Birational geometry of moduli of curves with an $S_3$-cover

TL;DR

This paper proves that the moduli space of curves with an $S_3$-cover is of general type for odd genus $g \,\geq\, 13$, and develops tools applicable to $G$-covers for any finite group.

Contribution

It establishes the general type of the moduli space of $S_3$-covers for high odd genus and introduces versatile tools for studying $G$-cover moduli spaces.

Findings

The moduli space $\\mathcal R_{g,S_3}^{S_3}$ is of general type for $g \geq 13$ and odd.

Developed new methods for analyzing the birational geometry of $G$-cover moduli spaces.

Provided a framework applicable to arbitrary finite groups $G$.

Abstract

We consider the space $\mathcal R_{g,S_3}^{S_3}$ of curves with a connected $S_3$-cover, proving that for any odd genus $g\geq 13$ this moduli is of general type. Furthermore we develop a set of tools that are essential in approaching the case of $G$-covers for any finite group $G$.

Equations

$$H^0\left({\hat{\mathcal{R}}}_{g,G},n{K}_{{\hat{\mathcal{R}}}_{g,G}}\right)=H^0\left({\overline{\mathcal{R}}}_{g,G}^{\mathrm{reg}},n{K}_{{\overline{\mathcal{R}}}_{g,G}^{\mathrm{reg}}}\right).$$

$$H^0\left({\hat{\mathcal{R}}}_{g,G},n{K}_{{\hat{\mathcal{R}}}_{g,G}}\right)=H^0\left({\overline{\mathcal{R}}}_{g,G}^{\mathrm{reg}},n{K}_{{\overline{\mathcal{R}}}_{g,G}^{\mathrm{reg}}}\right)$$

$$K_{{\overline{\mathcal{R}}}_{g,S_3}^{S_3}}=\alpha \cdot {\overline{\mathcal{U}}}_g+\beta \cdot ({\pi }^{\ast }{\mathcal{M}}_{g,i+1}^1)+E+\gamma \cdot \lambda \in {\mathrm{Pic}}_{\mathbb{Q}}({\overline{\mathcal{R}}}_{g,S_3}^{S_3}),$$

$$\begin{array}[]{ccc}\Sigma_{1},\Sigma_{2},\dots,\Sigma_{n}&\subset&\mathsf{C}\\ &&\downarrow\\ &&C\\ &&\downarrow\\ &&S.\end{array}$$

$$\left[\mathrm{Spec}A[x^{\prime }]/{\mathbf{\mu }}_r\right]\to \mathrm{Spec}A[x]\to \mathrm{Spec}A$$

$$\left[\mathrm{Spec}\left(\frac{A[x^{\prime },y^{\prime }]}{(x^{\prime }{y}^{\prime }-a)}\right)/{\mathbf{\mu }}_r\right]\to \mathrm{Spec}\left(\frac{A[x,y]}{(xy-a^{\ell })}\right)\to \mathrm{Spec}A$$

$$\xi :(x^{\prime },y^{\prime })\mapsto (\xi x^{\prime },{\xi }^m{y}^{\prime })$$

$$\mathrm{Spec}A[x^{\prime }]\to \mathrm{Spec}A[x]\to \mathrm{Spec}A,$$

$$\mathrm{Spec}\left(\frac{A[x^{\prime },y^{\prime }]}{(x^{\prime }{y}^{\prime }-a)}\right)\to \mathrm{Spec}\left(\frac{A[x,y]}{(xy-a^r)}\right)\to \mathrm{Spec}A,$$

$${\mathrm{Aut}}_{\mathrm{Adm}}(C,F)={\mathrm{Hom}}^G(\mathcal{T}(F),G).$$

$$\mathrm{Def}(\mathsf{C};\mathrm{Sing}\mathsf{C})=\mathrm{Def}(C;\mathrm{Sing}C)=\underset{i=1}{\overset{V}{⨁}}\mathrm{Def}({\overline{C}}_i;D_i)=\underset{i=1}{\overset{V}{⨁}}{H}^1({\overline{C}}_i,T_{{\overline{C}}_i}(-D_i)).$$

$$\mathrm{Def}(\mathsf{C})/\mathrm{Def}(\mathsf{C};\mathrm{Sing}\mathsf{C})=\underset{j=1}{\overset{\delta }{⨁}}{R}_j.$$

$${\alpha }_1{\beta }_1{\alpha }_1^{-1}{\beta }_1^{-1}\cdots {\alpha }_g{\beta }_g{\alpha }_g^{-1}{\beta }_g^{-1}\cdot {\gamma }_1\cdots {\gamma }_n=1.$$

$$\varpi :{\pi }_1(C_{\mathrm{gen}},p_{\ast })\to G.$$

$${\mathrm{Hom}}_{\text{mbox}s}({\pi }_1(C),{\mathbf{\mu }}_2)\cong {\mathrm{Hom}}_{\text{mbox}s}({\pi }_1(C),T).$$

$${\mathrm{Hom}}_{\text{mbox}s}({\pi }_1(C),{\mathbf{\mu }}_3)\twoheadrightarrow {\mathrm{Hom}}_{\text{mbox}s}({\pi }_1(C),N).$$

$${\mathrm{Hom}}_{\text{mbox}s}({\pi }_1(C),{\mathbf{\mu }}_3)/(\wedge 2)\cong {\mathrm{Hom}}_{\text{mbox}s}({\pi }_1(C),N).$$

$${\overline{\mathcal{M}}}_{i,1}\times {\overline{\mathcal{M}}}_{g-i,1}\to {\overline{\mathcal{M}}}_g,$$

$$C:=(C_1\bigsqcup C_2)/(q_1\sim q_2).$$

$${\delta }_i^{\prime }:=\sum {\delta }_i^{{\mathcal{H}}_1,{\mathcal{H}}_2},$$

$${\delta }_0^{\prime }$$

$${\delta }_{0,c_2}$$

$${\delta }_{0,c_3}$$

$${\pi }^{\ast }{\delta }_i={\delta }_i^{\prime }+3{\delta }_{i,c_3}^{S_3,S_3}.$$

$$\frac{\mathrm{Aut}(\mathsf{C},\varphi )}{\mathrm{Aut}(C)}=\mathrm{Aut}({\mathsf{C}}_q)={\mathbf{\mu }}_3.$$

$${\pi }^{\ast }{\delta }_1={\delta }_1^{\prime }+3{\delta }_{1,c_3}^{S_3,S_3}.$$

$${\pi }^{\ast }{\delta }_0={\delta }_0^{\prime }+2{\delta }_{0,c_2}+3{\delta }_{0,c_3}.$$

$$K_{{\overline{\mathcal{R}}}_{g,S_3}}=13\lambda -(2{\delta }_0^{\prime }+3{\delta }_{0,c_2}+4{\delta }_{0,c_3})-(3{\delta }_1^{\prime }+7{\delta }_{1,c_3}^{S_3,S_3})-\sum _{i=2}^{\lfloor g/2\rfloor }(2{\delta }_i^{\prime }+4{\delta }_{i,c_3}^{S_3,S_3}).$$

$$K_{{\overline{\mathcal{M}}}_g}=13\lambda -2{\delta }_0-3{\delta }_1-2{\delta }_2-\cdots -2{\delta }_{\lfloor g/2\rfloor }.$$

$$R={\delta }_{0,c_2}+2{\delta }_{0,c_3}+2\sum _{i=1}^{\lfloor g/2\rfloor }{\delta }_{i,c_3}^{S_3,S_3}.$$

$$[\overline{E}]=a\cdot \lambda -b_0^{\prime }\cdot {\delta }_0^{\prime }-b_{0,c_2}\cdot {\delta }_{0,c_2}-b_{0,c_3}\cdot {\delta }_{0,c_3}-b_1^{\prime }\cdot {\delta }_1^{\prime }-b_{1,c_3}^{S_3,S_3}\cdot {\delta }_{1,c_3}^{S_3,S_3}-\sum _{i=2}^{\lfloor g/2\rfloor }(b_i^{\prime }\cdot {\delta }_i^{\prime }+b_{i,c_3}^{S_3,S_3}\cdot {\delta }_{i,c_3}^{S_3,S_3})$$

Full text

**Keywords: **curves; moduli; covers; spin curves; principal bundles; admissible covers; twisted covers.

1. Introduction

The goal of this paper, alongside its companion [20], is to analyze the birational geometry of the moduli space of curves equipped with a $G$-cover, where $G$ is any finite group. More specifically, here we prove that the moduli space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ of genus $g$ connected twisted $S_{3}$-covers is of general type for any odd genus $g\geq 13$.

In a series of seminal works, Eisenbud, Harris and Mumford proved that the moduli $\mathcal{M}_{g}$ of genus $g$ smooth curves is a variety of general type for any genus $g>23$ (see [22, 21, 13]). The behavior is different for low genus, and in fact there is a large literature showing that $\mathcal{M}_{g}$ is unirational for $g\leq 14$ and rationally connected for $g=15$ (see for instance [3, 30, 8]). Recently, Farkas, Jensen and Payne showed that also $\mathcal{M}_{22}$ and $\mathcal{M}_{23}$ are of generaly type [15].

Many finite covers of $\mathcal{M}_{g}$ with a modular interpretation, have the same behavior: there is a transition genus such that for any higher value the variety is of general type. The interesting property is that in many cases the transition genus is strictly lower. Farkas and Ludwig proved in [16] that the space $\mathcal{R}_{g,\bm{\mu}_{2}}^{\bm{\mu}_{2}}$ of curves with a non-trivial $2$-torsion bundle is of general type for any genus $g\geq 14$ and $g\neq 15$; Chiodo, Eisenbud, Farkas and Schreyer (see [11]) proved the same in the case of $3$-torsion bundles for $g\geq 12$; Farkas and Verra approached in [18] the case of odd spin curves, which is of general type for $g\geq 12$.

In this paper we focus on the moduli $\mathcal{R}_{g,G}$ of curves with a principal $G$-bundle for $G$ finite group. We build its compactification $\overline{\mathcal{R}}_{g,G}$ as the moduli of twisted $G$-covers, that is balanced representable maps $\phi\colon\mathsf{C}\to BG$ where $\mathsf{C}$ is a Deligne-Mumford stack such that its coarse space is a stable curve $C$, with non-trivial stabilizer only at some nodes. We properly introduce twisted $G$-covers in §2.1, and in §2.2 we recall the equivalence with the notion of admissible $G$-cover. In order to evaluate the Kodaira dimension of $\overline{\mathcal{R}}_{g,G}$ it is necessary to consider a desingularization $\widehat{\mathcal{R}}_{g,G}\to\overline{\mathcal{R}}_{g,G}$, but it is difficult to evaluate the dimension of the space of $n$-canonical sections $H^{0}(\widehat{\mathcal{R}}_{g,G},nK)$. Following the approach of [22], we intend to focus in those cases where it is the same to evaluate the dimension of the space of $n$-canonical sections over the regular locus of $\overline{\mathcal{R}}_{g,G}$, that is those cases such that

[TABLE]

A pluricanonical form defined locally over $\overline{\mathcal{R}}^{\operatorname{reg}}_{g,G}$, extends uniquely to $\widehat{\mathcal{R}}_{g,G}$ in smooth points and in the so called canonical singularities (for a treatment of canonical and non-canonical singularities see [27]). Therefore, we need an analysis of the locus of non-canonical singularities, as done by Chiodo and Farkas in [12] for level $\ell$ curves (that is curves with an $\ell$-torsion bundle), and by the author in [19] for the moduli space $\mathcal{R}_{g,\ell}^{k}$ with a line bundle $L$ such that $L^{\otimes\ell}\cong\omega^{\otimes k}$.

We search for a result allowing a global unique extension of pluricanonical forms. Consider the natural project $\pi\colon\overline{\mathcal{R}}_{g,G}\to\overline{\mathcal{M}}_{g}$, we know from the author work [20] that the locus of non-canonical singularities is bipartitioned as $\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{R}}_{g,G}=T_{g,G}\cup J_{g,G}$, where the $T$-locus is defined as $T_{g,G}:=\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{R}}_{g,G}\cap\pi^{-1}\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{M}}_{g}$, while $J_{g,G}$ is the locus of ``new'' singularities that are not in the preimage of $\overline{\mathcal{M}}_{g}$ singularities. We recall that the $J$-locus is empty for $\overline{\mathcal{R}}_{g,S_{3}}$, a fact proven in [20, Theorem 5.2.1].

In Theorems 4.1.2 and 4.2.5 we prove the following.

Theorem**.**

Consider a desingularization $\widehat{\mathcal{R}}_{g,G}\to\overline{\mathcal{R}}_{g,G}$. If $G$ is abelian and $J_{g,G}=\varnothing$, then

[TABLE]

for $n$ sufficiently big and divisible. The same is true when $G=S_{3}$ is the symmetric group of order $3$.

The extension result is obtained via a non-trivial generalization of Harris-Mumford techniques used in [22], which is particularly tricky in the case $G=S_{3}$ because the covering $T_{g,S_{3}}\to\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{M}}_{g}$ is non-étale.

Finally, the main result of this paper concerns the component of $\overline{\mathcal{R}}_{g,S_{3}}$ parametrizing connected twisted $S_{3}$-covers, that we denote by $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$. The other components are finite covers of the well known moduli of level curves of order $2$ and $3$. In order to prove the bigness of the canonical divisor of $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, for every odd genus $g=2i+1$ we write it down as a linear combination

[TABLE]

where $\alpha,\beta,\gamma$ are real coefficients, $E$ is a boundary effective divisor and $\lambda$ is the Hodge class, which is known to be big. Furthermore, the divisor $\mathcal{U}_{g}$ is the jumping locus for the Koszul cohomology $K_{i,1}$ of a particular rank $2$ vector bundle (see Definition 5.2.4), its effectiveness is proved in Proposition 5.2.6, while $\pi^{*}\mathcal{M}^{1}_{g,i+1}$ is the lifting of an effective $\mathcal{M}_{g}$ divisor introduced by Harris and Mumford (see Definition 5.2.12). Therefore $\overline{\mathcal{U}}_{g}$ and $\pi^{*}\mathcal{M}^{1}_{g,i+1}$ are both effective. In Proposition 5.2.13 we prove that the coefficients $\alpha,\beta,\gamma$ can be chosen all positive for $i>5$, and this implies the bigness of $\lambda$ and the space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ being of general type.

Theorem**.**

*The moduli space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is of general type for every odd genus $g\geq 13$.

In §2 we introduce the compactification of $\mathcal{R}_{g,G}$ via twisted $G$-covers and admissible $G$-covers. In §3 we describe the boundary of $\overline{\mathcal{R}}_{g,S_{3}}$ and evaluate its canonical divisor. In §4 the extension of pluricanonical forms is shown and in §5 we build the Koszul divisor and conclude the proof.

Acknowledgements

This work is the completion of my PhD thesis at Institut de Mathématiques de Jussieu, thus I want to thank my PhD supervisor Alessandro Chiodo, for his patience and his advices. Furthermore, I want to thank Michele Bolognesi and Gavril Farkas for the careful reading and the comments on my thesis work. Finally, I am grateful to Claudio Fontanari, my post-doc tutor at Università degli Studi di Trento, for his important suggestions, and to Roberto Pignatelli for the fruitful conversations.

2. Twisted and admissible covers

In this section we recall the notions of twisted $G$-cover and admissible $G$-cover, and their equivalence in Theorem 2.2.15. In the following of the manuscript, we will adopt preferably the twisted cover notation in Section 4 and 5, and the admissible cover notation in Section 3.

In §2.2 we also introduce the notion of admissible $\mathcal{H}$-cover, where $\mathcal{H}$ is any subgroup conjugacy class of $G$. This concept is central in classifying the components and boundary divisors of any moduli space $\overline{\mathcal{R}}_{g,G}$, as we do in Section 3, in particular for the case $G=S_{3}$. In order to better understand these subjects, we also recall a series of group action tools.

Finally, in §2.3 we introduce a description of admissible $G$-covers via a monodromy-type approach. This description allows to treat admissible $G$-covers as particular classes of group morphisms.

2.1. Introduction to twisted $G$-covers

Definition 2.1.1** (Twisted curve).**

A twisted $n$-marked $S$-curve is a diagram

[TABLE]

1. (1)

$\mathsf{C}$ is a Deligne-Mumford stack, proper over $S$, and étale locally it is a nodal curve over $S$; 2. (2)

the $\Sigma_{i}\subset\mathsf{C}$ are disjoint closed substacks in the smooth locus of $\mathsf{C}\to S$ for all $i$; 3. (3)

$\Sigma_{i}\to S$ is an étale gerbe for all $i$; 4. (4)

$\mathsf{C}\to C$ exhibits $C$ as the coarse space of $\mathsf{C}$, and it is an isomorphism over $C_{\operatorname{gen}}$.

We recall that, given a scheme $U$ and a finite abelian group $\bm{\mu}$ acting on $U$, the stack $[U/\bm{\mu}]$ is the category of principal $\bm{\mu}$-bundles $E\to T$, for any scheme $T$, equipped with a $\bm{\mu}$-equivariant morphism $f\colon E\to U$. The stack $[U/\bm{\mu}]$ is a proper Deligne-Mumford stack and has a natural morphism to its coarse scheme $U/\bm{\mu}$.

By the definition of twisted curve we get the local pictures:

• •

At a marking, morphism $\mathsf{C}\to C\to S$ is locally isomorphic to

[TABLE]

for some normal ring $A$ and some integer $r>0$. Here $x=(x^{\prime})^{r}$, and $\bm{\mu}_{r}$ is the cyclic group of order $r$ acting on $\operatorname{Spec}A[x^{\prime}]$ by the action $\xi\colon x^{\prime}\mapsto\xi x^{\prime}$ for any $\xi\in\bm{\mu}_{r}$.

• •

At a node, morphism $\mathsf{C}\to C\to S$ is locally isomorphic to

[TABLE]

for some integer $r>0$ and $a\in A$. Here $x=(x^{\prime})^{\ell},\ y=(y^{\prime})^{\ell}$. The group $\bm{\mu}_{r}$ acts by the action

[TABLE]

where $m$ is an element of $\mathbb{Z}/r$ and $\xi$ is a primitive $r$th root of the unit. The action is called balanced if $m\equiv-1\mod r$. A curve with balanced action at every node is called a balanced curve.

Definition 2.1.2** (Twisted $G$-cover).**

Given an $n$-marked twisted balanced curve $(\Sigma_{1},\dots,\Sigma_{n};\ \mathsf{C}\to C\to S)$, a twisted $G$-cover is a representable stack morphism $\phi\colon\mathsf{C}\to BG$, i.e. an object of the category $\operatorname{Fun}(\mathsf{C},BG)$ which moreover is representable.

We observe that by adding the balancing hypothesis, we are using a slightly different notion of twisted $G$-cover with respect to [1].

Definition 2.1.3**.**

We consider the category $\mathcal{B}^{\operatorname{bal}}_{g,n}(G)$. The objects of $\mathcal{B}^{\operatorname{bal}}_{g,n}(G)$ are twisted $n$-marked $S$-curves of genus $g$ with a twisted $G$-cover, for any scheme $S$.

Consider two twisted $G$-covers $\phi^{\prime}\colon\mathsf{C}^{\prime}\to BG$ and $\phi\colon\mathsf{C}\to BG$ over the twisted $n$-marked curves $\mathsf{C}^{\prime}$ and $\mathsf{C}$ respectively. A morphism $(\mathsf{C}^{\prime},\phi^{\prime})\to(\mathsf{C},\phi)$ is a pair $(\mathsf{f},\alpha)$ such that $\mathsf{f}\colon\mathsf{C}^{\prime}\to\mathsf{C}$ is a morphism of $n$-marked twisted curves, and $\alpha\colon\phi^{\prime}\to\phi\circ\mathsf{f}$ is an isomorphism in $\operatorname{Fun}(\mathsf{C}^{\prime},BG)$.

2.2. Introduction to admissible $G$-covers

Definition 2.2.1** (Admissible cover).**

Given a nodal $S$-curve $X\to S$ with marked points, an admissible cover $u\colon F\to X$ is a morphism such that:

1. (1)

the composition $F\to S$ is a nodal $S$-curve; 2. (2)

given a geometric point $\bar{s}\in S$, every node of $F_{\bar{s}}$ maps via $u$ to a node of $X_{\bar{s}}$; 3. (3)

the restriction $F|_{X_{\operatorname{gen}}}\to X_{\operatorname{gen}}$ is an étale cover of degree $d$; 4. (4)

given a geometric point $\bar{s}\in S$, the local picture of $F_{\bar{s}}\xrightarrow{u}X_{\bar{s}}$ at a point of $F_{\bar{s}}$ mapping to a marked point of $X$ is isomorphic to

[TABLE]

for some normal ring $A$, an integer $r>0$ and $u^{*}x=(x^{\prime})^{r}$; 5. (5)

the local picture of $F_{\bar{s}}\xrightarrow{u}X_{\bar{s}}$ at a node of $F_{\bar{s}}$ is isomorphic to

[TABLE]

for some integer $r>0$ and an element $a\in A$, $u^{*}x=(x^{\prime})^{r}$ and $u^{*}y=(y^{\prime})^{r}$.

The category $\operatorname{Adm}_{g,n,d}$ of $n$-pointed stable curves of genus $g$ with an admissible cover of degree $d$, is a proper Deligne-Mumford stack.

Consider $F\to C$ an admissible cover of a nodal curve $C$, a $G$-action on $F$ such that the restriction $F|_{C_{\operatorname{gen}}}\to C_{\operatorname{gen}}$ is a principal $G$-bundle, a smooth point $p$ of $C$ and a preimage $\tilde{p}\in F$ of $p$. We denote by $H_{\tilde{p}}\subset G$ the stabilizer of $\tilde{p}$. The $G$-action induces a primitive character $\chi_{\tilde{p}}\colon H_{\tilde{p}}\to\operatorname{GL}(T_{\tilde{p}}F)=\mathbb{C}^{*}$. In the set of pairs $(H,\chi)$, with $H$ a $G$ subgroup and $\chi\colon H\to\mathbb{C}^{*}$ a character, we introduce the equivalence relation $(H,\chi)\sim(H^{\prime},\chi^{\prime})$ if and only if there exists $s\in G$ such that $H^{\prime}=sHs^{-1}$ and $\chi^{\prime}=\chi^{s}$, where $\chi^{s}\colon h\mapsto\chi(s^{-1}hs)$ is the conjugated character to $\chi$. Consider a point $\tilde{p}$ on $F$ with stabilizer $H_{\tilde{p}}$ and associated character $\chi_{\tilde{p}}$. We observe that for any point $s\cdot\tilde{p}$ of the same fiber, $H_{s\cdot\tilde{p}}=sH_{\tilde{p}}s^{-1}$ and $\chi_{s\cdot\tilde{p}}=\chi_{\tilde{p}}^{s}$. Therefore the equivalence class of the pair $(H_{\tilde{p}},\chi_{\tilde{p}})$ only depends on the point $p$.

Definition 2.2.2**.**

For any smooth point $\tilde{p}$ on $F$, we call local index the associated pair $(H_{\tilde{p}},\chi_{\tilde{p}})$. For any smooth point $p\in C$, the conjugacy class of the local index of any $\tilde{p}$ in $F_{p}$ is called the $G$-type at $p$, following the notation in [7]. We denote the $G$-type by $\llbracket H_{p},\chi_{p}\rrbracket$, where $H_{p}$ is the stabilizer of one of the points in $F_{p}$, and $\chi_{p}$ the associated character.

Lemma 2.2.3** (see [20, Lemma 2.3.3]).**

Consider $u\colon F\to C$ an admissible cover of a nodal curve $C$ such that the restriction $F|_{C_{\operatorname{gen}}}\to C_{\operatorname{gen}}$ is a principal $G$-bundle. If $\tilde{p}\in F$ is one of the preimages of a node or a marked point, then the stabilizer $H_{\tilde{p}}$ is a cyclic group.

Observe that the set of characters $\chi\colon\bm{\mu}_{r}\to\mathbb{C}^{*}$ of a cyclic group, is the group $\mathbb{Z}/r\mathbb{Z}$. In particular, the character associated to $k\in\mathbb{Z}/r\mathbb{Z}$ maps $\xi\mapsto\xi^{k}$ for any $\xi$ $r$th root of the unit. In the case of a node $\tilde{p}\in F$, we observe that $H_{\tilde{p}}$ acts independently on the two branches $U_{1}$ and $U_{2}$. We denote by $\chi_{\tilde{p}}^{(1)}$ and $\chi_{\tilde{p}}^{(2)}$ the characters of these actions.

Definition 2.2.4**.**

The $G$-action at node $\tilde{p}$ is balanced when $\chi_{\tilde{p}}^{(1)}=-\chi_{\tilde{p}}^{(2)}$, that is they are opposite as elements of $\mathbb{Z}/r\mathbb{Z}$.

Definition 2.2.5** (Admissible $G$-cover).**

Take $G$ finite group, an admissible cover $u\colon F\to C$ of a nodal curve $C$ is an admissible $G$-cover if

1. (1)

the restriction $u|_{C_{\operatorname{gen}}}\colon F|_{C_{\operatorname{gen}}}\to C_{\operatorname{gen}}$ is a principal $G$-bundle. This implies, by Lemma 2.2.3, that for every node or marked point $\tilde{p}\in F$, the stabilizer $H_{\tilde{p}}$ is a cyclic group; 2. (2)

the action of $H_{\tilde{p}}$ is balanced for every node $\tilde{p}\in F$.

This notion was developed by Abramovich, Corti and Vistoli in [1], and also by Jarvis, Kaufmann and Kimura in [23].

Definition 2.2.6**.**

We call $\operatorname{Adm}_{g,n}^{G}$ the stack of stable curves of genus $g$ with $n$ marked points and equipped with an admissible $G$-cover.

Remark 2.2.7**.**

For any cyclic subgroup $H\subset G$, we choose the privileged root $\exp(2\pi i/|H|)$. After this choice, The datum of $(H,\chi)$, is equivalent to the datum of the $H$ generator $h=\chi^{-1}(e^{2\pi i/|H|})$. As a consequence, the conjugacy class $\llbracket H,\chi\rrbracket$ is identified with the conjugacy class $\llbracket h\rrbracket$ of $h$ in $G$.

Definition 2.2.8**.**

Given an admissible $G$-cover $F\to C$ over an $n$-marked stable curve, the series $\llbracket h_{1}\rrbracket,\ \llbracket h_{2}\rrbracket,\dots,\ \llbracket h_{n}\rrbracket$, of the $G$-types of the singular fibers over the marked points, is called Hurwitz datum of the cover. The stack of admissible $G$-covers of genus $g$ with a given Hurwitz datum is denoted by $\operatorname{Adm}_{g,\llbracket h_{1}\rrbracket,\dots,\llbracket h_{n}\rrbracket}^{G}$.

Remark 2.2.9**.**

Given an admissible $G$-cover $F\to C$, if $p$ is a node of $C$ and $\tilde{p}$ one of its preimages on $F$, then the local index of $\tilde{p}$ and the $G$-type of $p$ are well defined once we fix a privileged branch of $p$. Switching the branches sends the local index and the $G$-type in their inverses.

We denote by $\mathcal{T}(F)$ the set of connected components of an admissible $G$-cover $F\to C$, and naturally it inherits the $G$-cover action $\psi$. The conjugacy class of the stabilizer $H_{E}\subset G$ of any connected component $E$, is independent of the choice of $E$.

We call $\mathcal{T}(G)$ the set of equivalence classes in $\operatorname{Sub}(G)$ with respect to conjugation. Then, for every admissible $G$-cover there exists a canonical class $\mathcal{H}$ in $\mathcal{T}(G)$ and a canonical surjective map $\mathcal{T}(F)\twoheadrightarrow\mathcal{H}$ sending any component in its stabilizer. About the theory of group actions that we use in this work, we also refer to [20, §2.1.2].

Definition 2.2.10**.**

Consider two subgroup conjugacy classes $\mathcal{H}_{1},\mathcal{H}_{2}$ in $\mathcal{T}(G)$, we say that $\mathcal{H}_{2}$ is a subclass of $\mathcal{H}_{1}$, denoted by $\mathcal{H}_{2}\leq\mathcal{H}_{1}$, if for one element $H_{2}\in\mathcal{H}_{2}$ (and hence for all), there exists $H_{1}\in\mathcal{H}_{1}$ such that $H_{2}$ is a subgroup of $H_{1}$. If the inclusion is strict, then $\mathcal{H}_{2}$ is a strict subclass of $\mathcal{H}_{1}$ and the notation is $\mathcal{H}_{2}<\mathcal{H}_{1}$.

Definition 2.2.11**.**

We denote by $\operatorname{Hom}^{G}(\mathcal{T}(F),G)$ the set of maps $v\colon\mathcal{T}(F)\to G$ such that $v(\psi(g,E))=g\cdot v(E)\cdot g^{-1}$.

Definition 2.2.12**.**

Consider a subgroup conjugacy class $\mathcal{H}$ of $G$. An admissible $\mathcal{H}$-cover is an admissible $G$-cover such that every connected component has stabilizer in $\mathcal{H}$.

Definition 2.2.13**.**

We denote by $\operatorname{Adm}^{G,\mathcal{H}}_{g}$ the stack of admissible $\mathcal{H}$-covers over stable curves of genus $g$, and we denote by $\operatorname{Adm}^{G,\mathcal{H}}_{g,\llbracket h_{1}\rrbracket,\dots,\llbracket h_{n}\rrbracket}$ the stack of admissible $\mathcal{H}$-cover with Hurwitz datum $\llbracket h_{1}\rrbracket,\dots,\llbracket h_{n}\rrbracket$ over the $n$ marked points.

Proposition 2.2.14** (see in [20, Proposition 2.3.14]).**

Consider $(C;p_{1},\dots,p_{n})$ a nodal $n$-marked curve, and $F\to C$ an admissible $G$-cover, then

[TABLE]

We introduced the two categories $\mathcal{B}_{g}^{\operatorname{bal}}(G)$ (see Definition 2.1.3) and $\operatorname{Adm}_{g}^{G}$ with the purpose of ``well'' defining the notion of principal $G$-bundle over stable non-smooth curves. These two categories are proven isomorphic in [1].

Theorem 2.2.15** (see [1, Theorem 4.3.2]).**

There exists a base preserving equivalence between $\mathcal{B}_{g}^{\operatorname{bal}}(G)$ and $\operatorname{Adm}_{g}^{G}$, therefore in particular they are isomorphic Deligne-Mumford stacks.

Remark 2.2.16**.**

From now on we will use the notation $\overline{\bm{\mathsf{R}}}_{g,G}=\mathcal{B}_{g}^{\operatorname{bal}}(G)=\operatorname{Adm}^{G}_{g}$ for the moduli stack of curves of genus $g$ equipped with an admissible $G$-cover, and $\overline{\mathcal{R}}_{g,G}$ for its coarse space. Analogously, we will use the notation $\overline{\bm{\mathsf{R}}}_{g,G}^{\mathcal{H}}=\operatorname{Adm}^{G,\mathcal{H}}_{g}$, and $\overline{\mathcal{R}}_{g,G}^{\mathcal{H}}$, for the moduli stack of admissible $\mathcal{H}$-covers over stable curves of genus $g$, and its coarse space.

Remark 2.2.17**.**

In the following, we will say that a twisted $G$-cover $(\mathsf{C},\phi)$ ``is'' an admissible $G$-cover $F\to C$ (or the other way around), meaning that $F\to C$ is the naturally associated admissible $G$-cover to $(\mathsf{C},\phi)$.

Remark 2.2.18**.**

We recall the local description of $\overline{\mathcal{R}}_{g,G}$, following [20, Remark 2.2.8]. For any twisted $G$-cover $(\mathsf{C},\phi)$, the local picture of $\overline{\mathcal{R}}_{g,G}$ at $[\mathsf{C},\phi]$ is $\operatorname{Def}(\mathsf{C},\phi)/\operatorname{Aut}(\mathsf{C},\phi)$, where the action of the automorphism group is induced by the universal property. If we consider the universal deformation $\operatorname{Def}(\mathsf{C};\operatorname{Sing}\mathsf{C})$ of $\mathsf{C}$ alongside with its nodes, this is naturally identified with $\operatorname{Def}(C;\operatorname{Sing}C)$ where $C$ is the coarse space of $\mathsf{C}$. If $C_{1},\dots,C_{V}$ are the irreducible components of $C$, $\overline{C}_{i}$ their normalizations, $D_{i}\subset\overline{C}_{i}$ the divisors of the the preimages of the nodes, then

[TABLE]

We denote by $q_{1},\dots,q_{\delta}$ the nodes of $C$, and we observe that we have a canonical splitting

[TABLE]

For every $j$, $R_{j}\cong\mathbb{A}^{1}$. These are the universal deformations (or smoothings) of nodes $q_{j}$ with the associated stabilizers at $\mathsf{C}$. We recall that for the coarse curve $C$, we have the splitting $\operatorname{Def}(C)/\operatorname{Def}(C;\operatorname{Sing}C)=\bigoplus_{j=1}^{\delta}M_{j}$, where again $M_{j}\cong\mathbb{A}^{1}$ and there exists a canonical morphisms $R_{j}\to M_{j}$ of degree $r_{j}$, the cardinality of the $q_{j}$ stabilizer, and branched at the origin. In particular we call $t_{j}$ the coordinate of $M_{j}$, and $\tilde{t}_{j}$ the coordinate of $R_{j}$ such that $t_{j}=(\tilde{t}_{j})^{r_{j}}$.

2.3. Monodromy description of admissible $G$-covers

Consider a smooth curve $C$ of genus $g$ and $n$ marked points $p_{1},\dots,p_{n}$, the fundamental group of $C_{\operatorname{gen}}=C\backslash\{p_{1},\dots,p_{n}\}$ has $2g+n$ generators $\alpha_{1},\alpha_{2},\dots\alpha_{g},\beta_{1}\dots,\beta_{g},\gamma_{1},\dots,\gamma_{n}$. These are represented in the figure below, where the arrows with the same label are identified respecting the orientation. As we can also see in the figure, these generators respect the following relation,

[TABLE]

This is called the canonical representation of the fundamental group of a genus $g$ smooth curve.

\alpha_{1}$$\beta_{1}$$\alpha_{1}$$\beta_{1}$$\alpha_{2}$$\cdots$$\beta_{g}$$\dots$$\gamma_{2}$$\gamma_{1}$$p_{2}$$p_{1}$$p_{*}

It is possible to describe admissible $G$-covers over smooth curves by the monodromy action, as done for example in [7, §2.3] and [28, §3.5]. Consider a smooth curve $C$, a generic point $p_{*}$ on it and an admissible $G$-cover $F\to C$. We denote the points of the fiber $F_{p_{*}}$ by $\tilde{p}_{*}^{(g)}$ for any $g\in G$, in such a way that $g\cdot\tilde{p}_{*}^{(1)}=\tilde{p}_{*}^{(g)}$. This induces a group morphism $\pi_{1}(C_{\operatorname{gen}},p_{*})\to G$. This monodromy morphism is well defined up to relabelling the points $\tilde{p}_{*}^{(g)}$, i.e. up to $G$ conjugation. The following proposition is a rephrasing of [7, Lemma 2.6].

Proposition 2.3.1**.**

Given a smooth $n$-marked curve $(C;p_{1},\dots,p_{n})$ and a point $p_{*}$ on its generic locus $C_{\operatorname{gen}}$, the set of isomorphism classes of admissible $G$-covers on $C$ is naturally in bijection with the set of conjugacy classes of maps

[TABLE]

Remark 2.3.2**.**

We also point out that, as represented in the figure, the monodromy of $\gamma_{i}$ at any point $p_{*}^{(g)}$, with $g\in G$, is given by a small circular lacet around the deleted point $p_{i}$. Therefore by definition of $G$-type, if $\llbracket h_{i}\rrbracket$ is the $G$-type of $p_{i}$, then $\llbracket\varpi(\gamma_{i})\rrbracket=\llbracket h_{i}\rrbracket$.

We consider now the case of an irreducible curve $C$ with one type [math] node, i.e. an autointersection node. If $\overline{C}$ is the $C$ normalization, we denote by $p_{1}$ and $p_{2}$ the preimages of the node on $\overline{C}$. An admissible $G$-cover $F\to C$ induces an admissible $G$-cover $\overline{F}\to\overline{C}$ on the $2$-marked genus $g-1$ curve $(\overline{C};p_{1},p_{2})$ and by what we said above, there exists an associated morphism $\varpi\colon\pi_{1}(\overline{C}_{\operatorname{gen}})\to G$ such that $\llbracket\varpi(\gamma_{1})\rrbracket=\llbracket\varpi(\gamma_{2})^{-1}\rrbracket$. As we obtain $C$ from $\overline{C}$ by gluing $p_{1}$ to $p_{2}$, $F$ is obtained by gluing the fibers $\overline{F}_{p_{1}}$ and $\overline{F}_{p_{2}}$.

Remark 2.3.3**.**

In the last case, consider a path $\gamma_{1,2}$ on $\overline{C}$ from $p_{1}$ to $p_{2}$. Gluing $\overline{F}_{p_{1}}$ to $\overline{F}_{p_{2}}$ is equivalent to lift $\gamma_{1,2}$ and therefore to give a monodromy factor $h_{\gamma}$ well defined up to conjugation. Taking the twisted $G$-cover point of view, a point of $\overline{F}_{p_{1}}$ has to be sent by $h_{\gamma}$ to a point with the same local index (see [20, §2.2.2]). For example if we take a point with local index $w\in\llbracket\varpi(\gamma_{1})\rrbracket$, then the local index of its image is $h_{\gamma}wh_{\gamma}^{-1}$. This means that $w$ and $h_{\gamma}$ must commute. This condition allows to define $h_{\gamma}$ up to conjugation and is also sufficient to make the gluing.

Remark 2.3.4**.**

If $C$ is a nodal curve with a type [math] node as above, the fundamental group of the normalization $\pi_{1}(\overline{C}_{\operatorname{gen}})$ is naturally a subgroup of $\pi_{1}(C)$, therefore the generators $\alpha_{1},\dots,\beta_{g-1}$ are $\pi_{1}(C)$ elements. Consider a slight deformation $F^{\prime}\to C^{\prime}$ of $F\to C$, smoothing the node, such that $C^{\prime}$ is a smooth curve. This induces a natural morphism $\pi_{1}(C)\to\pi_{1}(C^{\prime})$, in particular the $\alpha_{i},\beta_{i}$ from $\pi_{1}(\overline{C}_{\operatorname{gen}})$ are generators in the canonical representation of $\pi_{1}(C^{\prime})$, and we can complete this representation in such a way that the class of lacet $\gamma_{1}$ becomes the element $\alpha_{g}$ in $\pi_{1}(C^{\prime})$. The generator $\beta_{g}$ comes from the gluing data of the nodal fiber.

Following this monodromy approach, and in order to describe the components of $\overline{\bm{\mathsf{R}}}_{g,S_{3}}$ in §3.1.1, it is useful to introduce a notion of group morphism defined up to conjugation.

Definition 2.3.5**.**

Consider two groups $A,\ G$, and the equivalence relation $\sim$ such that for two morphisms $\varphi,\varphi^{\prime}\colon A\to G$, $\varphi\sim\varphi^{\prime}$ if there exists $g\in G$, $\varphi^{\prime}=g\cdot\varphi\cdot g^{-1}$.

For any conjugacy class $\mathcal{H}\in\mathcal{T}(G)$, we denote by $\operatorname{Hom}_{\mbox{s}}(A,\mathcal{H})$ the subset of $\operatorname{Hom}(A,G)/\sim$ induced by the morphisms whose image is a subgroup in $\mathcal{H}$.

We will use the notation $\varphi\colon A\overset{G}{\rightsquigarrow}\mathcal{H}$ for an element of $\operatorname{Hom}_{\mbox{s}}(A,\mathcal{H})$, that is a group morphism $A\to G$ defined up to conjugation whose image is in $\mathcal{H}$.

3. Structure of $\overline{\mathcal{R}}_{g,S_{3}}$

In the first section we study the boundary of $\overline{\mathcal{R}}_{g,S_{3}}$, i.e. the locus $\overline{\mathcal{R}}_{g,G}\backslash\mathcal{R}_{g,G}$. In the other two sections we describe the canonical divisor of this space as a combination of the Hodge divisor and boundary divisors.

3.1. Moduli boundary of $\overline{\mathcal{R}}_{g,S_{3}}$

For every subgroup conjugacy class $\mathcal{H}\in\mathcal{T}(G)$, we consider the moduli stack $\overline{\bm{\mathsf{R}}}_{g,G}^{\mathcal{H}}$ of admissible $\mathcal{H}$-covers over stable curves of genus $g$, i.e. admissible $G$-covers such that the stabilizer of every connected component of a cover is in the class $\mathcal{H}$ (see Definition 2.2.12 and Remark 2.2.16). These are all pairwise disconnected substacks of $\overline{\bm{\mathsf{R}}}_{g,G}$.

3.1.1. Components of $\overline{\mathcal{R}}_{g,S_{3}}$

We observe that $S_{3}$ has $6$ subgroups and $4$ subgroup classes.

• •

The trivial subgroup $(1)\subset S_{3}$. In this case $\overline{\mathcal{R}}_{g,S_{3}}^{1}$ is isomorphic to $\overline{\mathcal{M}}_{g}$.

• •

The three subgroups $T_{1},T_{2},T_{3}$ of order $2$, generated respectively by transpositions $(23),(13),(12)$. These subgroups all stay in the same conjugacy class $T=\{T_{1},T_{2},T_{3}\}$. We are going to show that the stack $\bm{\mathsf{R}}_{g,S_{3}}^{T}$ is isomorphic to $\bm{\mathsf{R}}_{g,\bm{\mu}_{2}}^{\bm{\mu}_{2}}\subset\bm{\mathsf{R}}_{g,2}^{0}$, the moduli stack of twisted curves equipped with a non-trivial square root of the trivial bundle.

Indeed, any admissible connected $\bm{\mu}_{2}$-cover $E\to C$ is equivalent to the data $(C,\rho)$ of a curve $C$ and $\rho\in\operatorname{Hom}_{\mbox{s}}(\pi_{1}(C),\bm{\mu}_{2})$, as stated by Proposition 2.3.1 and using the notation of Definition 2.3.5. There exists an isomorphism $\sigma_{2}\colon\bm{\mu}_{2}\overset{S_{3}}{\rightsquigarrow}T$ (see again Definition 2.3.5), and this induces a set bijection

[TABLE]

Therefore the map $\bm{\mathsf{R}}_{g,\bm{\mu}_{2}}^{\bm{\mu}_{2}}\to\bm{\mathsf{R}}_{g,S_{3}}^{T}$ defined by $(C,\rho)\mapsto(C,\sigma_{2}\circ\rho)$ is an isomorphism. We additionally observe that given an admissible $T$-cover $F\to C$ and any connected component $E\subset F$, the map $E\to C$ is an admissible connected $\bm{\mu}_{2}$-cover.

We observe that the isomorphism of moduli stacks extends to $\overline{\bm{\mathsf{R}}}_{g,S_{3}}^{T}\cong\overline{\bm{\mathsf{R}}}_{g,\bm{\mu}_{2}}^{\bm{\mu}_{2}}$, but the construction at the boundary introduces an additional complexity which is useless for the purpose of this work.

• •

The normal subgroup $N\subset S_{3}$, a cyclic group generated by the $3$-cycle $(123)$. With a little abuse of notation we call $N$ the class $\{N\}\in\mathcal{T}(S_{3})$. Consider the moduli stack $\bm{\mathsf{R}}_{g,\bm{\mu}_{3}}^{\bm{\mu}_{3}}\subset\bm{\mathsf{R}}_{g,\bm{\mu}_{3}}^{0}$ of twisted curves equipped with a non-trivial third root of the trivial bundle. We are going to prove that there exists a $2:1$ map $\bm{\mathsf{R}}_{g,\bm{\mu}_{3}}^{\bm{\mu}_{3}}\to\bm{\mathsf{R}}_{g,S_{3}}^{N}$.

Indeed, any admissible connected $\bm{\mu}_{3}$-cover $E\to C$ is equivalent to the data $(C,\rho)$ of a curve $C$ and an element $\rho\in\operatorname{Hom}_{\mbox{s}}(\pi_{1}(C),\bm{\mu}_{3})$. There exists an isomorphism $\sigma_{3}\colon\bm{\mu}_{3}\overset{S_{3}}{\rightsquigarrow}N$, and this induces a set surjection

[TABLE]

If $\wedge 2\colon\bm{\mu}_{3}\to\bm{\mu}_{3}$ is the second power map, then $\wedge 2$ also acts as an involution on $\operatorname{Hom}_{\mbox{s}}(\pi_{1}(C),\bm{\mu}_{3})$. We observe that if $\rho\colon\pi_{1}(C)\overset{\bm{\mu}_{3}}{\rightsquigarrow}\bm{\mu}_{3}$, then $\sigma_{3}\circ\rho=\sigma_{3}\circ(\wedge 2)\circ\rho$. In fact, the following is true

[TABLE]

Therefore the map $(C,\rho)\mapsto(C,\sigma_{3}\circ\rho)$ is the $2:1$ map we were searching. As before the map extends to $\overline{\bm{\mathsf{R}}}_{g,\bm{\mu}_{3}}^{\bm{\mu}_{3}}\to\overline{\bm{\mathsf{R}}}_{g,S_{3}}^{N}$.

• •

The group $S_{3}$ itself. In this case the stack $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is the moduli of curves equipped with a connected admissible $S_{3}$-cover. This is the ``really new'' component of moduli space $\overline{\mathcal{R}}_{g,S_{3}}$, and our analysis will focus on it.

Furthermore, we observe that there exists a canonical map $\overline{\bm{\mathsf{R}}}_{g,S_{3}}\to\overline{\bm{\mathsf{R}}}_{g,\bm{\mu}_{2}}$. Any admissible $G$-cover $F\to C$ is equivalent to the data of a curve $C$ plus an element of $\operatorname{Hom}_{\mbox{s}}(\pi_{1}(C),S_{3})$. The quotient $S_{3}\twoheadrightarrow S_{3}/N=\bm{\mu}_{2}$ induces a surjection $\operatorname{Hom}_{\mbox{s}}(\pi_{1}(C),S_{3})\twoheadrightarrow\operatorname{Hom}_{\mbox{s}}(\pi_{1}(C),\bm{\mu}_{2})$ and therefore the map above.

3.1.2. The boundary divisors

To classify the boundary divisors of $\overline{\mathcal{R}}_{g,G}$, that is the divisors filling the locus $\overline{\mathcal{R}}_{g,G}\backslash\mathcal{R}_{g,G}$, we start by recalling the boundary divisors of $\overline{\mathcal{M}}_{g}$.

Definition 3.1.1**.**

A node $q$ of a nodal curve $C$ is a disconnecting node if the partial normalization of $C$ at $q$ is a disconnected scheme. This disconnected scheme has two connected components noted $C_{1}$ and $C_{2}$. Consider $C$ of genus $g$ and $q$ a disconnecting node, $q$ is of type $i$, with $1\leq i\leq\lfloor g/2\rfloor$, if $C_{1}$ and $C_{2}$ have genus $i$ and $g-i$. If $q$ is a non-disconnecting node, then it is called a node of type [math].

Definition 3.1.2**.**

For every $i$ with $1\leq i\leq\lfloor g/2\rfloor$, the divisor $\Delta_{i}\subset\overline{\mathcal{M}}_{g}$ is the locus of curves with a disconnecting node of type $i$. For $i=0$, the divisor $\Delta_{0}\subset\overline{\mathcal{M}}_{g}$ is the locus of curves with a node of type [math], or equivalently the closure of the locus of nodal irreducible curves.

For every $i$ we denote by $\delta_{i}$ the class of $\Delta_{i}$ in $\operatorname{Pic}_{\mathbb{Q}}(\overline{\mathcal{M}}_{g})$. We consider the natural morphism $\pi\colon\overline{\mathcal{R}}_{g,G}\to\overline{\mathcal{M}}_{g}$ and look at the preimages $\pi^{-1}(\Delta_{i})$ for every $i$. The intersections of this preimages with the connected components of $\overline{\mathcal{R}}_{g,G}$ are the boundary components that we are going to consider.

We start by focusing on the loci of curves with a disconnecting node, i.e. the preimages of $\Delta_{i}$ with $i\neq 0$. In what follows for any curve $C$ in $\Delta_{i}$ we denote by $q$ the node who separates the components $C_{1}$, of genus $i$, and $C_{2}$, of genus $g-i$. As a consequence $C_{1}\sqcup C_{2}\to C$ is the partial normalization of $C$ at $q$, and we denote by $q_{1}$ and $q_{2}$ the preimages of $q$ respectively on $C_{1}$ and $C_{2}$.

There exists, at the level of the moduli space $\overline{\mathcal{M}}_{g}$, a natural gluing map

[TABLE]

defined in such a way that a pair of points $([C_{1},q_{1}],[C_{2},q_{2}])$, where $C_{1}$ and $C_{2}$ have genus $i$ and $g-i$ respectively, is sent on the point associated to the nodal curve

[TABLE]

We recall that any admissible $G$-cover on $C$ induces two admissible $G$-covers $F_{1}$ on $(C_{1};q_{1})$ and $F_{2}$ on $(C_{2};q_{2})$ such that the $G$-types $\llbracket h_{1}\rrbracket$ and $\llbracket h_{2}\rrbracket$ on $q_{1}$ and $q_{2}$ are one the inverse of the other, $\llbracket h_{1}\rrbracket=\llbracket h_{2}^{-1}\rrbracket$.

Definition 3.1.3**.**

Consider two subgroup conjugacy classes $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ in $\mathcal{T}(G)$, a conjugacy class $\llbracket h\rrbracket$ in $\llbracket G\rrbracket$ and $i$ such that $1\leq i\leq\lfloor g/2\rfloor$. We denote by $\Delta_{i,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ the locus in $\overline{\mathcal{R}}_{g,G}$ of curves $C$ with a node $q$ of type $i$ and with an admissible $G$-cover $F\to C$ such that if we call $F_{1}\to(C_{1};q_{1})$ and $F_{2}\to(C_{2};q_{2})$ the restrictions of $F$, then these are an admissible $\mathcal{H}_{1}$-cover of $(C_{1};q_{1})$ and an admissible $\mathcal{H}_{2}$-cover of $(C_{2};q_{2})$ respectively (see Definition 2.2.12), and the $G$-type at $q$, with respect to the branch of $C_{1}$, is $\llbracket h\rrbracket$. In order to simplify the notation we will omit the $G$-type when it is trivial.

The loci $\Delta_{i,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ are (not necessarily connected) divisors of the moduli space $\overline{\mathcal{R}}_{g,G}$.

For example in the case of $\overline{\mathcal{R}}_{g,S_{3}}$ we have the following classes:

• •

$\Delta_{i}^{1,S_{3}},\ \Delta_{i}^{S_{3},1},\ \Delta_{i}^{1,T},\ \Delta_{i}^{T,1},\ \Delta_{i}^{1,N},\ \Delta_{i}^{N,1}$ and $\Delta_{i}^{1,1}\cong\Delta_{i}$ are the cases of admissible $G$-covers which are trivial over $C_{1}$ or $C_{2}$;

• •

$\Delta_{i}^{T,T},\Delta_{i}^{T,N},\Delta_{i}^{N,T},\Delta_{i}^{T,S_{3}},\Delta_{i}^{N,N},\Delta_{i}^{N,S_{3}},\Delta_{i}^{S_{3},N}$ and $\Delta_{i}^{S_{3},S_{3}}$ are the other cases with trivial $G$-type at the node $q$;

• •

$\Delta_{i,c_{3}}^{S_{3},S_{3}}$ is the only case with non-trivial stabilizer at $q$. Here we denoted by $c_{3}$ the conjugacy class $\llbracket 123\rrbracket=\{(123),(132)\}$.

Remark 3.1.4**.**

We observe that by Equation (2.1) and Remark 2.3.2, it is possible to have non-trivial stabilizer at $q$ only if for both covers $F_{1}\to C_{1}$ and $F_{2}\to C_{2}$, the associated morphisms $\pi_{1}(C_{i})\to S_{3}$ are surjective, and therefore both are connected admissible $S_{3}$-covers.

Finally, we consider the preimage of $\Delta_{0}$, i.e. the locus of curves with a node of type [math]. We start by working on a covering stack of $\pi^{-1}(\Delta_{0})$.

Definition 3.1.5**.**

Category $\mathsf{D}_{0}$ has for objects the data of a curve $C$ with a node $q$ of type [math] with an admissible $G$-cover $F\to C$ and a privileged branch at $q$. Morphisms are admissible $G$-cover morphisms preserving the privileged branch.

Remark 3.1.6**.**

Category $\mathsf{D}_{0}$ is a Deligne-Mumford stack. Its coarse space $\mathcal{D}_{0}$ comes with a natural $2:1$ morphism $\gamma\colon\mathcal{D}_{0}\to\pi^{-1}(\Delta_{0})$.

In what follows we consider a curve $C$ with a node $q$ of type [math], we call $\mathsf{nor}\colon\overline{C}\to C$ the partial normalization of $C$ at $q$. Given any admissible $G$-cover $F\to C$, the pullback $\mathsf{nor}^{*}F$ over $\overline{C}$ is still an admissible $G$-cover.

Definition 3.1.7**.**

Consider two conjugacy classes $\mathcal{H}_{1},\mathcal{H}_{2}$ in $\mathcal{T}(G)$ such that $\mathcal{H}_{2}\leq\mathcal{H}_{1}$ (see Definition 2.2.10). The category $\mathsf{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ is a full subcategory of $\mathsf{D}_{0}$. Its objects are stable curves $C$ with a node $q$ of type [math], a privileged branch at $q$ and an admissible $\mathcal{H}_{1}$-cover $F\to C$ such that the pullback $\mathsf{nor}^{*}F\to\overline{C}$ is an admissible $\mathcal{H}_{2}$-cover on $\overline{C}$, and the $G$-type of $F$ at $q$ with respect to the privileged branch is $\llbracket h\rrbracket$. In case $h=1$ we will omit its notation. The category $\mathsf{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ is again a Deligne-Mumford stack and we denote its coarse space by $\mathcal{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$.

Remark 3.1.8**.**

By Equation (2.1) and Remark 2.3.2, there exists a compatibility condition for the stack $\mathsf{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ to be non-empty. There must exist an element $h$ in the class $\llbracket h\rrbracket$ and a subgroup $H_{2}$ in the class $\mathcal{H}_{2}$ such that $h$ lies in the subgroup of commutators of $H_{2}$.

Remark 3.1.9**.**

There exists a natural automorphism $\operatorname{inv}\colon\mathsf{D}_{0}\to\mathsf{D}_{0}$, which sends any curve with an admissibile $G$-cover to the same curve and $G$-cover but changing the privileged branch at the node. This sends isomorphically $\mathsf{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ in $\mathsf{D}_{0,\llbracket h^{-1}\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$.

Definition 3.1.10**.**

For any group $G$ we consider the inverse relation in the set of conjugacy classes $\llbracket G\rrbracket$, that is $\llbracket h\rrbracket\sim\llbracket h^{\prime}\rrbracket$ if and only if $\llbracket h\rrbracket=\llbracket h^{\prime}\rrbracket$ or $\llbracket h\rrbracket=\llbracket h^{\prime-1}\rrbracket$. Then we define the set $\underline{\llbracket G\rrbracket}$ as $\underline{\llbracket G\rrbracket}:=\llbracket G\rrbracket/\sim$. we denote by $\underline{\llbracket h\rrbracket}$ the class in $\underline{\llbracket G\rrbracket}$ of any element $h\in G$.

We observe that any point $[C,F]$ of $\mathcal{D}_{0}$ and its image $\operatorname{inv}([C,F])$ are sent by $\gamma\colon\mathcal{D}_{0}\to\pi^{-1}(\Delta_{0})$ to the same point. This means that the image of $\mathcal{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ via $\gamma$ depends only on $\mathcal{H}_{1},\mathcal{H}_{2}$ and on the class $\underline{\llbracket h\rrbracket}\in\underline{\llbracket G\rrbracket}$. Equivalently, for all ${\llbracket h\rrbracket}\in{\llbracket G\rrbracket}$, $\mathcal{D}_{0,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ and $\mathcal{D}_{0,\llbracket h^{-1}\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ have the same image.

Definition 3.1.11**.**

For every class $\mathcal{H}_{1},\mathcal{H}_{2}$ in $\mathcal{T}(G)$ and $\underline{\llbracket h\rrbracket}$ in $\underline{\llbracket G\rrbracket}$, the locus $\Delta_{0,\underline{\llbracket h\rrbracket}}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ is the image of $\mathcal{D}_{0,\underline{\llbracket h\rrbracket}}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ via $\gamma$ for any $G$-type $\llbracket h\rrbracket\in\underline{\llbracket h\rrbracket}$. As before we will omit to note $\underline{\llbracket h\rrbracket}$ in the case of the trivial class $\underline{\llbracket 1\rrbracket}$.

As before, we list the divisors in $\pi^{-1}(\Delta_{0})$:

• •

$\Delta_{0}^{N,1},\ \Delta_{0}^{T,1}$ and $\Delta_{0}^{1,1}\cong\Delta_{0}$ are the divisor of admissible $G$-covers $F\to C$ such that $\mathsf{nor}^{*}F\to\overline{C}$ is trivial;

• •

$\Delta_{0}^{T,T},\ \Delta_{0}^{S_{3},T},\ \Delta_{0}^{N,N},\Delta_{0}^{S_{3},N}$ and $\Delta_{0}^{S_{3},S_{3}}$ are the other cases with trivial associated $G$-type at node $q$;

• •

with another small abuse of notation, we define $c_{2}:=\underline{\llbracket 12\rrbracket}=\{(12),(13),(23)\}$ and $c_{3}:=\underline{\llbracket 123\rrbracket}=\{(123),(132)\}$. Then, $\Delta_{0,c_{2}}^{T,T},\ \Delta_{0,c_{2}}^{S_{3},S_{3}},\ \Delta_{0,c_{3}}^{N,N}$ and $\Delta_{0,c_{3}}^{S_{3},S_{3}}$ are the cases with non-trivial stabilizer at $q$.

Observe that the divisors $\Delta_{0,c_{2}}^{S_{3},T}$ and $\Delta_{0,c_{3}}^{S_{3},N}$ are both empty as a direct consequence of Remark 2.3.4.

3.2. The canonical divisor

To evaluate the canonical divisor we start by evaluating the pullbacks of the $\overline{\mathcal{M}}_{g}$ boundary divisors. We denote by $\delta_{i,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ the class of the divisor $\Delta_{i,\llbracket h\rrbracket}^{\mathcal{H}_{1},\mathcal{H}_{2}}$ in the ring $\operatorname{Pic}_{\mathbb{Q}}(\overline{\mathcal{R}}_{g,S_{3}})$. Furthermore, we introduce some other notation to simplify the formulas that will follow.

• •

For any $i\geq 1$ we define

[TABLE]

where the sum is over al the divisors with trivial $S_{3}$-type at the node.

• •

For $i=0$ we define

[TABLE]

Given the classes $\delta_{i}$ in $\operatorname{Pic}_{\mathbb{Q}}(\overline{\mathcal{M}}_{g})$, and the natural morphism $\pi\colon\overline{\mathcal{R}}_{g,S_{3}}\to\overline{\mathcal{M}}_{g}$, we consider the pullbacks $\pi^{*}\delta_{i}$ to the ring $\operatorname{Pic}_{\mathbb{Q}}(\overline{\mathcal{R}}_{g,S_{3}})$.

Lemma 3.2.1**.**

If $i>1$, then

[TABLE]

Proof.

This relation is true also in $\operatorname{Pic}_{\mathbb{Q}}(\overline{\bm{\mathsf{R}}}_{g,S_{3}})$. We consider a general $S_{3}$-cover $(\mathsf{C},\phi)$ in the divisor $\Delta_{i,c_{3}}^{S_{3},S_{3}}$, and we denote by $C$ the coarse space of the curve and by $q$ the type $i$ node. By construction we have

[TABLE]

This implies that the morphism $\overline{\bm{\mathsf{R}}}_{g,S_{3}}\to\overline{\bm{\mathsf{M}}}_{g}$ is $3$-ramified over $\Delta_{i,c_{3}}^{S_{3},S_{3}}$. Over $\Delta_{i}$ but outside $\Delta_{i,c_{3}}^{S_{3},S_{3}}$ the morphism is étale, and the thesis follows.∎

Lemma 3.2.2**.**

If $i=1$, then

[TABLE]

Proof.

The case of a curve $\mathsf{C}$ with an elliptic tail, is different from the case $i>1$ because there exists an elliptic tail involution on $\mathsf{C}$. This involution always lifts to an admissible $S_{3}$-cover. Indeed, an admissible $S_{3}$-cover over an elliptic curve $(E,p_{*})$ is the datum of a morphism $\pi_{1}(E,p_{*})\to S_{3}$ defined up to conjugation, and this is always preserved by the involution. Thus, $\pi^{*}\Delta_{1}=\Delta_{1}^{\prime}+\Delta_{1,c_{3}}^{S_{3},S_{3}}$. To obtain the result we observe that, as in the case of Lemma 3.2.1, the morphism $\overline{\bm{\mathsf{R}}}_{g,S_{3}}\to\overline{\bm{\mathsf{M}}}_{g}$ is $3$-ramified over $\Delta_{1,c_{3}}^{S_{3},S_{3}}$.∎

Lemma 3.2.3**.**

If $i=0$, then

[TABLE]

Proof.

Similarly to what observed in the case of Lemma 3.2.1, the morphism $\overline{\bm{\mathsf{R}}}_{g,S_{3}}\to\overline{\bm{\mathsf{M}}}_{g}$ is $2$-ramified over $\Delta_{0,c_{2}}$, $3$-ramified over $\Delta_{0,c_{3}}$ and étale elsewhere over $\Delta_{0}$.∎

From the Harris and Mumford work [22], we know the evaluation of the canonical divisor $K_{\overline{\mathcal{M}}_{g}}$ of the moduli space $\overline{\mathcal{M}}_{g}$. Knowing that the morphism $\pi\colon\overline{\mathcal{R}}_{g,S_{3}}\to\overline{\mathcal{M}}_{g}$ is étale outside the divisors $\Delta_{i}$ and a sublocus of codimension greater than $1$, we can now evaluate the canonical divisor $K_{\overline{\mathcal{R}}_{g,S_{3}}}$.

Lemma 3.2.4**.**

On the smooth variety $\overline{\mathcal{R}}_{g,S_{3}}^{\operatorname{reg}}$, the sublocus of regular points, we have the following evaluation of the canonical divisor,

[TABLE]

Proof.

As proved in [22, Theorem 2 bis, p.52], on $\overline{\mathcal{M}}_{g}^{\operatorname{reg}}$ we have

[TABLE]

We know, from the description above, that the ramification divisor of $\pi\colon\overline{\mathcal{R}}_{g,S_{3}}\to\overline{\mathcal{M}}_{g}$ is

[TABLE]

By Hurwitz formula we have $K_{\overline{\mathcal{R}}_{g,S_{3}}}=\pi^{*}K_{\overline{\mathcal{M}}_{g}}+R$, this is true on $\pi^{-1}(\overline{\mathcal{M}}_{g}^{\operatorname{reg}})$, but this locus and $\overline{\mathcal{R}}_{g,S_{3}}^{\operatorname{reg}}$ differ by a codimension $2$ locus, therefore the evaluation is unchanged on $\overline{\mathcal{R}}_{g,S_{3}}^{\operatorname{reg}}$. Precisely, if $\pi$ is non-étale in a point $[C,F]\in\overline{\mathcal{R}}_{g,S_{3}}$ outside the $\pi^{-1}(\Delta_{i})$, then necessarily $C$ is an irreducible curve with a non-trivial automorphism group. The thesis follows by the evaluations of the divisors $\pi^{*}\delta_{i}$ of Lemmata 3.2.1, 3.2.2 and 3.2.3.∎

3.3. The subspace of covers on irreducible curves

We consider the full substack $\widetilde{\bm{\mathsf{R}}}_{g,S_{3}}$ of $\overline{\bm{\mathsf{R}}}_{g,S_{3}}$ of irreducible stable curves of genus $g$ with an admissible $S_{3}$-cover. We denote by $\widetilde{\mathcal{R}}_{g,S_{3}}$ its coarse space. We will prove that the bigness of the canonical divisor $K_{\overline{\mathcal{R}}_{g,S_{3}}}$ over $\widetilde{\mathcal{R}}_{g,S_{3}}$ implies the bigness over $\overline{\mathcal{R}}_{g,S_{3}}$. As a consequence, it suffices to prove that $K_{\overline{\mathcal{R}}_{g,S_{3}}}$ is big over $\widetilde{\mathcal{R}}_{g,S_{3}}$, to prove $\overline{\mathcal{R}}_{g,S_{3}}$ being of general type.

We build some pencils filling up specific divisors of $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$. To do this we follow [16] and [17]. Given a general K3 surface $X$ of degree $2i-2$ in $\mathbb{P}^{i}$, the map $\operatorname{Bl}_{i^{2}}(X)\to\mathbb{P}^{1}$ is a family of genus $i$ stable curves lying on $X$, where $\operatorname{Bl}_{i^{2}}(X)$ is the blowup of $X$ in $i^{2}$ points. This induces a pencil $B\subset\overline{\mathcal{M}}_{i}$. Moreover, there exists at least one section $\sigma$ on $B$, therefore for every genus $g>i$ we can glue along $\sigma$ a fixed $1$-marked curve $(C_{2},p_{*})$ of genus $g-i$, thus inducing a pencil $B_{i}\subset\overline{\mathcal{M}}_{g}$. The pencils $B_{i}$ fill up the divisors $\Delta_{i}$ except for $i=10$, if $i=10$ then the $B_{10}$ fill up a divisor $\mathcal{Z}$ in $\Delta_{10}$ which is the locus of smooth curves of genus $10$ lying on a K3 surface and attached to a curve of genus $g-10$. We denote by $\mathcal{Z}^{\prime}$ the preimage of $\mathcal{Z}$ in $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$.

Lemma 3.3.1**.**

Consider an effective divisor $E$ in $\operatorname{Pic}_{\mathbb{Q}}(\mathcal{R}_{g,S_{3}}^{S_{3}})$ such that

[TABLE]

in $\operatorname{Pic}_{\mathbb{Q}}(\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}})$, where the $a$ and $b$ are rational coefficients.

If $a\leq 13$, $b_{0}^{\prime}\geq 2$, $b_{0,c_{2}}\geq 3$ and $b_{0,c_{3}}\geq 4$, then $b_{i}^{\prime}\geq 3$ and $b_{i,c_{3}}^{S_{3},S_{3}}>7$ for all $i\geq 1$ and $i\neq 10$. The same is true for $i=10$ if $\overline{E}$ does not contain $\mathcal{Z}^{\prime}$.

Proof.

By [17, Lemma 2.4] we have

[TABLE]

We introduce some pencils lying in the preimage of $B_{i}$ with respect to the natural projection $\pi\colon\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}\to\overline{\mathcal{M}}_{g}$.

The pencil $A_{i}^{T,N}$ is the preimage of $B_{i}$ in $\Delta_{i}^{T,N}$. It is obtained by taking any $1$-marked stable curve $C_{1}$ of genus $i$ with an admissible $T$-cover, and gluing it to a fixed general $1$-marked curve $C_{2}$ of genus $g-i$ with an admissible $N$-cover. We remark that the gluing is uniquely defined.

The pencil $A_{i,c_{3}}^{S_{3},S_{3}}$ is the preimage of $B_{i}$ in $\Delta_{i,c_{3}}^{S_{3},S_{3}}$. It is obtained by taking any $1$-marked stable curve $(C_{1},p_{*}^{\prime})$ of genus $i$ with an admissible connected $S_{3}$-cover and $S_{3}$-type equal to $c_{3}$ at $p_{*}^{\prime}$, and gluing it to a fixed general $1$-marked curve $(C_{2},p_{*})$ of genus $g-i$ with an admissible connected $S_{3}$ cover and the same $S_{3}$-type at $p_{*}$. Again, the gluing is unique.

We can write down some intersection numbers for the $A_{i}^{T,N}$:

• •

$A_{i}^{T,N}\cdot\lambda=(2^{2i}-1)(i+1)$, which is true because $\pi^{*}\lambda_{\overline{\mathcal{M}}_{g}}=\lambda_{\overline{\mathcal{R}}_{g,S_{3}}}$ and moreover $\pi_{*}A_{i}^{T,N}=B_{i}\cdot\deg(A_{i}^{T,N}/B_{i})$;

• •

$A_{i}^{T,N}\cdot\delta_{0}^{\prime}=(2^{2i-1}+1)(6i+18),\ A_{i}^{T,N}\cdot\delta_{0,c_{2}}=(2^{2i-1}-2)(6i+18),\ A_{i}^{T,N}\cdot\delta_{0,c_{3}}=0$. The third equality is clear. The second one is obtained by counting the admissible $T$-covers over a curve of genus $i-1$ and by multiplying for the $2$ possible gluing factors at the node. The first equality is obtained by difference;

• •

$A_{i}^{T,N}\cdot\delta_{j,c_{3}}^{S_{3},S_{3}}=0$ for all $j$ and $A_{i}^{T,N}\cdot\delta_{j}^{\prime}=0$ for all $j\neq i$ and $A_{i}^{T,N}\cdot\delta_{i}^{\prime}=-(2^{2i}-1)$ because it is the same of $A_{i}^{T,N}\cdot\pi^{*}\delta_{i}$.

Consider an effective divisor $E$ of $\mathcal{R}_{g,S_{3}}^{S_{3}}$, as the pencils $A_{i}^{T,N}$ fill up the boundary divisor $\Delta_{i}^{T,N}$ for $i\neq 10$, we have $A_{i}^{T,N}\cdot\overline{E}\geq 0$ that is, by the relations above,

[TABLE]

which implies $b_{i}^{\prime}\geq 3$ for all $i\geq 1$. The same is true for $i=10$ if $\overline{E}$ does not contain the locus $\mathcal{Z}^{\prime}$.

We define $d:=\deg(A_{i,c_{3}}^{S_{3},S_{3}})$. As in the previous case, also for $A_{i,c_{3}}^{S_{3},S_{3}}$ we have the equalities

• •

$A_{i,c_{3}}^{S_{3},S_{3}}\cdot\lambda=d\cdot(i+1)$;

• •

$A_{i,c_{3}}^{S_{3},S_{3}}\cdot\delta_{i,c_{3}}^{S_{3},S_{3}}=-d$;

• •

$A_{i,c_{3}}^{S_{3},S_{3}}\cdot\delta_{j}^{\prime}=0$ for all $j$ and $A_{i,c_{3}}^{S_{3},S_{3}}\cdot\delta_{j,c_{3}}^{S_{3},S_{3}}=0$ for all $j\neq i$.

Similarly we have $A_{i,c_{3}}^{S_{3},S_{3}}\cdot(\delta_{0}^{\prime}+\delta_{0,c_{2}}+\delta_{0,c_{3}})=A_{i,c_{3}}^{S_{3},S_{3}}\cdot\pi^{*}\delta_{0}=d\cdot(6i+18)$.

The following inequality is also true

[TABLE]

For any effective divisor $E$ of $\mathcal{R}_{g,S_{3}}^{S_{3}}$, as the pencils $A_{i,c_{3}}^{S_{3},S_{3}}$ fill up a connected component of the boundary divisor $\Delta_{i,c_{3}}^{S_{3},S_{3}}$ for $i\neq 10$, we have $A_{i,c_{3}}^{S_{3},S_{3}}\cdot\overline{E}\geq 0$. This implies, by the inequalities above,

[TABLE]

[TABLE]

The same is true if $i=10$ and $\overline{E}$ does not contain the locus $\mathcal{Z}^{\prime}$.

It remains to prove the inequality (3.1). Consider a $1$-marked curve $(C_{1},p_{*}^{\prime})$ of genus $i$ with an autointersection node $q$, we call $\overline{C}_{1}$ the partial normalization at $q$. By Remarks 2.3.2 and 2.3.3, an admissible $S_{3}$-cover on $(C_{1},p_{*}^{\prime})$ is equivalent to an admissible $S_{3}$-cover on $(\overline{C}_{1},p_{*}^{\prime})$ plus the data of the $S_{3}$-type $\llbracket h_{1}\rrbracket$ at $q$ and a gluing factor $h_{\gamma}$ at the same node (defined up to conjugation), such that $h_{\gamma}$ is in the centralizer of $h_{1}$.

Consider the pencil $A_{i,c_{3}}^{S_{3},S_{3}}$ obtained by joining any such curve $(C_{1},p_{*}^{\prime})$ to a fixed general $1$-marked curve $(C_{2},p_{*})$ of genus $g-i$ with a connected admissible $S_{3}$-cover on it. Any curve in the (finite) intersection $A_{i,c_{3}}^{S_{3},S_{3}}\cap\Delta_{0,c_{2}}$ induces a curve in the intersection $A_{i,c_{3}}^{S_{3},S_{3}}\cap\Delta_{0}^{\prime}$ by putting a trivial $S_{3}$-type at $q$ instead of $\llbracket h_{1}\rrbracket$. The same is true for $A_{i,c_{3}}^{S_{3},S_{3}}\cap\Delta_{0,c_{3}}$. All the points of $A_{i,c_{3}}^{S_{3},S_{3}}\cap\Delta_{0}^{\prime}$ are obtained at least once via these operations of $S_{3}$-type trivialization. This proves the inequality.∎

Proposition 3.3.2**.**

If

[TABLE]

on $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, where $a^{\prime}$ is a positive coefficient and $E^{\prime}$ an effective divisor not containing $\mathcal{Z}^{\prime}$, then the canonical divisor is big over the space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3},\operatorname{reg}}$.

Proof.

The equation in the hypothesis implies that the canonical divisor is big on $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, because $\lambda$ is a big divisor. If we consider the cloture $\overline{E}^{\prime}$ of $E^{\prime}$ on the space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3},\operatorname{reg}}$, we observe that after Lemma 3.2.4, it respects the hypothesis of Lemma 3.3.1, and therefore there exists another effective boundary divisor $E^{\prime\prime}\in\operatorname{Pic}_{\mathbb{Q}}(\overline{\mathcal{R}}_{g,S_{3}}^{S_{3},\operatorname{reg}})$ such that $K_{\overline{\mathcal{R}}_{g,S_{3}}}=a^{\prime}\cdot\lambda+\overline{E}^{\prime}+E^{\prime\prime}$ on $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3},\operatorname{reg}}$, and the proof is completed.∎

4. Extension of pluricanonical forms

We recall that to every twisted $G$-cover $(\mathsf{C},\phi)$ is uniquely associated an admissible $G$-cover $F\to C$ and vice versa, see Remark 2.2.17. In this and the following sections, we will mainly use the notation of twisted $G$-covers.

In order to evaluate the Kodaira dimension of any moduli space $\overline{\mathcal{R}}_{g,G}$, we want to prove an extension result of pluricanonical forms, as done for example by Harris and Mumford for $\overline{\mathcal{M}}_{g}$ (see [22]) and by Chiodo and Farkas for $\overline{\mathcal{R}}_{g,\ell}^{0}$ with $\ell<5$ and $\ell=6$ (see [12]). In particular given a desingularization $\widehat{\mathcal{R}}_{g,G}\to\overline{\mathcal{R}}_{g,G}$, and denoting by $\overline{\mathcal{R}}_{g,G}^{\operatorname{reg}}$ the sublocus of regular points, we know that $H^{0}\left(\widehat{\mathcal{R}}_{g,G},nK_{\widehat{\mathcal{R}}_{g,G}}\right)\subset H^{0}\left(\overline{\mathcal{R}}_{g,G}^{\operatorname{reg}},nK_{\overline{\mathcal{R}}_{g,G}}\right)$ and we would like to prove

[TABLE]

for $n$ sufficiently big and divisible. This condition is verified locally for smooth points and canonical singularities, it remains to treat the non-canonical locus $\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{R}}_{g,G}$.

First, we recall the structure of $\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{M}}_{g}$. Consider a curve $[C]\in\overline{\mathcal{M}}_{g}$, then $[C]$ is a non-canonical singularity if and only if $C$ admits an elliptic tail automorphism of order $6$. This means that $C$ has an irreducible component which is an elliptic curve $E$ such that $E\cap\overline{C\backslash E}$ is a single point and $E$ admits an automorphism of order $6$. In the following, we denote by $(E,p_{*})$ the elliptic tail of a curve $C$, where $p_{*}$ is the preimage of the node. Moreover we define $C_{1}:=\overline{C\backslash E}$ and we mark the preimage $p_{*}^{\prime}$ of the node on $C_{1}$. With this notation $C=C_{1}\cup E$.

Following again [20], we introduce the $T$-curves over $\mathcal{R}_{g,G}$.

Definition 4.0.1** ($T$-curve).**

A twisted $G$-cover $(\mathsf{C},\phi)$ is a $T$-curve if there exists an automorphism $\mathsf{a}\in\underline{\operatorname{Aut}}(\mathsf{C},\phi)$ such that its coarsening $a$ is an elliptic tail automorphism of order $6$. The locus of $T$-curves in $\overline{\mathcal{R}}_{g,G}$ is denoted by $T_{g,G}$.

In order to introduce $J$-curves, we recall the definition of the age invariant associated to a linear automorphism. Given a finite order automorphism $\mathsf{h}\in\operatorname{GL}(m)$, we can diagonalize it as $\mathsf{h}=\operatorname{Diag}\left(\xi_{r_{1}}^{a_{1}},\dots,\xi_{r_{m}}^{a_{m}}\right)$, where $\xi_{r_{i}}$ is an $r_{i}$th root of the unit. Then, its age is defined as $\operatorname{age}(\mathsf{h}):=\sum(a_{i}/r_{i})$. A finite subgroup of $\operatorname{GL}(m)$ with no quasireflections is junior if it contains a non-trivial element $\mathsf{h}$ with $\operatorname{age}(\mathsf{h})<1$. While the notion of age depends on the choice of the roots $\xi_{r_{i}}$, the notion of junior group is independent from this choice. For a wider introduction to the age invariant see [20, §5.1.1].

Definition 4.0.2** ($J$-curve).**

A twisted $G$-cover $(\mathsf{C},\phi)$ is a $J$-curve if the group

[TABLE]

which is the group of ghosts quotiented by its subgroup of quasireflections, is junior (see [20, Definition 4.1.2] for the notion of ghost automorphism). The locus of $J$-curves in $\overline{\mathcal{R}}_{g,G}$ is denoted by $J_{g,G}$.

As proved in [20, Theorem 5.1.8], $\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{R}}_{g,G})$ is the union of $T_{g,G}$ and $J_{g,G}$. We consider the cases where the second one is empty, and therefore we treat the $T$-locus by generalizing the Harris-Mumford technique for $\overline{\mathcal{M}}_{g}$. In particular, we focus in the case of $G$ abelian group and in the case of $G=S_{3}$ the symmetric group of order $3$.

4.1. The case $G$ abelian group

Lemma 4.1.1**.**

Consider a twisted $G$-cover $(\mathsf{C},\phi)$ with $G$ finite abelian group. If $(\mathsf{C},\phi)$ is a $T$-curve, then the restriction of the cover to the elliptic tail is trivial.

Proof.

By Proposition 2.3.1, the set of admissible $G$-covers over the elliptic curve $(E,p_{*})$ (that is the set of twisted $G$-covers over the same curve) is in bijection with the set of maps $\varpi\colon\pi_{1}(E_{\operatorname{gen}},p_{*})\to G$. If $(E,p_{*})$ admits an order $6$ automorphism $\mathsf{a}_{6}$, then $E\cong\mathbb{C}/(\mathbb{Z}\oplus\mathbb{Z}\cdot\Omega)$, where $\Omega$ is a primitive $6$th root of the unit and $p_{*}$ is the origin. Therefore $\mathsf{a}_{6}$ acts on $E$ as multiplication by $\Omega$. The fundamental group $\pi_{1}(E,p_{*})\subset\pi_{1}(E_{\operatorname{gen}},p_{*})$ is generated by $a$ and $b$ which are the classes of the two laces $\gamma_{a}$ and $\gamma_{b}$ such that

[TABLE]

We have as a consequence $\mathsf{a}_{6}(a)=b$, and $\mathsf{a}_{6}(b)=ba^{-1}$. Therefore if we call $\varpi^{\prime}$ the map $\mathsf{a}_{6}^{*}\varpi$, by Proposition 2.3.1, $\mathsf{a}_{6}$ lifts to the cover if and only if $\varpi^{\prime}=\varpi$. This is true if and only if $\varpi\equiv 1$, i.e. the restriction of $(\mathsf{C},\phi)$ to the elliptic tail must be trivial.∎

Theorem 4.1.2**.**

In the case of a moduli space $\overline{\mathcal{R}}_{g,G}$ of twisted $G$-covers with $G$ finite abelian group, we consider a desingularization $\widehat{\mathcal{R}}_{g,G}\to\overline{\mathcal{R}}_{g,G}$. If the locus $J_{g,G}\subset\overline{\mathcal{R}}_{g,G}$ is empty, then

[TABLE]

for $n$ sufficiently big and divisibile.

In [22] the same is proved for the moduli space $\overline{\mathcal{M}}_{g}$. The idea is the following. Consider a general non-canonical singularity of $\overline{\mathcal{M}}_{g}$, that is a point $[C]$ where $C=C_{1}\cup E$, $C_{1}$ is smooth of genus $g-1$ and without automorphisms, and $E$ is an elliptic tail admitting an order $6$ automorphism. Consider the operation of gluing any elliptic tail $E^{\prime}$ at $C_{1}$ along the same node. This gives an immersion

[TABLE]

and the image of $\Psi$ passes through the point $[C]$. Furthermore, there exists a neighborhood $S=S([C])$ of $\operatorname{Im}\Psi$ in $\overline{\mathcal{M}}_{g}$ with the following properties:

1. (1)

it exists a smooth $(3g-3)$-dimensional variety $B$ and a birational morphism $g\colon S\to B$; 2. (2)

it exists a subvariety $Z\subset B$ of codimension $2$ such that $g^{-1}(B\backslash Z)\cong B\backslash Z$; 3. (3)

as $B\backslash Z\subset S\subset\overline{\mathcal{M}}_{g}$, we have $B\backslash Z\subset\overline{\mathcal{M}}_{g}^{0}\subset\overline{\mathcal{M}}_{g}^{\operatorname{reg}}$, where $\overline{\mathcal{M}}_{g}^{0}$ is the subspace of stable curves with trivial automorphism group.

[TABLE]

This allows to conclude. Indeed, for every pluricanonical form $\omega$ on $S([C])^{\operatorname{reg}}$, we consider its restriction to $B\backslash Z$, this extends to the smooth variety $B$ and pullbacks to a desingularization $\widehat{S}([C])\to S([C])\to B$. We use the Ludwig approach for $\overline{\mathcal{R}}_{g,\bm{\mu}_{2}}$ developed in [24]. In order to complete this, we need a generalization of the age tools and the age criterion [20, Proposition 5.1.4].

Proposition 4.1.3** (see Appendix $1$ to $\S 1$ of [22]).**

Consider a complex vector space $V\cong\mathbb{C}^{n}$, $\mathfrak{G}\subset\operatorname{GL}(V)$ finite subgroup, a desingularization $\widehat{V/\mathfrak{G}}\to V/\mathfrak{G}$ and a $\mathfrak{G}$-invariant pluricanonical form $\omega$ on $V$. Consider an element $\mathsf{h}$ in $\mathfrak{G}$, $V^{0}\subset V$ the subset where $\mathfrak{G}$ acts freely and $\operatorname{Fix}(\mathsf{h})\subset V$ the fixed point set of $\mathsf{h}$. By an abuse of notation we denote by $\operatorname{Fix}(\mathsf{h})$ also the image of the fixed point set in $V/\mathfrak{G}$. Let $U\subset V/\mathfrak{G}$ be an open subset such that $V^{0}/\mathfrak{G}\subset U$ and such that for every $\mathsf{h}$ with $\operatorname{age}\mathsf{h}<1$ (with respect to some primitive root of the unit), the intersection $U\cap\operatorname{Fix}(\mathsf{h})$ is non-empty. We denote by $\widehat{U}\subset\widehat{V/\mathfrak{G}}$ the preimage of $U$ under the desingularization.

If $\omega$, as a meromorphic form on $\widehat{V/\mathfrak{G}}$, is holomorphic on $\widehat{U}$, then it is holomorphic on $\widehat{V/\mathfrak{G}}$.

Proof of Theorem 4.1.2.

Consider a pluricanonical form $\omega$ on $\overline{\mathcal{R}}_{g,G}^{\operatorname{reg}}$. We show that $\omega$ lifts to a desingularization of an open neighborhood of every point $[\mathsf{C},\phi]$ of $\overline{\mathcal{R}}_{g,G}$.

If $[\mathsf{C},\phi]$ is a canonical singularity this is obvious by definition.

If $[\mathsf{C},\phi]$ is a non-canonical singularity, at first we consider the case of a general non-canonical singularity. As $\operatorname{Sing}^{\operatorname{nc}}\overline{\mathcal{R}}_{g,G}=T_{g,G}$, then by Lemma 4.1.1 a general point $[\mathsf{C},\phi]\in T_{g,G}$ is a $T$-curve $\mathsf{C}$ whose coarse space $C$ has two irreducible components $(E,p_{*})$, an elliptic tail, and $(C_{1},p_{*}^{\prime})$ of genus $g-1$. Moreover, if $(\mathsf{C}_{1},\phi_{1})$ and $(\mathsf{E},\phi_{\mathsf{E}})$ are the restrictions, then $(\mathsf{E},\phi_{\mathsf{E}})$ is the trivial cover of $E$, $\mathsf{C}_{1}=C_{1}$ and $\underline{\operatorname{Aut}}(\mathsf{C}_{1},\phi_{1})$ is trivial.

Once we fix the twisted $G$-cover $(C_{1},\phi_{1})$, we consider the morphism $\Psi_{1}\colon\overline{\mathcal{M}}_{1,1}\to\overline{\mathcal{R}}_{g,G}$ sending any point $[E^{\prime}]$ of $\overline{\mathcal{M}}_{1,1}$ to the point $[C,\phi]$ obtained by joining $C_{1}$ and $E^{\prime}$ along their marked points, and by considering the $G$-cover $\phi$ such that $\phi|_{C_{1}}=\phi_{1}$ and $\phi|_{E^{\prime}}$ is trivial.

Following [24], we see that the projection $\pi\colon\overline{\mathcal{R}}_{g,G}\to\overline{\mathcal{M}}_{g}$ sends $\operatorname{Im}\Psi_{1}$ isomorphically on $\operatorname{Im}\Psi$, and $\pi|_{\operatorname{Im}\Psi}$ is a local isomorphism. Indeed, $\operatorname{Def}(C;\phi)=\operatorname{Def}(C)$ and for every point of $\operatorname{Im}\Psi_{1}$ the automorphism group $\underline{\operatorname{Aut}}(C,\phi)$ is isomorphic to $\operatorname{Aut}(C)$. Therefore if we consider the neighborhood $S([C])$ of $[C]\in\operatorname{Im}\Psi$ introduced by Harris and Mumford, then up to shrinking $\pi^{-1}S([C])\cong S([C])$, we have a neighborhood of $[C,\phi]$ with the same properties.

It remains to consider the case of any non-canonical singularity $[\mathsf{C},\phi]$. Here $\mathsf{C}$ is a twisted curve such that $\mathsf{C}=\mathsf{C}_{1}\cup\bigcup\mathsf{E}^{(i)}$, where the $\mathsf{E}^{(i)}$ are all the elliptic tails of $\mathsf{C}$ admitting an elliptic tail automorphism of order $6$. Again we follow the last part of the Ludwig's demonstration of [24, Theorem 4.1]. We consider for each $i$ a small deformation $(\mathsf{C}^{(i)},\phi^{(i)})$ of $(\mathsf{C},\phi)$ which fixes the $i$th elliptic tail. That is, $\mathsf{C}^{(i)}=\mathsf{C}_{1}^{(i)}\cup\mathsf{E}^{(i)}$ where $\mathsf{C}_{1}^{(i)}$ is irreducible. Moreover, the twisted $G$-cover admits no non-trivial automorphism over $\mathsf{C}_{1}^{(i)}$ and it is unchanged over $\mathsf{E}^{(i)}$. By the previous point we consider $S^{(i)}:=S([\mathsf{C}^{(i)},\phi^{(i)}])$. Up to shrinking the open subsets $S^{(i)}$, they are all disjoint. Given the point $[\mathsf{C},\phi]$ of $\overline{\mathcal{R}}_{g,G}$, we consider the local picture of its universal deformation $V:=\operatorname{Def}(\mathsf{C},\phi)\cong\mathbb{C}^{3g-3}$, and recall that the local picture at $[\mathsf{C},\phi]$ is the same of $V/\underline{\operatorname{Aut}}(\mathsf{C},\phi)$ at the origin. We define

[TABLE]

If $V^{0}$ is the $V$ subset where $\underline{\operatorname{Aut}}(\mathsf{C},\phi)$ acts freely, let $U\subset S([\mathsf{C},\phi])$ be the set

[TABLE]

If $\omega$ is a pluricanonical holomorphic form on $S^{\operatorname{reg}}$, then it extends to $\widehat{S}^{(i)}$ by definition of the neighborhoods $S^{(i)}$. Moreover, by applying Proposition 4.1.3 to the subset $U$, $\omega$ extends to $\widehat{\frac{V}{\underline{\operatorname{Aut}}(\mathsf{C},\phi)}}$, and therefore to the whole $\widehat{S}$.∎

Remark 4.1.4**.**

If we consider $G=\bm{\mu}_{\ell}$, we observe that this is the case (treated in [12]) of the moduli space $\overline{\mathcal{R}}_{g,\ell}^{0}$ of curves equipped with a line bundle which is an $\ell$th root of the trivial bundle. The proof above applies with minor adjustments also to the case of the moduli space $\overline{\mathcal{R}}_{g,\ell}^{k}$ for any $k$ (see [19]), that is the moduli space of curves with an $\ell$th root of $\omega^{k}$. Therefore the extension result is true also in this case if the $J$-locus $J_{g,\ell}^{k}$ of $\overline{\mathcal{R}}_{g,\ell}^{k}$ is empty.

4.2. The case $G=S_{3}$

We know from [20] that in $\overline{\mathcal{R}}_{g,S_{3}}$ the non-canonical singular locus coincides with the $T$-locus.

Lemma 4.2.1**.**

Consider a twisted $S_{3}$-cover $(\mathsf{C},\phi)$ which is a $T$-curve. If $\mathsf{E}$ is an elliptic tail admitting an elliptic tail automorphism $\mathsf{a}_{6}$ of order $6$, then the restriction $(\mathsf{E},\phi_{\mathsf{E}})$ is a trivial cover or an admissible $N$-cover, i.e. in this last case it has two connected components and trivial $S_{3}$-type on the marked point $p_{*}$.

Proof.

We follow the same approach of Lemma 4.1.1. By Proposition 2.3.1, the set of admissible $S_{3}$-covers on $E$ is in bijection with the conjugacy classes of maps $\varpi\colon\pi_{1}(E_{\operatorname{gen}},p_{*})\to S_{3}$. As before we note that if $(E,p_{*})$ is an elliptic curve admitting and order $6$ automorphism, then $E\cong\mathbb{C}/(\mathbb{Z}\oplus\mathbb{Z}\cdot\Omega)$, where $\Omega$ is a primitive $6$th root of the unit and $p_{*}$ is the origin. Here $\mathsf{a}_{6}$ acts as multiplication by $\Omega$. If $a$ and $b$ are defined as before, then again $\mathsf{a}_{6}(a)=b$ and $\mathsf{a}_{6}(b)=ba^{-1}$. If $\varpi^{\prime}=\mathsf{a}_{6}^{*}\varpi$, then by Proposition 2.3.1, $\mathsf{a}_{6}$ lifts to the cover if and only if $\varpi^{\prime}=h\cdot\varpi\cdot h^{-1}$ for some element $h\in S_{3}$. This is true in the case $\varpi\equiv 1$ and for every group morphism $\varpi\colon\pi_{1}(E)\twoheadrightarrow N\subset S_{3}$, where $N$ is the normal non-trivial subgroup of $S_{3}$, that is the group generated by any $3$-cycle. By definition of admissible $N$-cover, the thesis follows.∎

As observed in §3.1.1, there exists an isomorphism, $\overline{\mathcal{R}}_{1,S_{3}}^{1}\cong\overline{\mathcal{M}}_{1,1}$. We remarked in the same section that there exists a natural $2:1$ morphism $\overline{\mathcal{R}}_{1,\bm{\mu}_{3}}^{\bm{\mu}_{3}}\to\overline{\mathcal{R}}_{1,S_{3}}^{N}$. As the natural morphism $\overline{\mathcal{R}}_{1,\bm{\mu}_{3}}^{\bm{\mu}_{3}}\to\overline{\mathcal{M}}_{1,1}$ is $8:1$, we have that $\Psi_{N}\colon\overline{\mathcal{R}}_{1,S_{3}}^{N}\to\overline{\mathcal{M}}_{1,1}$ is a $4:1$ morphism.

Remark 4.2.2**.**

Consider $(E,p_{*})$ a general elliptic curve, we list the four preimages of $[E,p_{*}]\in\overline{\mathcal{M}}_{1,1}$ with respect to $\Psi_{N}$. We characterize every class with a representative $\varpi\colon\pi_{1}(E,p_{*})\twoheadrightarrow N$ of the correspondent conjugacy class of morphisms. We recall that $\pi_{1}(E,p_{*})$ is a free abelian group generated by $a$ and $b$.

[TABLE]

We recall that $\overline{\mathcal{M}}_{1,1}\cong\mathbb{P}^{1}$, $\overline{\mathcal{R}}_{1,S_{3}}^{N}$ is a connected curve and we are interested in finding the branch points for $\Psi_{N}$. We observe that the only automorphism of a general elliptic curve is the natural involution $i$, and this always lifts to any admissible $S_{3}$-cover. Following Remark 2.2.18, we detect a branch point in three cases.

• •

The elliptic curve $E_{4}$ admitting an automorphism $\mathsf{a}_{4}$ of order $4$. Automorphism $\mathsf{a}_{4}$ acts on $\pi_{1}(E_{4},p_{*})$ by sending $a\mapsto b$ and $b\mapsto a^{-1}$, therefore it does not lift to any admissible $N$-cover: in particular it exchanges classes $(i)$ and $(iv)$, and also classes $(ii)$ and $(iii)$. Therefore over $[E_{4}]\in\overline{\mathcal{M}}_{1,1}$ we have two branch points of order $2$ that we denote by $[E_{4}^{\prime}]$ and $[E_{4}^{\prime\prime}]$.

• •

The elliptic curve $E_{6}$ admitting an automorphism $\mathsf{a}_{6}$ of order $6$. Automorphism $\mathsf{a}_{6}$ acts on $\pi_{1}(E_{6},p_{*})$ by sending $a\mapsto b$ and $b\mapsto ba^{-1}$, therefore it lifts to the cover $(iii)$. Moreover, it sends $(i)$ to $(ii)$, $(ii)$ to $(iv)$ and $(iv)$ to $(i)$. Therefore over $[E_{6}]\in\overline{\mathcal{M}}_{1,1}$ we have one branch point $[E_{6}^{\prime}]$ of order $3$, and moreover we have another preimage point $[E_{6}^{\prime\prime}]$.

• •

The curve $E_{0}$ with an autointersection node $q_{1}$, whose normalization $\overline{E}_{0}$ is a rational curve. By Remark 2.2.18, $[E_{0}]\in\overline{\mathcal{M}}_{1,1}$ has two preimages via $\Psi_{N}$, one is obtained by putting an order $3$ stabilizer at $q_{1}$, we call this preimage $[E_{0}^{\prime}]$ and it is an order $3$ branch point. The other preimage, called $[E_{0}^{\prime\prime}]$, is associated to an admissible $N$-cover with trivial stabilizer at $q_{1}$.

Lemma 4.2.3**.**

The moduli space $\overline{\mathcal{R}}_{1,S_{3}}^{N}$ is isomorphic to $\mathbb{P}^{1}$.

Proof.

This is a consequence of Hurwitz formula plus the observations we just listed.∎

We consider a general genus $g-1$ curve with one marked point $(C_{1},p_{*}^{\prime})$ and a twisted $S_{3}$-cover $(C_{1},\phi_{1})$ with trivial $S_{3}$-type at $p_{*}^{\prime}$. For any twisted $G$-cover $(\mathsf{E},\phi_{\mathsf{E}})$ which is an admissible $N$-cover over an elliptic curve $(E,p_{*})$, we glue $(C_{1},\phi_{1})$ and $(\mathsf{E},\phi_{\mathsf{E}})$ along their marked points, to get a node $q$. We obtain a map $\Upsilon\colon\overline{\mathcal{R}}_{1,S_{3}}^{N}\to\overline{\mathcal{R}}_{g,S_{3}}$ which is an isomorphism into the image and fits into the diagram

[TABLE]

Therefore the projection $\pi|_{\operatorname{Im}\Upsilon}\colon\operatorname{Im}\Upsilon\to\operatorname{Im}\Psi$ is the $4:1$ morphism we described above.

Remark 4.2.4**.**

Before stating the main theorem of this section, we describe the local picture of $\pi$ at the points of $\operatorname{Im}\Upsilon$. We call $(\mathsf{C},\phi)$ the admissibile $S_{3}$-cover obtained via the gluing. By Remark 2.2.18, the local picture of $\overline{\mathcal{R}}_{g,S_{3}}$ at $[\mathsf{C},\phi]$ is

[TABLE]

Here $\tilde{t}_{q}$ and $\tilde{t}_{E}$ are the (non-canonical) coordinates associated to the smoothing of node $q$ and to the deformation of $(\mathsf{E},\phi_{\mathsf{E}})$. At the same time the local picture of $\overline{\mathcal{M}}_{g}$ at $[C]$ is

[TABLE]

Here $t_{q}$ and $t_{E}$ are associated to the smoothing of node $q$ and the deformation of the elliptic curve $E$. As the $S_{3}$-type at $q$ is trivial, $\tilde{t}_{q}=t_{q}$, while for the other coordinate there are different cases. As the automorphism group acts non-trivially only on $t_{q},t_{E}$ and $\tilde{t}_{E}$, we focus on these coordinates. Moreover we recall that the canonical elliptic tail involution of $E$, acts trivially.

We consider again the case treated by Harris and Mumford in [22], the birational morphism $g\colon S([C])\to B$ at a general non-canonical singularity $[C]$. This means that the associated curve $C$ is the junction of $(C_{1},p_{*}^{\prime})$, a genus $g-1$ smooth automorphism free curve, and $(E,p_{*})$, an elliptic tail. By construction $S([C])\cong S_{1}\times\operatorname{Def}(C_{1},p_{*}^{\prime})$ and $B\cong B_{1}\times\operatorname{Def}(C_{1},p_{*}^{\prime})$, where $S_{1}$ is a singular surface, $B_{1}$ is a smooth surface and $g$ is the identity on the component $\operatorname{Def}(C_{1},p_{*}^{\prime})$. We denote by $D$ the projection of $\operatorname{Im}\Psi$ on $S_{1}$. In particular the coordinates $t_{q},t_{E}$ span exactly the tangent space to $S_{1}$ at any point of $D$.

If $[\mathsf{C},\phi]$ is a general non-canononical singularity or $\overline{\mathcal{R}}_{g,S_{3}}$, then the curve coarse space $C$ is a union $C_{1}\cup E$ as before. We denote by $S([\mathsf{C},\phi])$ the component of $\pi^{-1}(S([C]))$ containing $[\mathsf{C},\phi]$. By Lemma 4.2.1, the restriction $\phi_{\mathsf{E}}$ is trivial or an admissible $N$-cover, we focus in the second case and we observe that as a consequence $\operatorname{Im}\Upsilon\subset S([\mathsf{C},\phi])$. Following the construction above, $S([\mathsf{C},\phi])=S_{2}\times\operatorname{Def}(C_{1},p_{*}^{\prime})$, where $S_{2}$ is a singular surface. We call $D^{\prime}$ the projection of $\operatorname{Im}\Upsilon$ on $S_{2}$. With an abuse of notation we call $\pi\colon S_{2}\to S_{1}$ the natural projection.

In the following list we describe the local picture of $S_{2}\to S_{1}$ at any point. For the sake of simplicity we identify $D^{\prime}$ with $\overline{\mathcal{R}}_{1,S_{3}}^{N}$ and $D$ with $\overline{\mathcal{M}}_{1,1}$. We use the notation $\left(\frac{a_{1}}{r_{1}},\cdots,\frac{a_{m}}{r_{m}}\right)$ to denote the diagonal matrix $\operatorname{Diag}(\xi_{r_{1}}^{a_{1}},\dots,\xi_{r_{m}}^{a_{m}})$, where $\xi_{r}=\exp(2\pi i/r)$ is the privileged $r$th root of the unit.

• •

At the two points $[E_{4}^{\prime}],[E_{4}^{\prime\prime}]\in S_{2}$, the automorphism $\mathsf{a}_{4}$ does not lift. It acts as $t_{q}\mapsto\xi_{2}t_{q}$ and $t_{E}\mapsto\xi_{2}t_{E}$ locally at $[E_{4}]\in S_{1}$, therefore the local picture of $\pi$ at $[E_{4}^{\prime}]$ and $[R_{4}^{\prime\prime}]$ is the canonical projection $\mathbb{C}^{2}\to\mathbb{C}^{2}/\left(\frac{1}{2},\frac{1}{2}\right)$.

• •

At the two points $[E_{6}^{\prime}],[E_{6}^{\prime\prime}]\in S_{2}$, the automorphism $\mathsf{a}_{6}$ does not lift to $[E_{6}^{\prime}]$ but it lifts to $[E_{6}^{\prime\prime}]$. It acts as $t_{q}\mapsto\xi_{3}t_{q}$ and $t_{E}\mapsto\xi_{3}t_{E}$ at $[E_{6}]\in S_{1}$, therefore the local picture of $\pi$ at $[E_{6}^{\prime}]$ is the projection $\mathbb{C}^{2}\to\mathbb{C}^{2}/\left(\frac{1}{3},\frac{1}{3}\right)$, and the local picture of $\pi$ at $[E_{6}^{\prime\prime}]$ is the identity $\mathbb{C}^{2}/\left(\frac{1}{3},\frac{1}{3}\right)\to\mathbb{C}^{2}/\left(\frac{1}{3},\frac{1}{3}\right)$.

• •

At the two points $[E_{0}^{\prime}],[E_{0}^{\prime\prime}]\in S_{2}$, we have that $t_{E}=\tilde{t}_{E}^{3}$ in the case of $[E_{0}^{\prime}]$ because of the definition of admissible $S_{3}$-cover. Instead in the case of $[E_{0}^{\prime\prime}]$, $t_{E}=\tilde{t}_{E}$. In both cases there are no additional automorphisms, therefore the local picture of $\pi$ at $[E_{0}^{\prime}]$ is the projection $\mathbb{C}^{2}\to\mathbb{C}^{2}/\left(1,\frac{1}{3}\right)\cong\mathbb{C}^{2}$, and the local picture at $[E_{0}^{\prime\prime}]$ is the identity $\mathbb{C}^{2}\to\mathbb{C}^{2}$.

• •

Elsewhere on $D^{\prime}$ and on the whole surface $S_{2}$, the projection $\pi$ is étale, therefore its local picture is the identity $\mathbb{C}^{2}\to\mathbb{C}^{2}$.

Theorem 4.2.5**.**

Consider a desingularization $\widehat{\mathcal{R}}_{g,S_{3}}\to\overline{\mathcal{R}}_{g,S_{3}}$ of the moduli space of genus $g$ curves equipped with an admissible $S_{3}$-cover. Then,

[TABLE]

for $n$ sufficiently big and divisible.

Proof.

We are going to prove the result for a general non-canonical singularity of $\overline{\mathcal{R}}_{g,S_{3}}$. As a consequence, the extension of pluricanonical forms is true also for any non-canonical singularity via a patchwork of ``good'' neighborhoods analogous to what we do in the proof of Theorem 4.1.2.

Consider a general non-canonical singularity $[\mathsf{C},\phi]\in\overline{\mathcal{R}}_{g,S_{3}}$, therefore $\mathsf{C}=C_{1}\cup\mathsf{E}$ as described before and by Lemma 4.2.1 the restriction $\phi_{\mathsf{E}}$ is trivial or it is an admissible $N$-cover. In the first case, $[\mathsf{C},\phi]$ is in a component of $\pi^{-1}\left(\operatorname{Im}\Psi\right)$ which is isomorphic to $\operatorname{Im}\Psi$ and the construction of $S([\mathsf{C},\phi])$ is the same of Theorem 4.1.2. In the second case, $[\mathsf{C},\phi]$ is in $\operatorname{Im}\Upsilon$ and we are going to show a neighborhood $S([\mathsf{C},\phi])=S_{2}\times\operatorname{Def}(C_{1},p_{*}^{\prime})$ and a birational morphism $S_{2}\to B_{2}$ such that $B_{2}$ has at most canonical singularities.

Consider the blowup at the points $[E_{4}],[E_{6}]\in D\subset S_{1}$. We denote by $A_{4}$ and $A_{6}$ the two corresponding exceptional curves, by the description of singularities we gave above, the autointersection numbers of these divisors are $A_{4}^{2}=-2$ and $A_{6}^{2}=-3$. We denote by $S_{1}^{*}$ the blown up surface and by $\overline{D}$ the strict transform of $D$ after the blowups. We know by the Harris-Mumford result resumed above, that there exists a contraction of the curve $A_{4}\cup A_{6}\cup\overline{D}$, and the contracted surface is smooth. By [6, Theorem II.11] we must have a sequence of $(-1)$-curve contractions, and this happens if and only if $\overline{D}^{2}=-1$, and all the three curves are rational.

Consider the blowup at the points $[E_{4}^{\prime}],[E_{4}^{\prime\prime}],[E_{6}^{\prime}],[E_{6}^{\prime\prime}]\in D^{\prime}\subset S_{2}$. We denote by $A_{4}^{\prime},A_{4}^{\prime\prime},A_{6}^{\prime},A_{6}^{\prime\prime}$ the corresponding exceptional divisors, by $S^{*}_{2}$ the blown up surface and by $\overline{D}^{\prime}$ the strict transform of $D^{\prime}$. From the description of the singularities we gave above, we know $(A_{4}^{\prime})^{2}=(A_{4}^{\prime\prime})^{2}=(A_{6}^{\prime})^{2}=-1$ and $(A_{6}^{\prime\prime})^{2}=-3$. Moreover, after the blowups there exists a morphism $\tilde{\pi}\colon S^{*}2\to S^{*}1$ of degree $4$ and whose ramification locus is $2A_{6}^{\prime}$ (plus some component of codimension $2$). Therefore $\tilde{\pi}^{*}\overline{D}=4\overline{D}^{\prime}$, and so $(\overline{D}^{\prime})^{2}=-4$. By costruction all the $A_{i}^{\prime}$ and $A_{i}^{\prime\prime}$ are rational curves and $\overline{D}^{\prime}$ is a rational curve too by Lemma 4.2.3. We obtain that there exists a birational morphism $S_{2}\to B_{2}$ contracting the curve $A_{4}^{\prime}\cup A_{4}^{\prime\prime}\cup A_{6}^{\prime}\cup A_{6}^{\prime\prime}\cup D^{\prime}$. Indeed, by [5, Theorem III.5.1] there exists such a contraction and $B_{2}$ has a singularity of type $\mathbb{C}^{2}/\left(\frac{1}{2},\frac{1}{2}\right)$, that is a canonical singularity.

We obtained a contraction $g^{\prime}\colon S([\mathsf{C},\phi])\to B^{\prime}=B_{2}\times\operatorname{Def}(C_{1},p_{*}^{\prime})$ such that $B^{\prime}$ has only canonical singularities, moreover there exists a locus $Z^{\prime}\subset B^{\prime}$ of codimension $2$ such that $(g^{\prime})^{-1}(B^{\prime}\backslash Z^{\prime})\cong B^{\prime}\backslash Z^{\prime}$. By construction $(g^{\prime})^{-1}(B^{\prime}\backslash Z^{\prime})$ is naturally a subset of $\overline{\mathcal{R}}_{g,S_{3}}$ and this allows to conclude as in the previous case. ∎

5. Evaluating the Kodaira dimension

In order to calculate the Kodaira dimension, we need to develop some calculations in the tautological ring of the moduli space. In §5.1 we develop Grothendieck Riemann-Roch type calculations for vector bundles. In §5.2 we apply it to evaluate the canonical divisor $K_{\overline{\mathcal{R}}_{g,S_{3}}}$ and prove its bigness over $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, the connected component of $\overline{\mathcal{R}}_{g,S_{3}}$ of connected $S_{3}$-covers.

5.1. Adapted Grothendieck Riemann-Roch

5.1.1. Tautological classes

In this section we recall some well known tautological classes in the Chow ring $A^{*}(\overline{\mathcal{M}}_{g,n})$ of the moduli space of curves, and their generalizations to $\overline{\mathcal{R}}_{g,G}$. For a wider survey of the tautological relations and the tautological rings structure see [4, §17], [26] and [14].

There exists two natural morphisms ``coming from the geometry of curves'' on the moduli spaces $\overline{\mathcal{M}}_{g,n}$. With this we mean that we can define them using the modular interpretation of the space.

• •

The forgetful morphism is a morphism

[TABLE]

sending any geometric point $[C;p_{1},\dots,p_{n}]$ to the same marked stable curve without the last point, $[C;p_{1},\dots,p_{n-1}]$.

• •

The gluing morphisms are of two types

[TABLE]

In the first case $[C;p_{1},\dots,p_{n_{1}+1}]\times[C^{\prime};p_{1}^{\prime},\dots,p_{n_{2}+1}^{\prime}]$ is sent two the junction of $C$ and $C^{\prime}$ along the marked points $p_{n_{1}+1}$ and $p_{n_{2}+1}^{\prime}$. The new curve maintains all the other marked points. In the second case $[C;p_{1},\dots,p_{n+2}]$ is sent to the quotient curve $C/(p_{n+1}\sim p_{n+2})$ with the same other marked points.

Definition 5.1.1**.**

The system of tautological rings $R^{*}(\overline{\mathcal{M}}_{g,n})\subset A^{*}(\overline{\mathcal{M}}_{g,n})$ with $g,n$ varying on the non-negative integers, is the smallest system of $\mathbb{Q}$-algebras closed under the pushforwards of the forgetful morphisms and the gluing morphisms.

We define $n$ tautological $\psi$-classes inside the Chow ring $A^{*}(\overline{\mathcal{M}}_{g,n})$. We consider the universal family $u\colon\mathcal{C}_{g,n}\to\overline{\mathcal{M}}_{g,n}$, where $\mathcal{C}_{g,n}$ is a Deligne-Mumford stack such that every geometric fiber of $u$ is isomorphic to the associated $n$-marked stable curve, and there exist $n$ sections $\sigma_{1},\dots,\sigma_{n}\colon\overline{\mathcal{M}}_{g,n}\to\mathcal{C}_{g,n}$. For every $i=1,\dots,n$ the line bundle $\mathcal{L}_{i}$ over $\overline{\mathcal{M}}_{g,n}$ is the $i$th cotangent line bundle, $\mathcal{L}_{i}:=\sigma_{i}^{*}\left(T_{u}^{\vee}\right)$. Then we define

[TABLE]

Remark 5.1.2**.**

The $\psi$-classes are in the tautological ring $R^{1}(\overline{\mathcal{M}}_{g,n})$, as showed for example in [14].

There are two other type of classes that are very important for our analysis and belong to the tautological ring. To introduce $\kappa$-classes we consider the log-canonical line bundle on $\mathcal{C}_{g,n}$, $\omega_{u}^{\log}:=\omega_{u}(\sigma_{1}+\dots+\sigma_{n})$. Therefore we have

[TABLE]

As before the class $\kappa_{d}$ is well defined in the Chow ring $A^{d}(\overline{\mathcal{M}}_{g,n})$. We state without proof the known fact that $\kappa_{d}=\mu_{*}(\psi_{n+1}^{d+1}$) and being $\psi_{n+1}$ in the tautological ring, the $\kappa$-classes is contained in the tautological ring too.

The Hodge bundle over $\overline{\mathcal{M}}_{g,n}$ is the rank $g$ vector bundle $\mathcal{E}:=u_{*}\omega_{u}$, i.e. the vector bundle whose fiber at $[C;p_{1},\dots,p_{n}]$ is $H^{0}(C;\omega_{C})$. The Hodge class is

[TABLE]

The Hodge class is proved to be in the tautological ring $R^{1}(\overline{\mathcal{M}}_{g,n})$ in [25].

We define the universal family also on $\overline{\mathcal{R}}_{g,G}$ and we use the same notation $u\colon\mathcal{C}_{g,G}\to\overline{\mathcal{R}}_{g,G}$. We will see in the next section that this family is equipped with a universal twisted $G$-cover. On $A^{*}\left(\overline{\mathcal{R}}_{g,G}\right)$ it is possible to define as before the $\kappa$ classes and the Hodge class $\lambda$.

5.1.2. Using Grothendieck Riemann-Roch in the Chow ring of $\mathcal{R}_{g,G}$

The universal family of curves over $\mathcal{C}_{g,G}\to\overline{\mathcal{R}}_{g,G}$ is equipped with a universal twisted $G$-cover $\Phi\colon\mathcal{C}_{g,G}\to BG$. In particular for every geometric point $[\mathsf{C},\phi]$ of $\overline{\mathcal{R}}_{g,G}$, the restriction of $\Phi$ to the associated geometric fiber is isomorphic to the twisted $G$-cover $\phi\colon\mathsf{C}\to BG$.

We consider the singular locus $\mathcal{N}\subset\mathcal{C}_{g,G}$ of the universal family, whose points are the nodes of $\mathcal{C}_{g,G}$ fibers. Furthermore, we consider the stack $\mathcal{N}^{\prime}$ whose points are nodes equipped with the choice of a privileged branch. There exists a natural étale double cover $\mathcal{N}^{\prime}\to\mathcal{N}$, and an involution $\varepsilon\colon\mathcal{N}^{\prime}\to\mathcal{N}^{\prime}$ associated to the branch switch. There exists also a natural decomposition of $\mathcal{N}^{\prime}$: given a conjugacy class $\llbracket h\rrbracket$ in $\llbracket G\rrbracket$, we denote by $\mathcal{N}_{i,\llbracket h\rrbracket}^{\prime}\subset\mathcal{N}^{\prime}$ the substack of nodes such that the associated privileged branch is in a component of genus $i$, and it has $\llbracket h\rrbracket$ as $G$-type. In the case of a node of type [math], the component is $\mathcal{N}^{\prime}_{0,\llbracket h\rrbracket}$. Therefore

[TABLE]

We denote the natural projection by $j\colon\mathcal{N}^{\prime}\to\overline{\mathcal{R}}_{g,G}$. Furthermore, we denote by $j_{i,\llbracket h\rrbracket}$ the restriction of the map $j$ to the component $\mathcal{N}^{\prime}_{i,\llbracket h\rrbracket}$. We finally define the classes $\psi$ and $\psi^{\prime}$ on $\mathcal{N}^{\prime}$:

[TABLE]

In order to evaluate the Chern character of some line bundle pushforwards, we state a generalization of the Grothendieck Riemann-Roch formula, following the approach of Chiodo in [9]. In the case treated by Chiodo, this vector bundle is the universal root of the trivial (or canonical) line bundle. Here we generalize to the case of any vector bundle coming from a representation of the group $G$.

Consider $W$, a dimension $w$ representation of group $G$, then $W$ can be regarded as a vector bundle on $BG$. We consider the universal cover $\Phi\colon\mathcal{C}_{g,G}\to BG$, then the pullback

[TABLE]

is a vector bundle of rank $w$ on the universal family $\mathcal{C}_{g,G}$. From now on we will use the more compact notation $W_{\mathcal{C}}$.

We observe that the projection $j_{i,\llbracket h\rrbracket}\colon\mathcal{N}^{\prime}_{i,\llbracket h\rrbracket}\to\overline{\mathcal{R}}_{g,G}$ is locally isomorphic to $B\bm{\mu}_{r}\to\operatorname{Spec}\mathbb{C}$ at every point of $\overline{\mathcal{R}}_{g,G}$, where $r=r(h)$ is the order of class $\llbracket h\rrbracket$ in $G$. We follow the approach of [29, §2.2] to decompose the restriction of $W_{\mathcal{C}}$ to any locus $\mathcal{N}_{i,\llbracket h\rrbracket}^{\prime}$. The local picture of $\mathcal{C}_{g,G}$ at any point of $\mathcal{N}_{i,\llbracket h\rrbracket}^{\prime}$ is a Deligne-Mumford stack $[U/\bm{\mu}_{r}]$ where $U$ is an affine scheme, therefore the vector bundle restriction $W_{\mathcal{C}}|_{\mathcal{N}_{i,\llbracket h\rrbracket}^{\prime}}$ is a $\bm{\mu}_{r}$-equivariant vector bundle. Following [20, §2.3.2], $\bm{\mu}_{r}$ acts naturally on $W_{\mathcal{C}}|_{\mathcal{N}_{i,\llbracket h\rrbracket}^{\prime}}$, so there exists a decomposition in subbundles,

[TABLE]

where $W_{k}$ is the eigen-subbundle with eigenvalue $\xi_{r}^{k}$ for all $k=0,\dots,r-1$. We denote by $w_{i,\llbracket h\rrbracket}(k)$ the rank of $W_{k}$, and clearly these integers satisfy the equation $\sum_{k}w_{i,\llbracket h\rrbracket}(k)=w$. We also recall the Bernoulli polynomials $B_{d}(x)$ defined by the generating function

[TABLE]

The Bernoulli numbers $B_{d}:=B_{d}(0)$ are the evaluations of the Bernoulli polynomials at [math]. With this setting we can state the following.

Proposition 5.1.3**.**

On $\overline{\mathcal{R}}_{g,G}$ we have the following evaluation for the degree $d$ component of the Chern character of $Ru_{*}W_{\mathcal{C}}$.

[TABLE]

[TABLE]

This formula follows directly from Tseng formula (7.3.6.1) in [29]. In Tseng notation the morphism $ev_{n+1}$ is the $u$ morphism, the representation $W$ is denoted by $F$ and moreover

[TABLE]

is the first term in our formula, the one with $\kappa$ classes. The $\psi_{i}$-classes terms are associated to marked points and therefore are absent in our formula. Finally, the terms $A_{m}$ (see [29, Definition 4.1.2]) give the last term of our formula.

5.2. Bigness of the canonical divisor

We consider the moduli space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ of genus $g$ curves equipped with a connected admissible $S_{3}$-cover. This is a component of $\overline{\mathcal{R}}_{g,S_{3}}$.

The goal of this section is to prove that the canonical divisor of $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is big over the subspace $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}\subset\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ of covers on irreducible curves, for every odd genus $g>11$. By Proposition 3.3.2, this implies that $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is of general type for every odd genus $g\geq 13$.

The approach follows the strategy of [11] for $\overline{\mathcal{R}}_{g,\bm{\mu}_{3}}^{\bm{\mu}_{3}}$. We write down the canonical divisor as a sum

[TABLE]

Here $\mathcal{U},\bm{\mathsf{M}},E$ are effective divisors, $\lambda$ is the hodge class, $\alpha,\beta$ positive coefficients and $\gamma$ a strictly positive coefficient.

5.2.1. Basic notions of syzygy theory

The divisor $\mathcal{U}$ in equation (5.1) will be defined following the approach of Chiodo-Eisenbud-Farkas-Schreyer paper [11]. It is the locus of curves with ``extra'' syzygies with respect to a particular vector bundle, and it has a determinantal structure over an open subset of $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$. To properly define this, we recall some fundamental notions of syzygy theory over stable curves, following the notations of Aprodu-Farkas paper [2].

Consider a finitely generated graded module $N$ over the polynomial ring $S=\mathbb{C}[x_{0},\dots,x_{n}]$. The module has a minimal free resolution $0\leftarrow N\leftarrow F_{0}\leftarrow F_{1}\leftarrow\cdots$, where

[TABLE]

The numbers $b_{i,j}$ are well defined and are called the Betti numbers of $N$, moreover we have

[TABLE]

Remark 5.2.1**.**

In an irreducible flat family of modules $N_{t}$, the Betti numbers $b_{i,j}(N_{t})$ are semicontinuous, and simultaneously take minimal values on an open set. The jumping locus for their values is where one of the values $b_{i,j}(N_{t})$ is bigger than this minimum.

For every stable curve we consider a line bundle $L\in\operatorname{Pic}(C)$, a sheaf $\mathcal{F}$ on $C$, the polynomial ring $S:=\operatorname{Sym}H^{0}(C;L)$ and the graded $S$-module

[TABLE]

From now on we will use the Green notation by calling

[TABLE]

One fundamental idea of syzygy theory is that the vector space $K_{i,j}(C;\mathcal{F},L)$ can be evaluated via a minimal $S$ resolution of $\mathbb{C}$, where the latter is seen as a graded $S$-module. To do this we introduce the vector bundle $M_{L}$ of rank $H^{0}(L)-1$ over $C$, via the following short exact sequence

[TABLE]

Using this vector bundle we can state the following lemma

Lemma 5.2.2** (see [2, Theorem 2.6]).**

Consider $C$ a stable curve, $L$ a line bundle on it, $\mathcal{F}$ a coherent sheaf on it and $m$ a positive integer. Then,

[TABLE]

5.2.2. The Koszul divisor

The symmetric group $S_{3}$ has $3$ irreducible representations.

1. (1)

The trivial representation $\mathbb{I}\colon S_{3}\to\operatorname{GL}(1;\mathbb{C})=\mathbb{C}^{*}$; 2. (2)

the parity representation $\epsilon\colon S_{3}\to\mathbb{C}^{*}$, which sends even elements to $1$ and odd elements to $-1$; 3. (3)

given a vector space $R\cong\mathbb{C}^{2}$, the representation $\rho\colon S_{3}\to\operatorname{GL}(R)\cong\operatorname{GL}(2;\mathbb{C})$ such that

[TABLE]

Remark 5.2.3**.**

In particular, if we consider the tautological representation over the vector space $P:=\langle v_{1},v_{2},v_{3}\rangle_{\mathbb{C}}$ such that $S_{3}$ acts naturally by permutation, then $P$ is the direct sum of the trivial representation and $R$, $P=\mathbb{C}_{\mathbb{I}}\oplus R$. At the same time, if we consider the regular representation $\mathbb{C}[S_{3}]$ of dimension $|S_{3}|=6$, we have $\mathbb{C}[S_{3}]=\mathbb{C}_{\mathbb{I}}\oplus\mathbb{C}_{\epsilon}\oplus R^{\oplus 2}$.

As explained above, we consider $R$ as a line bundle over the stack $BS_{3}$. Given a twisted $G$-cover $[\mathsf{C},\phi]$, the pullback $R_{\mathsf{C}}:=\phi^{*}R$ is a rank $2$ vector bundle over $\mathsf{C}$. In particular if $\mathsf{C}=C$ is a scheme theoretic curve, $R_{C}$ is a usual scheme theoretic vector bundle.

We consider the case of odd genus $g=2i+1$, and we focus on the Koszul cohomology $K_{i,1}$. We introduce the locus $\mathcal{U}_{g}$ as the locus with non-zero cohomology $K_{i,1}$. Supposing that the minimal value assumed by $\dim K_{i,1}$ on $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is [math], this is therefore the jumping locus for $K_{i,1}$. In particular we will show that $\mathcal{U}_{g}$ is a virtual divisor, i.e. $\mathcal{U}_{g}$ is an effective divisor if and only if $\dim K_{i,1}$ takes value [math] on a general curve, or equivalently on at least one curve.

Definition 5.2.4**.**

Given an odd genus $g=2i+1$,

[TABLE]

We want to show that $\mathcal{U}_{g}$ is a virtual divisor. For any stable curve $C$, by Lemma 5.2.2 applied in the case $L=\omega_{C}$ with the vector bundle $R_{C}$ as the sheaf $\mathcal{F}$, we have

[TABLE]

We can reformulate the definition of $\mathcal{U}_{g}$ with another scheme theoretic condition. Consider the Equation (5.2) in the case of the canonical line bundle $L=\omega_{C}$. As $\omega_{C}$ is a line bundle, by a well known property we have the short exact sequence

[TABLE]

Passing to the long exact sequence we have

[TABLE]

Proposition 5.2.5**.**

The two vector spaces

[TABLE]

have the same dimension. As a consequence any point $[C,\phi]$ of $\mathcal{R}_{g,S_{3}}^{S_{3}}$ is in $\mathcal{U}_{g}$ if and only if the associated $\Lambda$ morphism is not an isomorphism.

Proof.

We start by proving that

[TABLE]

We observe that this two terms fit in the long exact sequence

[TABLE]

and because the last term is [math], it suffices to prove that $H^{1}(\omega_{C}\otimes R_{C})=H^{0}(R^{\vee})=0$.

Consider the group quotient $S_{3}/N=\bm{\mu}_{2}$, it induces a twisted $\bm{\mu}_{2}$-cover $C\xrightarrow{\phi}BS_{3}\to B\bm{\mu}_{2}$. This is equivalent to a principal $\bm{\mu}_{2}$-bundle $\varphi\colon F\to C$. As $R_{C}$ is defined via the representation $\rho\colon S_{3}\to\operatorname{GL}(R)\cong\operatorname{GL}(2;\mathbb{C})$, its pullback on $F$ is given by the restriction $\rho|_{N}$ which is a representation of $N\cong\bm{\mu}_{3}$, which decomposes as direct sum of two irreducible $\bm{\mu}_{3}$ representations of rank $1$. We observe in particular that $\varphi^{*}R_{C}^{\vee}=\eta\oplus\eta^{\otimes 2}$ where $\eta$ is a non-trivial line bundle such that $\eta^{\otimes 3}\cong\mathcal{O}_{F}$. Therefore $H^{0}(\eta)=H^{0}(\eta^{\otimes 2})=0$ and a fortiori $H^{0}(R_{C}^{\vee})=0$.

Since the first cohomology group is trivial, we have that the Euler characteristic of both vector bundles coincides with the dimension of their spaces of global sections. In particular, if $\mathcal{E}$ is one of these vector bundles, $H^{0}(\mathcal{E})=\chi(\mathcal{E})=\deg(\mathcal{E})+\operatorname{rk}(\mathcal{E})(1-g)$.

• •

If $\mathcal{E}=\bigwedge^{i}H^{0}(\omega_{C})\otimes H^{0}(\omega_{C}\otimes R_{C})$, then

[TABLE]

The first term is simply $\binom{g}{i}$ because $h^{0}(\omega_{C})=g$. For the second term we have $h^{0}(\omega_{C}\otimes R_{C})=\chi(\omega_{C}\otimes R_{C})=\deg(\omega_{C}\otimes\mathcal{R}_{C})+2-2g=2g-2$. Therefore

[TABLE]

• •

If $\mathcal{E}=H^{0}(\bigwedge^{i-1}M_{\omega}\otimes\omega_{C}^{\otimes 2}\otimes R_{C})$, then

[TABLE]

Knowing that $\operatorname{rk}(\mathcal{E})=2\cdot\binom{g-1}{i-1}$, we have

[TABLE]

and this completes the proof.

∎

Proposition 5.2.6**.**

Consider a general hyperelliptic curve $(C,p)$ of genus $g\geq 2$ with a Weierstrass point. Then there exists a twisted $S_{3}$-cover $\phi\colon C\to BG$ such that, if $R_{C}$ is the vector bundle of rank $2$ associated to the $R$ representation via $\phi$, then $H^{0}(C,R_{C}((g-1)p))=0$.

This proves the effectiveness of $\mathcal{U}_{g}$ by following the approach of [11, Theorem 2.3]: via the proposition we prove that $[C,\phi]$ is a point outside $\mathcal{U}_{g}$ in the case of genus $g=2i+1$, and therefore that $\mathcal{U}_{g}$ is effective. Indeed, $M_{\omega_{C}}=\mathcal{O}_{C}(-2p)^{\oplus g-1}$ and therefore

[TABLE]

This means that $[C,\phi]\notin\mathcal{U}_{g}$ if and only if $H^{0}(R_{C}((g-1)p))=0$.

Proof.

We recall that $R_{C}$ is obtained by pulling back via $\phi$ the irreducible representation $\rho\colon S_{3}\to\operatorname{GL}(R)\cong\operatorname{GL}(2;\mathbb{C})$. Consider the normal subgroup $N\subset S_{3}$, and the group quotient $S_{3}/N=\bm{\mu}_{2}$. This induces a twisted $\bm{\mu}_{2}$-cover $C\xrightarrow{\phi}BS_{3}\to B\bm{\mu}_{2}$. As $C$ is a scheme theoretic curve, this is simply a principal $\bm{\mu}_{2}$-bundle $\pi\colon F\to C$. We use the notation $W:=R_{C}((g-1)p)$ and we want to show that $\pi^{*}W$ decomposes as direct sum of two line bundles. By construction the image of $F\to C\to BS_{3}$ is $BN$, that is $\pi^{*}W$ is induced by a representation of $N\cong\bm{\mu}_{3}$. The group $\bm{\mu}_{3}$ has $3$ irreducible representations: the trivial one, the identity $\eta\colon\bm{\mu}_{3}\hookrightarrow\mathbb{C}^{*}$ and $\eta^{\otimes 2}$. The restriction $\rho|_{N}$ decomposes as $\eta\oplus\eta^{\otimes 2}$. With a little abuse of notation we denote by $\eta$ and $\eta^{\otimes 2}$ the order $3$ line bundles induced by these representations. Moreover, we call $A_{1}:=\eta((g-1)p)$ and $A_{2}:=\eta^{\otimes 2}((g-1)p)$.

By construction we have the decomposition $\pi^{*}W=\pi^{*}A_{1}\oplus\pi^{*}A_{2}$. We observe that $H^{0}(W)=H^{0}(\pi^{*}W)_{+}$, that is the space of sections which are invariant by the natural involution of $F$. If $H^{0}(\pi^{*}W)_{+}\neq 0$, then one between $H^{0}(\pi^{*}A_{1})_{+}$ and $H^{0}(\pi^{*}A_{2})_{+}$ is non-empty, but $H^{0}(\pi^{*}A_{j})_{+}=H^{0}(A_{j})$ and by [11, Theorem 2.3] there exists $(C,\phi)$ such that both $H^{0}(A_{j})$ are empty.

To prove this last point, observe that by [11, Theorem 2.3] there exists a point a point $[C,\nu]$ outside the divisor $\mathcal{U}_{g,3}\subset\overline{\mathcal{R}}_{g,\bm{\mu}_{3}}^{\bm{\mu}_{3}}$, where $\nu$ is a non-trivial third root of $\mathcal{O}_{C}$, or equivalently a non-trivial principal $\bm{\mu}_{3}$-bundle on $C$. As the locus of hyperelliptic curves with a Weiestrass point and a third root, is not included in $\mathcal{U}_{g,3}$, then by dimensional considerations there exists an hyperelliptic curve $C$ such that $[C,\nu]\notin\mathcal{U}_{g,3}$ for any $\nu$. And this completes the proof.∎

We use the result of Proposition 5.2.5 to evaluate the class of $\mathcal{U}_{g}$ in the Chow ring, and also the class of its closure on the space $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$. Consider the universal family $u\colon\mathcal{C}_{g,S_{3}}\to\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ and the universal rank $2$ vector bundle $R_{\mathcal{C}}$ associated to the representation $R$. Introduce the vector bundle $\mathfrak{M}_{u}$ on $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ defined by the short exact sequence

[TABLE]

The geometric fiber of $\mathfrak{M}_{u}$ over any point $[\mathsf{C},\phi]$ of $\mathcal{R}_{g,S_{3}}^{S_{3}}$ is the previously defined vector bundle $M_{\omega}$.

Definition 5.2.7**.**

We introduce a series of sheaves on $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$,

[TABLE]

with $j\geq 0$ and $b\geq 1$ integers.

Remark 5.2.8**.**

To prove that these sheaves are locally free on $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, by Grauert's Theorem it suffices to prove that $h^{1}(M_{\omega}\otimes\omega_{\mathsf{C}}^{\otimes b}\otimes R_{\mathsf{C}})=0$ for every twisted $S_{3}$-cover $(\mathsf{C},\phi)$ in $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$. As showed in the proof of Proposition 5.2.5, this reduces to prove $h^{0}((\omega_{\mathsf{C}}^{\vee})^{\otimes(b-1)}\otimes R_{\mathsf{C}}^{\vee})=0$ for all $b\geq 1$. In the proof of Proposition 5.2.5 we showed this equality for $b=1$ and a scheme theoretic curve $\mathsf{C}=C$, but the same proof works for every $b\geq 1$ and for every twisted curve $\mathsf{C}$ such that its coarse space is irreducible.

By Proposition 5.2.5, on $\mathcal{R}_{g,S_{3}}^{S_{3}}$ the locus $\mathcal{U}_{g}$ is the degeneration locus of a morphism between the vector bundles $\mathbb{E}_{i-1,2}$ and $\bigwedge^{i}\mathbb{E}\otimes\mathbb{E}_{0,1}$, where $\mathbb{E}$ is the Hodge bundle $u_{*}\omega_{u}$. In the following we will evaluate this degeneration locus on the space $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$. We denote by $\overline{\mathcal{U}}_{g}$ the closure of the locus $\mathcal{U}_{g}$ on the space $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$.

Lemma 5.2.9**.**

Given an odd genus $g=2i+1$, the class of $\overline{\mathcal{U}}_{g}$ in $\operatorname{Pic}_{\mathbb{Q}}(\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}})$ is

[TABLE]

Proof.

Given two vector bundles $\mathcal{A}$ and $\mathcal{B}$ over $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, the morphism vector bundle $\operatorname{Hom}(\mathcal{A},\mathcal{B})$ is isomorphic to $\mathcal{B}\otimes\mathcal{A}^{\vee}$ and therefore $c_{1}(\operatorname{Hom}(\mathcal{A},\mathcal{B}))=\operatorname{rk}(\mathcal{A})c_{1}(\mathcal{B})-\operatorname{rk}(\mathcal{B})c_{1}(\mathcal{A})$. To conclude we observe that by the definition of the vector bundles $\mathbb{E}_{j,b}$, they fit in the short exact sequences

[TABLE]

for all $j\geq 0$ and $b\geq j$.∎

Lemma 5.2.10**.**

The first Chern class of $\mathbb{E}_{0,b}$ is

[TABLE]

Proof.

This is a direct application of Proposition 5.1.3 in the evaluation of $\operatorname{ch}_{1}(Ru_{*}W_{\mathcal{C}})$. By [10] we have

[TABLE]

To complete the proof we only need the eigenvalues decomposition of $R((12))$ and $R((123))$, where $R\colon S_{3}\to\operatorname{GL}(\mathbb{C},2)$ is the irreducible $S_{3}$ representation of dimension $2$.∎

With these lemmata we can develop the calculations to evaluate $[\overline{\mathcal{U}}_{g}]$ in terms of the Hodge class and the boundary classes.

Proposition 5.2.11**.**

In the Picard group $\operatorname{Pic}_{\mathbb{Q}}(\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}})$ we have,

[TABLE]

Proof.

From the result of Lemma 5.2.10, we have that

[TABLE]

[TABLE]

This, thanks to Lemma 5.2.9, allows to conclude the evaluation.∎

We are ready to introduce the divisor $\bm{\mathsf{M}}$ of Equation (5.1) over $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$, with $g=2i+1$ odd genus. Harris and Mumford introduced in [22] the following divisor.

Definition 5.2.12**.**

If $W^{r}_{d}(C)$ is the set of complete linear series over $C$ of degree $d$ and dimension at least $r$, it defines the locus

[TABLE]

In the same paper they proved

[TABLE]

where $\widetilde{\mathcal{M}}_{g}\subset\overline{\mathcal{M}}_{g}$ is the locus of irreducible stable curves, and $c^{\prime}$ is a positive coefficient. Then,

[TABLE]

where $\pi$ is the natural projection $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}\to\widetilde{\mathcal{M}}_{g}$.

Summarizing,

[TABLE]

for every odd genus $g=2i+1$, with $\alpha$ and $\beta$ positive coefficients and $E$ an effective sum of boundary divisors.

Proposition 5.2.13**.**

In Equation (5.4) the $\gamma$ coefficient can be chosen strictly positive for any $i>5$.

Proof.

By scaling appropriately every coefficient, the equation is equivalent to choosing a real number $s\in[0,1]$ such that

[TABLE]

[TABLE]

[TABLE]

For $E$ to be an effective divisor we must have $s\cdot\left(\frac{6i+1}{4i}\right)+(1-s)\cdot 2\geq\frac{3}{2}$ and $s\cdot\left(\frac{5i+2}{3i}\right)+(1-s)\cdot 3\geq 2$, and therefore $s\leq\frac{3i}{4i-2}$ is a necessary and sufficient condition for $E$ to be an effective boundary divisor.

To complete the proof we evaluate the $\gamma$ coefficient,

[TABLE]

After calculations this gives $\gamma=\frac{i-11}{2(i+1)}+s\cdot\frac{4i-2}{i(i+1)}$, which means a maximal possible value of

[TABLE]

which is positive if and only if $i>5$.∎

After this proposition, the canonical divisor over $\widetilde{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is big for every odd genus $g\geq 13$. Then, thanks to Proposition 3.3.2, the moduli space $\overline{\mathcal{R}}_{g,S_{3}}^{S_{3}}$ is of general type for every odd genus $g\geq 13$.