ is a positive integer that cannot take the value $2$ and $h(\varepsilon ', \varepsilon ')\in {\mathcal E}\theta \cap {\mathbb Q}=3{\mathbb Z}$ . This implies that $\lambda \overline \lambda =1$ , meaning that $\lambda $ is a unit.

Assertions A4 and A6 of [Reference Heckman and Looijenga4] state that any $(-6)$ -vector in L can be written as the sum of two $(-3)$ -vectors with inner product $\theta $ and that the $(-6)$ -vectors make up a single $\Gamma $ -orbit. In particular, the hyperballs in ${\mathbb B}$ defined by such vectors define an irreducible totally geodesic divisor ${\mathcal D}^\Theta $ in ${\mathbb B}_\Gamma $ . We can therefore complete Theorem 3.4 as follows (see also [Reference Heckman and Looijenga4, Th. 8.2]).