The birational geometry of $\overline{{\mathcal {R}}}_{g,2}$ and Prym-canonical divisorial strata

Abstract

We prove that the moduli space of double covers ramified at two points ${\mathcal {R}}_{g,2}$ is uniruled for $3\le g\le 6$ and of general type for $g\ge 16$. Furthermore, we consider Prym-canonical divisorial strata in the moduli space $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ parametrizing n-pointed Prym curves, and we compute their classes in $\textrm{Pic}_{\mathbb {Q}}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$.

1 Introduction

In his fundamental paper [42], Mumford initiated the study of double covers as a way of understanding polarized Abelian varieties. It is then natural to consider the moduli space ${\mathcal {R}}_{g,2n}$ parametrizing double covers ramified at 2n points, and describe its birational geometry. The classical case ${\mathcal {R}}_g$ when the cover is unramified has received considerable attention. When the genus g is small, it is known that ${\mathcal {R}}_g$ is rational for $g =2,3,4$ (cf. [17], the references therein and [14]), unirational for $g = 5, 6, 7$ (cf. [19, 32, 39, 50, 51] and [29]) and uniruled for $g =8$ (cf. [29]). The situation changes for higher genus and we know that ${\mathcal {R}}_g$ is of general type when $g\ge 13, g\ne 16$ (cf. [11] and [26]). Apart from one exotic case in genus 2, see [33], the only other way to obtain principally polarized Abelian varieties is by considering double covers ramified at two points.

By the theory of double covers, the moduli space ${\mathcal {R}}_{g,2}$ can be alternatively described as

$$\begin{aligned} {\mathcal {R}}_{g,2} {:=}\left\{ [C,x+y, \eta ] \ | \ [C,x,y]\in {\mathcal {M}}_{g,2} \ \textrm{and} \ \eta \in \textrm{Pic}^{-1}(C), \ \eta ^{\otimes 2}\cong {\mathcal {O}}_C(-x-y) \right\} \end{aligned}$$

where x and y correspond to the two branch points of the associated cover and their order is irrelevant. We will call such a triple $[C, x+y,\eta ]$ a 2-branched Prym curve.

One important feature of the moduli space ${\mathcal {R}}_{g,r}$ is that it comes with the Prym map

$$\begin{aligned} {\mathcal {P}}_{g,r}:{\mathcal {R}}_{g,r} \rightarrow {\mathcal {A}}^{\delta }_{g-1+\frac{r}{2}} \end{aligned}$$

to the moduli space of Abelian varieties of dimension $g-1+\frac{r}{2}$ equipped with a polarization of type $\delta =(1,\ldots ,1,2,\ldots ,2)$ where 2 appears g times. This map received considerable attention in recent years, see [38, 44, 45]; adding to the vast literature on the Prym map in the unbranched case, see [8, 20] and [18] among many others.

Our interest in the case $r=2$ is motivated by the fact that $r=0$ and $r=2$ are the only two cases when ${\mathcal {P}}_{g,r}$ provides a correspondence between double covers and principally polarized Abelian varieties, as first pointed out in [42]. Our main result is the following:

Theorem 1.1

The moduli space ${\mathcal {R}}_{g,2}$ is of general type for $g\ge 16$ and ${\mathcal {R}}_{13,2}$ has non-negative Kodaira dimension.

There are three main ideas of the proof. First, we consider a suitable compactification $\overline{{\mathcal {R}}}_{g,2}$ of ${\mathcal {R}}_{g,2}$, following the method outlined in [15] and [6]. Secondly, we show that the canonical class $K_{\overline{{\mathcal {R}}}_{g,2}}$ is big and lastly, we show that the singularities of $\overline{{\mathcal {R}}}_{g,2}$ are mild enough in order to extend holomorphically the pluricanonical forms of $\overline{{\mathcal {R}}}^{\textrm{reg}}_{g,2}$ to any desingularisation. For this last step, we follow closely [36] and [26].

To prove that $K_{\overline{{\mathcal {R}}}_{g,2}}$ is big, we will use pullbacks of divisors through the map $\overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{2g}$ retaining the source of the double cover. The image of the map is not contained in any Brill–Noether or Gieseker–Petri divisor, see Theorem 4.1. This is in sharp contrast with the situation in the unramified case, where the Brill–Noether properties of a generic double cover depend on the parity of the genus of the base, see [2, Theorem 0.4].

Next, we are interested in the birational geometry of ${\mathcal {R}}_{g,2}$ when the genus g is small. We have that:

Theorem 1.2

The moduli space ${\mathcal {R}}_{g,2}$ is uniruled for $3\le g\le 6$.

This result is obtained by relating the moduli space ${\mathcal {R}}_{g,2}$ to strata parametrizing divisors of quadratic differentials, which we know from [7] to be uniruled when $3\le g\le 6$.

In the second part of this paper, we investigate further the relation between Prym curves and quadratic differentials. To set things up, we introduce the moduli space ${\mathcal {C}}^n{\mathcal {R}}_g$ parametrizing tuples $[X, x_1,\ldots , x_n, \eta ]$ where $[X,\eta ]$ is an element of the Prym variety ${\mathcal {R}}_g$ and $x_1,\ldots ,x_n$ are distinct points on X. For a positive partition ${\underline{d}} = (d_1,\ldots , d_n)$ of $g-1$, we consider the divisor $PD_{\underline{d}}$ in ${\mathcal {C}}^n{\mathcal {R}}_g$ defined as:

$$\begin{aligned} PD_{{\underline{d}}} {:=}\left\{ [X,x_1,\ldots , x_n,\eta ] \in {\mathcal {C}}^n{\mathcal {R}}_g \ | \ h^0\left( X, \omega _X\otimes \eta \left( -\sum _{i=1}^nd_ix_i\right) \right) \ge 1 \right\} \end{aligned}$$

We consider a suitable compactification $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ of ${\mathcal {C}}^n{\mathcal {R}}_g$ and compute the class of the divisor ${\overline{PD}}_{{\underline{d}}}$ in this space. We obtain:

Theorem 1.3

Let ${\underline{d}} = (d_1,\ldots ,d_n)$ a partition of $g-1$ with all entries positive. The class of the Prym-canonical divisorial stratum ${\overline{PD}}_{{\underline{d}}}$ in $\textrm{Pic}_{{\mathbb {Q}}}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$ is given by:

$$\begin{aligned}{}[{\overline{PD}}_{{\underline{d}}}]= & {} -\lambda + \sum _{i=1}^n\frac{d_i(d_i+1)}{2}\psi _i +\frac{1}{4}\delta _0^{\textrm{ram}}-\sum _{\begin{array}{c} 1\le i \le g-1 \\ d_S\ge i-1 \end{array}} \left( {\begin{array}{c}d_S-i+2\\ 2\end{array}}\right) \delta _{i,S} -\cdots \\{} & {} \cdots -\sum _{\begin{array}{c} 1\le i \le g \\ d_S \le i-1 \end{array}} \left( {\begin{array}{c}i-d_S\\ 2\end{array}}\right) (\delta _{i,S:g-i} + \delta _{i,S}) \end{aligned}$$

where $d_S {:=}\sum _{i\in S} d_i$ and $\delta _{0,S:g} {:=}0$.

Note that the coefficients of $\delta _0'$ and $\delta _0''$ are 0. For the definition of the classes appearing in Theorem 1.3 we refer to Sect. 6. These divisors can be seen as a Prym analogue of the canonical divisorial strata appearing in [35, 40, 43] and [30]. Moreover, these divisors are closely related to the divisorial strata of quadratic differentials, see [41, Proposition 1.4]. The study of such divisors led to important results in understanding the geometric aspects of the moduli space $\overline{{\mathcal {M}}}_{g,n}$ such as the Kodaira dimension (cf. [35]) and the effective cone (cf. [41]).

To prove Theorem 1.3, we consider suitable maps $\pi _1:\overline{{\mathcal {M}}}_{g-i,n+1-s} \rightarrow \overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ and $\pi _2:\overline{{\mathcal {C}}^{n-s+1}{\mathcal {R}}}_{g-i} \rightarrow \overline{{\mathcal {C}}^n{\mathcal {R}}}_g$. Understanding the pullbacks at the level of rational Picard groups is enough to compute all coefficients of $[{\overline{PD}}_{{\underline{d}}}]$ but the one of $\delta _0^{\textrm{ram}}$. Lastly, we use [41, Proposition 1.4] to conclude the theorem.

As torsion classes are irrelevant to us, the Picard groups will be considered over ${\mathbb {Q}}$ throughout the paper.

2 A compactification of the moduli space ${\mathcal {R}}_{g,2}$

We want to compactify the moduli space ${\mathcal {R}}_{g,2}$ parametrizing smooth 2-branched Prym curves. These are double covers of smooth genus g curves, branched at two unordered points. The way we do this is similar to the approaches in [15] and [6] and it inspires us to consider the following definitions:

Definition 2.1

Let $[X, x+y]$ be a pointed semistable Deligne-Mumford curve (with the two points unordered) and let E be an irreducible component of X. We say that E is exceptional if E is smooth, rational, the points x, y are not on E and $|E \cap \overline{X{\setminus } E}| = 2$. We say that $[X, x+y]$ is quasistable if any two distinct exceptional components do not intersect.

We are now ready to extend the definition of a 2-branched Prym curve to singular curves.

Definition 2.2

We define a 2-branched Prym curve of genus g to be the data $[X, x+y, \eta , \beta ]$, where $[X, x+y]$ is a genus g quasistable curve, $\eta \in \text {Pic}(X)$ and $\beta :\eta ^{\otimes 2} \rightarrow {\mathcal {O}}_X(-x-y)$ is a morphism of invertible sheaves satisfying:

• 1. The sheaf $\eta$ has total degree $-1$ and has degree 1 on each exceptional component,

• 2. The morphism $\beta$ is non-zero at a general point of a non-exceptional component of X.

In the above setting, consider $E_1,\ldots , E_n$ the exceptional components of $[X, x+y]$ and let ${\tilde{X}} {:=}\overline{X{\setminus } \cup _{i=1}^n E_i}$. We denote by $q_i^1$ and $q_i^2$ the intersection of $E_i$ with ${\tilde{X}}$ and we get an isomorphism

$$\begin{aligned} \beta _{{\tilde{X}}}:\eta ^{\otimes 2}_{|{\tilde{X}}} \rightarrow {\mathcal {O}}_{{\tilde{X}}}\left( -x-y-\sum _{i=1}^{n}(q_i^1 +q_i^2) \right) \end{aligned}$$

In particular, when X is smooth, we obtain that $\eta$ is a root of order 2 of ${\mathcal {O}}_X(-x-y)$. Next, we define the notion of isomorphism between two 2-branched Prym curves.

Definition 2.3

We say that two 2-branched Prym curves $[X, x+y, \eta , \beta ]$ and $[X', x'+y', \eta ', \beta ']$ are isomorphic if there exists an isomorphism $\sigma :X \rightarrow X'$ such that

• 1. it sends $x+y$ to $x'+y'$

• 2. there exists an isomorphism $\tau :\sigma ^*\eta ' \rightarrow \eta$ making the following diagram commutative

  

Moreover, we say that an automorphism is inessential if it induces the identity on the stable model of $[C, x+y]$.

The results of [15] can be easily adapted to our situation and we obtain a compactification $\overline{{\mathcal {R}}}_{g,2}$ of ${\mathcal {R}}_{g,2}$, parametrizing isomorphism classes of 2-branched Prym curves. As in [15], we obtain that the space $\overline{{\mathcal {R}}}_{g,2}$ is normal and projective. Moreover, it is irreducible as it is birational to the irreducible divisor $\Delta _0^{\textrm{ram}}$ in $\overline{{\mathcal {R}}}_{g+1}$ (see [9, page 9] for irreducibility).

The compactification $\overline{{\mathcal {R}}}_{g,2}$ can be alternatively described as the admissible cover compactification of the moduli space ${\mathcal {R}}_{g,2}$ of double covers branched at two points, see [1, Sect. 4]. Seen as birational to a boundary divisor of $\overline{{\mathcal {R}}}_{g+1}$ via the map

$$\begin{aligned} i_{g,2}:\overline{{\mathcal {R}}}_{g,2} \rightarrow \Delta _0^{\textrm{ram}}\subseteq \overline{{\mathcal {R}}}_{g+1}, \end{aligned}$$

the space $\overline{{\mathcal {R}}}_{g,2}$ is exhibited as a compactification of the moduli space of admissible double covers of an irreducible genus $g+1$ nodal curve with the additional data of a ramified node of the source curve.

We consider the map forgetting the 2-branched Prym structure

$$\begin{aligned} \pi _{g,2}:\overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2} \end{aligned}$$

and we will describe the boundary divisors of $\overline{{\mathcal {R}}}_{g,2}$ lying above each boundary component of $\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}$.

• 1. Consider a generic element $[X/t_1\sim t_2, x+y]$ of the divisor $\Delta _0$ in $\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}$. Over $\Delta _0$ we have two divisors: $\Delta _0'$ and $\Delta _0^{\textrm{ram}}$.

  • The divisor $\Delta _0'$ contains the pairs $[X/t_1\sim t_2,x+y, \eta ]$ such that for the normalization map $\nu :X \rightarrow X/t_1\sim t_2$ we have $(\nu ^{*}\eta )^{\otimes 2} \cong {\mathcal {O}}_X(-x-y)$.

  • The divisor $\Delta _0^{\textrm{ram}}$ contains the pairs $[X\cup R/{t_1\sim r_1, t_2\sim r_2}, x+y, \eta ]$, where R is an exceptional component, $\eta _{|X}^{\otimes 2} \cong {\mathcal {O}}_X(-x-y-t_1-t_2)$ and $\eta _{|R} \cong {\mathcal {O}}_R(1)$.

  It is immediate to see that $\deg (\Delta _0'/\Delta _0) = 2^{2g-1}$ and $\deg (\Delta _0^{\textrm{ram}}/\Delta _0) = 2^{2g-2}$. Furthermore $\Delta _0^{\textrm{ram}}$ is the ramification divisor of $\pi _{g,2}$ and has ramification order 2. To see this, the same methods used by Bernstein in [9, Chapter 2] can be adapted for the moduli $\overline{{\mathcal {R}}}_{g,2}$ of 2-branched Prym curves. Alternatively, this can be seen by viewing $\Delta _0^{\text {ram}}\subseteq \overline{{\mathcal {R}}}_{g,2}$ as a codimension 2 locus in the boundary of $\overline{{\mathcal {R}}}_{g+1}$ and applying [6, Proposition 11].

  We remark that $\Delta _0^{\text {ram}}$ is the ramification divisor of

  $$\begin{aligned} \pi _{g,2}:\overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2} \end{aligned}$$

  viewed as a morphism of coarse moduli spaces. At the level of moduli stacks the ramification locus contains other boundary divisors, see the subsequent discussion after Remark 3.1.

• 2. Consider $[X\cup _{x'\sim y'} Y, x+y]$ a generic element of $\Delta _{i,\left\{ 1\right\} }$, where $g(X) = i$, $g(Y) = g-i$, with the points $x, x'$ on X and $y,y'$ on Y. Then there is a unique divisor in $\overline{{\mathcal {R}}}_{g,2}$ lying above $\Delta _{i,\left\{ 1\right\} }$, which we will denote $\Delta _{i:g-i}$. A generic element of $\Delta _{i:g-i}$ is a tuple $[X\cup _{x'\sim r_1} R\cup _{r_2\sim y'} Y, x+y, \eta ]$, where R is an exceptional component, $\eta _{|R} \cong {\mathcal {O}}_R(1)$, $\eta _{|X}^{\otimes 2} \cong {\mathcal {O}}_X(-x-x')$ and $\eta _{|Y}^{\otimes 2} \cong {\mathcal {O}}_X(-y-y')$.

• 3. Consider $[X\cup _{x'\sim y'}Y, x_1+x_2]$ a generic element of the boundary divisor $\Delta _{i,\left\{ 1,2\right\} }$, where $g(X) = i$, $g(Y) = g-i$ and $x', x_1, x_2 \in X$. Then there are two divisors $\Delta _{i:g-i,\left\{ {\mathcal {O}}\right\} }$ and $\Delta _{i:g-i,\left\{ \eta \right\} }$ lying in $\overline{{\mathcal {R}}}_{g,2}$ above $\Delta _{i,\left\{ 1,2\right\} }$.

  • The divisor $\Delta _{i:g-i,\left\{ {\mathcal {O}}\right\} }$ contains the pairs $[X\cup _{x'\sim y'}Y, x_1+x_2, \eta ]$ satisfying that $\eta ^{\otimes 2}_{|X} \cong {\mathcal {O}}_X(-x_1-x_2)$ and $\eta _{|Y} \cong {\mathcal {O}}_Y$.

  • The divisor $\Delta _{i:g-i,\left\{ \eta \right\} }$ contains the pairs $[X\cup _{x'\sim y'}Y, x_1+x_2, \eta ]$ satisfying that $\eta ^{\otimes 2}_{|X} \cong {\mathcal {O}}_X(-x_1-x_2)$ and $\eta _{|Y} \in \text {Pic}(Y)[2]{\setminus } \left\{ {\mathcal {O}}_Y\right\}$.

Remark 2.4

All the boundary divisors of $\overline{{\mathcal {R}}}_{g,2}$ described above are irreducible.

The remark follows immediately for almost all boundary divisors by simply noting that ${\mathcal {M}}_g$, ${\mathcal {R}}_g$ and ${\mathcal {R}}_{g,2}$ are irreducible. We exemplify this for the divisor $\Delta _0'$. To see it is irreducible, we consider the moduli space $\widetilde{{\mathcal {C}}}^2{\mathcal {R}}_{g-1,2}$ parametrizing tuples $[C,x+y, \eta , p_1, q_1]$ where $[C,x+y, \eta ] \in {\mathcal {R}}_{g-1,2}$ and $p_1,q_1$ are two points on the double cover ${\widetilde{C}}$ associated to $[C,x+y, \eta ]$. We assume $p_1, q_1$ are not in the same fiber over C and denote $p_2$, $q_2$ the two points in the same fiber as $p_1$ and $q_1$ respectively. By glueing the points $p_i$ and $q_i$ for $i =1,2$ together (as well as their images in C together), we obtain a dominant map

$$\begin{aligned} \widetilde{{\mathcal {C}}}^2{\mathcal {R}}_{g-1,2} \rightarrow \Delta _0'\subseteq \overline{{\mathcal {R}}}_{g,2} \end{aligned}$$

The fiber of $\widetilde{{\mathcal {C}}}^2{\mathcal {R}}_{g-1,2}$ over a generic $[C,x+y,\eta ] \in {\mathcal {R}}_{g-1,2}$ is birational to ${\widetilde{C}}\times {\widetilde{C}}$. Because ${\mathcal {R}}_{g-1,2}$ and ${\widetilde{C}}\times {\widetilde{C}}$ are irreducible, we conclude that the same is true for $\widetilde{{\mathcal {C}}}^2{\mathcal {R}}_{g-1,2}$. This concludes that $\Delta _0'$ is irreducible.

The only divisor for which Remark 2.4 requires more attention is $\Delta ^{\textrm{ram}}_0$. That $\Delta ^{\textrm{ram}}_0$ is irreducible is deduced from the following proposition.

Proposition 2.5

The moduli space ${\mathcal {R}}_{g,2n}$ is irreducible for all $g \ge 2, n\ge 0$.

Proof

We will prove the proposition using an inductive argument. The cases $n = 0$ and $n = 1$ are already covered, hence we can assume that $n\ge 2$. For a given g, consider the smallest n for which ${\mathcal {R}}_{g,2n}$ is not irreducible.

We consider the moduli space ${\mathcal {R}}_{g,2n}'$ parametrizing pairs $[C, x_1,\ldots , x_{2n}, \eta ]$ where $[C] \in {\mathcal {M}}_g$, the points $x_1,\ldots , x_{2n} \in C$ are pairwise distinct and $\eta$ is a degree $-n$ line bundle satisfying $\eta ^{\otimes 2} \cong {\mathcal {O}}_C(-x_1-\cdots - x_{2n})$. Because the approach in [15] applies with little change to this case, we obtain a compactification $\overline{{\mathcal {R}}}_{g,2n}'$ and, in particular a map

$$\begin{aligned} \pi :{\mathcal {R}}_{g,2n-2}' \times {\mathcal {M}}_{0,4} \rightarrow \overline{{\mathcal {R}}}_{g,2n}' \end{aligned}$$

sending a pair $([C, x_1, \ldots , x_{2n-3}, t_1, \eta _C], [{\mathbb {P}}^1, t_2, x_{2n-2}, x_{2n-1}, x_{2n}])$ to the point $[C\cup _{t_1\sim r_1} R \cup _{r_2\sim t_2} {\mathbb {P}}^1, x_1,\ldots , x_{2n}, \eta ]$, where R is an exceptional component and the line bundle $\eta$ is defined by

$$\begin{aligned} \eta _{|C} \cong \eta _C, \ \ \eta _{|{\mathbb {P}}^1} \cong {\mathcal {O}}_{{\mathbb {P}}^1}(-2) \ \ \text {and} \ \eta _{|R} \cong {\mathcal {O}}_R(1) \end{aligned}$$

We know from the approach in [15] that $\overline{{\mathcal {R}}}_{g,2n}'$ is given locally as the quotient of the base of a universal deformation by the automorphism group of the 2n-branched Prym curve (where the branch points are ordered). Because a generic element in ${\mathcal {M}}_{0,4}$ and in ${\mathcal {R}}_{g,2n-2}'$ has no non-trivial automorphisms it follows that a generic element in $\text {Im}(\pi )$ has no inessential automorphisms.

In particular, a generic point in $\text {Im}(\pi )$ is a nonsingular point of the moduli space $\overline{{\mathcal {R}}}_{g,2n}'$. If we consider the finite map of degree $2^{2g}$

$$\begin{aligned} \overline{{\mathcal {R}}}_{g,2n}' \rightarrow \overline{{\mathcal {M}}}_{g,2n} \end{aligned}$$

obtained by forgetting the 2n-Prym structure, we observe that $\text {Im}(\pi )$ has degree $2^{2g}$ over the divisor $\Delta _{0, \left\{ 2n-2, 2n-1, 2n\right\} }$ in $\overline{{\mathcal {M}}}_{g,2n}$.

Because $\text {Im}(\pi )$ is irreducible (from the induction hypothesis), contains a smooth point of $\overline{{\mathcal {R}}}'_{g,2n}$ and has degree $2^{2g}$ over its image in $\overline{{\mathcal {M}}}_{g,2n}$, it follows immediately that $\overline{{\mathcal {R}}}'_{g,2n}$ is irreducible (otherwise $\text {Im}(\pi )$ would be in the intersection of all irreducible components and hence it would be impossible to contain smooth points).

Because we have an obvious surjective map ${\mathcal {R}}'_{g,2n} \rightarrow {\mathcal {R}}_{g,2n}$, the conclusion follows. $\square$

3 Maps between moduli spaces

As easily remarked, there is an obvious map $i_{g,2}:\overline{{\mathcal {R}}}_{g,2} \rightarrow \Delta _0^{\textrm{ram}}\subseteq \overline{{\mathcal {R}}}_{g+1}$ obtained by glueing an exceptional component to the two marked points. This map fits into a commutative diagram

We are interested in describing the pullback map $i_{g,2}^{*}:\text {Pic}(\overline{{\mathcal {R}}}_{g+1}) \rightarrow \text {Pic}(\overline{{\mathcal {R}}}_{g,2})$. For this, we first set some notations.

The Picard group of $\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}$ injects in $\text {Pic}(\overline{{\mathcal {M}}}_{g,2})$ as the subgroup of ${\mathbb {Z}}_2$-invariant classes. Hence, Pic$(\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2})$ is generated by $\psi {:=}\psi _1 + \psi _2$, the class $\lambda$ and the boundary divisors (for which we preserve the notation from $\overline{{\mathcal {M}}}_{g,2}$). We denote again by $\psi$ and $\lambda$, the pullbacks by $\pi _{g,2}$ of the respective classes.

Remark 3.1

The pullback map $i'^{*}$ at the level of Picard groups is explicitly known, see [4, Lemma 1.3]. Moreover, because $\pi _{g,2}$ and $\pi _{g+1}$ are forgetful maps, ramified only above their respective divisor $\Delta _0^{\text {ram}}$, the maps $\pi _{g,2}^{*}$ and $\pi _{g+1}^{*}$ at the level of Picard groups are straight-forward to compute. Using the formula $i_{g,2}^{*}\circ \pi _{g+1}^{*} = \pi _{g,2}^{*}\circ i'^{*}$, we conclude that:

$$\begin{aligned} i_{g,2}^{*}\lambda= & {} \lambda , \ \ i_{g,2}^{*}\delta _0^{\textrm{ram}} = -\frac{1}{2}\psi +\delta _0^{\textrm{ram}} + \sum \delta _{i:g-i}, \ \ i_{g,2}^{*}\delta _0'' = 0, \ \ i_{g,2}^{*}\delta _0' = \delta _0'\\ i_{g,2}^{*}\delta _i= & {} \delta _{i-1:g-i+1, \left\{ {\mathcal {O}}\right\} } \ \ \text {and} \ i_{g,2}^{*}\delta _{i:g+1-i} = \delta _{i-1:g-i+1, \left\{ \eta \right\} } + \delta _{g-i:i, \left\{ \eta \right\} } \end{aligned}$$

We remark that the computation above is done at the level of moduli stacks (not coarse moduli spaces). In this situation we have $\pi _{g,2}^{*}\delta _{i,\left\{ 1\right\} } = 2\delta _{i:g-i} = [\Delta _{i:g-i}]$. The equality $\pi _{g,2}^{*}\delta _{i,\left\{ 1\right\} } = [\Delta _{i:g-i}]$ is true because $\pi _{g,2}$ is unramified over $\Delta _{i,\left\{ 1\right\} }$ at the level of coarse moduli spaces. A generic element of $\Delta _{i:g-i}$ admits an order two inessential automorphism, coming from the fact that one can interchange the sheets of the cover on the two components independently. This implies that, the map ${{\textbf {R}}}_{g,2} \rightarrow {\mathcal {R}}_{g,2}$ from the moduli stack to the coarse moduli space is ramified with order two above the divisor $\Delta _{i:g-i}$. We conclude that $2\delta _{i:g-i} = [\Delta _{i:g-i}]$.

Having an element $[C, x+y, \eta ,\beta ] \in \overline{{\mathcal {R}}}_{g,2}$, we obtain a degree 2 map $\pi :{\tilde{C}} \rightarrow C$ that is ramified only above x, y and eventually above the nodes of C. Seeing the space $\overline{{\mathcal {R}}}_{g,2}$ as parametrizing such admissible covers $\pi :{\tilde{C}} \rightarrow C$, we get a map

$$\begin{aligned} \chi :\overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{2g,2/{\mathbb {Z}}_2} \end{aligned}$$

sending $[\pi :{\tilde{C}} \rightarrow C]$ to $[{\tilde{C}}, {\tilde{x}}+ {\tilde{y}}]$ where ${\tilde{x}}$ and ${\tilde{y}}$ are the two smooth ramification points of ${\tilde{C}}$. Forgetting the points we obtain a map

$$\begin{aligned} \chi _{g,2}:\overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{2g} \end{aligned}$$

This is a pointed version of the map $\chi _{g+1}$ considered in [9] and [26]. In fact, we have the obvious commutative diagram

Our next task is to describe the map $\chi ^*_{g,2}:\text {Pic}(\overline{{\mathcal {M}}}_{2\,g}) \rightarrow \text {Pic}(\overline{{\mathcal {R}}}_{g,2})$. Because we know the maps $i_{g,2}^{*}, i'^{*}$ and $\chi _{g+1}^{*}$ at the level of Picard groups (cf. [4, Lemma 1.3], [9, Lemma 3.1.3] and [26, Proposition 4.1]), we can immediately see that

$$\begin{aligned} \chi _{g,2}^{*}\lambda = 2\lambda - \frac{1}{4} \delta _0^{\textrm{ram}} - \frac{1}{4} \sum \delta _{i:g-i} + \frac{1}{8}\psi \end{aligned}$$

To compute the pullback of the boundary divisors, it suffices to consider generic test curves and apply the same method as in [9] to reduce the problem to a simple count of the number of nodes. We will use local equations around nodes to deduce the multiplicities, in the spirit of [31, Theorem 6] and [16, Sect. 7].

To exemplify this, we consider the double cover associated to a generic 2-branched Prym curve in $\Delta _0^{\text {ram}}$. We consider

to be a 1-dimensional smoothing of this double cover. The local equation around a node of the central fiber of ${\widetilde{C}}$ is of the form $xy = t$, where t is the local coordinate of $\Delta _{t}$. Around the corresponding node in the central fiber of ${\mathcal {C}}$, the local equation is $uv = t^2$ and the map $\pi :{\widetilde{C}}\rightarrow {\mathcal {C}}$ is locally given by $u = x^2, v = y^2$. This implies $B\cdot \delta _0^{\text {ram}} = 2$.

We view ${\widetilde{f}}:{\widetilde{C}} \rightarrow \Delta _t$ as a curve in $\overline{{\mathcal {M}}}_{2g}$, after collapsing the exceptional component of ${\widetilde{C}}$ we notice that we have a unique node of the central fibre with local equation $xy = t^2$ around it. This implies $\chi _{g,2*}B\cdot \delta _0 = 2$. But $\chi _{g,2*}B\cdot \delta _0 = B\cdot \chi _{g,2}^*\delta _0$.

Hence the coefficient of $\delta ^{\textrm{ram}}_0$ in $\chi _{g,2}^{*}\delta _0$ is 1. Proceeding as in this example we obtain:

$$\begin{aligned} \chi _{g,2}^{*}\delta _0&= \delta _0^{\textrm{ram}} + 2\delta _0' + 2\sum \delta _{i:g-i, \left\{ \eta \right\} } \\ \chi _{g,2}^{*}\delta _i&= 2\delta _{g-i:i, \left\{ {\mathcal {O}}\right\} } \ \ \text {if} \ i \ \text {is odd} \end{aligned}$$

and

$$\begin{aligned} \chi _{g,2}^{*}\delta _i = 2\delta _{g-i:i, \left\{ {\mathcal {O}}\right\} } + \delta _{\frac{i}{2}:g-\frac{i}{2}} \ \ \text {if} \ i \ \text {is even} \end{aligned}$$

In fact, we can compute the pullback $\chi ^{*}:\text {Pic}(\overline{{\mathcal {M}}}_{2\,g,2/{\mathbb {Z}}_2}) \rightarrow \text {Pic}(\overline{{\mathcal {R}}}_{g,2})$ and obtain for $i\le g$:

$$\begin{aligned} \chi ^{*}\delta _{i,\left\{ 1\right\} }= & {} {\left\{ \begin{array}{ll} \delta _{\frac{i}{2}:g-\frac{i}{2}} &{} \text {if} \ i \ \text {is even} \\ 0 &{} \text {if} \ i \ \text {is odd} \end{array}\right. }\\ \chi ^{*}\delta _{i,\emptyset }= & {} 2\delta _{g-i:i,\left\{ {\mathcal {O}}\right\} } \ \textrm{and} \ \chi ^{*}\delta _{i,\left\{ 1,2\right\} } = 0 \end{aligned}$$

We can easily check that the commutativity $\chi ^{*}\circ i'^{*} = i^{*}\circ \chi _{g+1}^{*}$ is respected.

Remark 3.2

The formula for $\chi _{g,2}^{*}\lambda$ can be alternatively computed by considering a family in $\overline{{\mathcal {R}}}_{g,2}$ given as

We consider $D_2\subseteq \tilde{{\mathcal {C}}}$ the image of the two sections and let $C_2$ be its image in ${\mathcal {C}}$.

We have

$$\begin{aligned} f_{*}\circ \pi _{*} \left( c_1(\omega _{{\tilde{f}}})\cdot c_1(\omega _{{\tilde{f}}})\right) = f_*\circ \pi _*\left( [\pi ^*c_1(\omega _f)+D_2 ]\cdot [\pi ^*c_1(\omega _f)+D_2 ]\right) \end{aligned}$$

Using that $\pi ^{*}(C_2) = 2D_2$ and the push-pull formula, this is furthermore equal to

$$\begin{aligned} 2f_*\left( c_1(\omega _f)\cdot c_1(\omega _f)\right) + 2f_*\left( C_2\cdot c_1(\omega _f)\right) + \frac{1}{2}f_*(C_2\cdot C_2) \end{aligned}$$

We use the notation $\delta$ for the sum of the boundary divisors of the respective moduli stack. Next, we compute the canonical class of the variety $\overline{{\mathcal {R}}}_{g,2}$.

Proposition 3.3

The canonical class $K_{\overline{{\mathcal {R}}}_{g,2}}$ is equal to

$$\begin{aligned} K_{\overline{{\mathcal {R}}}_{g,2}}= & {} \psi +13\lambda - 2\delta - \delta _0^{\textrm{ram}} - \delta _{0:g,\left\{ {\mathcal {O}}\right\} } - \delta _{0:g, \left\{ \eta \right\} }\\{} & {} - 2\sum \delta _{i:g-i} - \delta _{g-1:1, \left\{ \eta \right\} } - \delta _{g-1:1, \left\{ {\mathcal {O}}\right\} } \end{aligned}$$

Proof

We know that

$$\begin{aligned} K_{\overline{{\mathcal {M}}}_{g,2}} = \psi +13\lambda - 2\delta - \delta _{1,\emptyset } \end{aligned}$$

We consider the map $\pi :\overline{{\mathcal {M}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}$, which we know is ramified only along the divisor $\delta _{0,\left\{ 1,2\right\} }$. Using the Riemann-Hurwitz formula for $\pi$ we deduce that

$$\begin{aligned} \pi ^{*}K_{\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}} = \psi +13\lambda - 2\delta - \delta _{0,\left\{ 1,2\right\} } - \delta _{1,\emptyset } \end{aligned}$$

Consequently we get:

$$\begin{aligned} K_{\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}} = \psi +13\lambda - 2\delta - \delta _{0,\left\{ 1,2\right\} }- \delta _{1,\emptyset } \end{aligned}$$

Since the map $\pi _{g,2}:\overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}$ is ramified only along the divisor $\Delta _0^{\textrm{ram}}$, the Riemann-Hurwitz formula implies

$$\begin{aligned} K_{\overline{{\mathcal {R}}}_{g,2}}= & {} \psi +13\lambda - 2\delta - \delta _0^{\textrm{ram}} - \delta _{0:g,\left\{ {\mathcal {O}}\right\} } - \delta _{0:g, \left\{ \eta \right\} }\\{} & {} - 2\sum \delta _{i:g-i} - \delta _{g-1:1, \left\{ \eta \right\} } - \delta _{g-1:1, \left\{ {\mathcal {O}}\right\} } \end{aligned}$$

Again, we used that the ramification of $\pi _{g,2}$ is well-understood. This is sufficient to describe the map $\pi _{g,2}^*$ at the level of Picard groups, see [26, Sect. 1] for the analogous result in the unramified case. $\square$

4 The geometry of $\overline{{\mathcal {R}}}_{g,2}$

One way of obtaining effective divisors of small slope on $\overline{{\mathcal {R}}}_{g,2}$ is by pullback from $\overline{{\mathcal {M}}}_{2g}$. Let [D] be the class of an effective divisor D in $\overline{{\mathcal {M}}}_{2\,g}$. If the image of $\chi _{g,2}$ is not contained in D, we conclude that $\chi _{g,2}^{*}([D])$ is the class of an effective divisor in $\overline{{\mathcal {R}}}_{g,2}$.

Consequently, we prove next that $\chi _{g,2}(\overline{{\mathcal {R}}}_{g,2})$ is not contained in some well-known divisors of small slope in $\overline{{\mathcal {M}}}_{2g}$.

Theorem 4.1

The image of $\chi _{g,2}:\overline{{\mathcal {R}}}_{g,2}\rightarrow \overline{{\mathcal {M}}}_{2g}$ is not contained in any Brill–Noether or Gieseker–Petri divisor.

Proof

A point $[C\cup _{p\sim x}{{\mathbb {P}}}^1, y+z]$ in $\Delta _{0:g,\left\{ {\mathcal {O}}\right\} }$ is mapped by $\chi _{g,2}$ to $[C_1\cup _{p_1\sim p_2} C_2]$ in $\overline{{\mathcal {M}}}_{2\,g}$, where $[C_1,p_1]$ and $[C_2, p_2]$ are two copies of $[C,p] \in \overline{{\mathcal {M}}}_{g,1}$. We show that we can choose [C, p] in such a way that $[C_1\cup _{p_1\sim p_2} C_2]$ is not contained in any Brill–Noether divisor.

Let [C, p] be the curve obtained by glueing a copy [E, z] of a generic elliptic curve to each point $y_i$ of a curve $[{{\mathbb {P}}}^1, p, y_1,\ldots ,y_g]\in {\mathcal {M}}_{0,g+1}$.

Next, we consider the moduli space $\overline{{\mathcal {M}}}_{0, 2g}$ and the map

$$\begin{aligned} i:\overline{{\mathcal {M}}}_{0, 2g} \rightarrow \overline{{\mathcal {M}}}_{2g} \end{aligned}$$

obtained by glueing a copy of [E, z] at each marking, where [E, z] is again a generic elliptic curve.

Next, observe that for the curve [C, p] we have $[C_1\cup _{p_1\sim p_2} C_2] \in i(\overline{{\mathcal {M}}}_{0, 2\,g})$. Our conclusion follows from Proposition 4.1 in [23] which says that $i(\overline{{\mathcal {M}}}_{0, 2g})$ does not meet any Brill–Noether divisor.

The Gieseker–Petri case can be treated analogously. Using Theorem A$'$ in [21] we see that we can construct a curve in $\chi _{g,2}(\overline{{\mathcal {R}}}_{g,2})$ not contained in any Gieseker–Petri divisor. $\square$

Remark 4.2

The case of the Gieseker–Petri divisor $E^1_{g+1}$ admits another proof, which we will now present.

Proof

We remark that $E^1_{g+1}$ is the closure of the branch locus of a proper and finite map

$$\begin{aligned} \pi :\overline{{\mathcal {G}}}^1_{g+1} \rightarrow \overline{{\mathcal {M}}}_{2g}^{\text {ct}} \end{aligned}$$

where $\overline{{\mathcal {G}}}^1_{g+1}$ is the map parametrizing limit $g^1_{g+1}$’s on curves of compact type. The degree of this map is known to be the Catalan number $C_g = \frac{1}{g+1}\left( {\begin{array}{c}2\,g\\ g\end{array}}\right)$.

We consider the curve $[C_1\cup _{p_1\sim p_2}C_2]$ obtained by glueing together two copies of a generic $[C,p] \in {\mathcal {M}}_{g,1}$.

Our goal is to show that every crude limit linear series in $G^1_{g+1}([C_1\cup _{p_1\sim p_2}C_2])$ is refined and that the number of $g^1_{g+1}$ over this curve is $\frac{1}{g+1}\left( {\begin{array}{c}2g\\ g\end{array}}\right)$. If we show this, it will follow from Corollary 3.5 in [22] that this curve is not contained in $E^1_{g+1}$.

We consider a crude limit linear series in $G^1_{g+1}([C_1\cup _{p_1\sim p_2}C_2])$. This is simply a collection $\left\{ L_1, V_1\right\}$ and $\left\{ L_2, V_2\right\}$ of $g^1_{g+1}$ on the two respective components, satisfying the inequalities

$$\begin{aligned} a_0 + b_1 \ge g+1 \ \ \text {and} \ a_1+b_0\ge g+1 \end{aligned}$$

where $(a_0,a_1)$ and $(b_0, b_1)$ are the two vanishing sequences at the points $p_1$ and $p_2$ respectively. Because $[C,p]\in {\mathcal {M}}_{g,1}$ is generic, Theorem 4.5 in [22] implies that $a_0+a_1 \le g+1$ and $b_0+b_1 \le g+1$.

It follows that

$$\begin{aligned} a_0+b_1 = a_1+b_0 = a_0+a_1= b_0+b_1 = g+1 \end{aligned}$$

and implicitly all the crude limit linear series in $G^1_{g+1}([C_1\cup _{p_1\sim p_2}C_2])$ are refined.

We count the number of such limit linear series. Denoting $d= g+1-a_0$ we observe that $\left\{ L_1(-a_0p_1), V_1(-a_0p_1)\right\}$ and $\left\{ L_2(-a_0p_2), V_2(-a_0p_2)\right\}$ are $g^1_d$’s having order $2d-g-1$ at $p_1$ and $p_2$ respectively.

Theorem A in [31] implies there are $(2d-g-1)\frac{g!}{d!(g-d+1)!}$ such $g^1_d$’s over a generic curve $[C,p] \in {\mathcal {M}}_{g,1}$. Consequently, we get in this way

$$\begin{aligned} \sum \limits _{d = \big \lceil {\frac{g+1}{2}}\big \rceil }^{g+1} \frac{(2d-g-1)^2}{(g+1)^2} \left( {\begin{array}{c}g+1\\ d\end{array}}\right) ^2 \end{aligned}$$

distinct limit linear series in $G^1_{g+1}([C_1\cup _{p_1\sim p_2}C_2])$. We can rewrite this sum as

$$\begin{aligned} \sum \limits _{d = \big \lceil {\frac{g+1}{2}}\big \rceil }^{g+1}\left[ \left( {\begin{array}{c}g\\ d\end{array}}\right) -\left( {\begin{array}{c}g\\ d-1\end{array}}\right) \right] ^2 = \frac{1}{2}\sum \limits _{d = 0}^{g+1}\left[ \left( {\begin{array}{c}g\\ d\end{array}}\right) -\left( {\begin{array}{c}g\\ d-1\end{array}}\right) \right] ^2 \end{aligned}$$

Using the Narayana-Catalan identity (cf. A46 in [48])

$$\begin{aligned} \sum \limits _{d=1}^g\left( {\begin{array}{c}g\\ d\end{array}}\right) \left( {\begin{array}{c}g\\ d-1\end{array}}\right) = \frac{g}{g+1} \left( {\begin{array}{c}2g\\ g\end{array}}\right) \end{aligned}$$

and the classical identity

$$\begin{aligned} \sum \limits _{d=0}^g \left( {\begin{array}{c}g\\ d\end{array}}\right) ^2 = \left( {\begin{array}{c}2g\\ g\end{array}}\right) \end{aligned}$$

we conclude that the cardinality of $G^1_{g+1}([C_1\cup _{p_1\sim p_2}C_2])$ is

$$\begin{aligned} \sum \limits _{d = \big \lceil {\frac{g+1}{2}}\big \rceil }^{g+1} \frac{(2d-g-1)^2}{(g+1)^2} \left( {\begin{array}{c}g+1\\ d\end{array}}\right) ^2 = \frac{1}{g+1}\left( {\begin{array}{c}2g\\ g\end{array}}\right) \end{aligned}$$

as required. $\square$

We remark that Theorem 4.1 is in stark contrast with the known results for the map $\chi _g:\overline{{\mathcal {R}}}_g\rightarrow \overline{{\mathcal {M}}}_{2\,g-1}$ for which we know that transversality with Brill–Noether loci is not always satisfied (see [2, Theorem 0.4] and [10, Theorem 1.4]).

Proof of Theorem 1.1

For $g\ge 22$, the existing literature implies that $\overline{{\mathcal {R}}}_{g,2}$ is of general type: We consider the chain of forgetful morphisms

$$\begin{aligned} \overline{{\mathcal {R}}}_{g,2} \rightarrow \overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2} \rightarrow \overline{{\mathcal {M}}}_g \end{aligned}$$

The first map is dominant with finite fibers. Then [49, Theorem 6.10] implies the following inequality of Kodaira dimensions

$$\begin{aligned} \kappa ( \overline{{\mathcal {R}}}_{g,2}) \ge \kappa (\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}). \end{aligned}$$

Hence if $\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}$ is of general type, then so is $\overline{{\mathcal {R}}}_{g,2}$. We look at the second map, whose general fiber is the second symmetric power of a generic curve C of genus g. We know that $\text {Sym}^2(C)$ is of general type for $g\ge 3$, see [47, Sect. 2]. The main result of [52] implies

$$\begin{aligned} \kappa (\overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2}) \ge \kappa (\text {Sym}^2(C)) + \kappa (\overline{{\mathcal {M}}}_g) = 2 + \kappa (\overline{{\mathcal {M}}}_g) \end{aligned}$$

In particular, if $\overline{{\mathcal {M}}}_g$ is of general type, then so is $\overline{{\mathcal {R}}}_{g,2}$. This is true for $g\ge 22$, see [23] and [25].

We are left to treat the cases $g=13$ and $16\le g \le 21$. In order to prove that $\overline{{\mathcal {R}}}_{g,2}$ is of general type, it is sufficient to write the canonical divisor as the sum of an effective divisor and a divisor that is big and nef. One such divisor we can consider is $\psi$, see [28, Proposition 1.2].

Our goal is to write $K_{\overline{{\mathcal {R}}}_{g,2}}$ as the sum of an effective divisor and a positive contribution of $\psi$. To write $K_{\overline{{\mathcal {R}}}_{g,2}}$ in such a way, we need an effective divisor whose class

$$\begin{aligned} a_\psi \cdot \psi + a_\lambda \cdot \lambda - \sum \limits _{\delta = \text {boundary class}}b_\delta \cdot \delta \end{aligned}$$

satisfies the inequalities

$$\begin{aligned} \frac{a_\lambda }{a_\psi } > 13 \ \text {and} \ \frac{a_\lambda }{b_\delta } < \frac{13}{c_\delta } \ \forall \ \delta \ \text {a boundary class}, \end{aligned}$$

where $c_\delta$ is minus the coefficient of $\delta$ in $K_{\overline{{\mathcal {R}}}_{g,2}}$, as appearing in Proposition 3.3. The value $\frac{a_\lambda }{b_\delta }$ is called the slope of $\delta$ (for the corresponding effective divisor).

Most of these slope conditions are automatically satisfied by the divisors we will use. For brevity, we will omit writing the whole class for these effective divisors and only write $``\cdots ''$ instead. We will only write down the coefficients for which is not clear without computation that the slope condition is satisfied.

Because we do not have a uniform way of finding suitable effective divisors satisfying all the numerical requirements, these low genus cases will be treated one by one.

To conclude Theorem 1.1 we first set up some notations. We denote by $\pi _g:\overline{{\mathcal {R}}}_g \rightarrow \overline{{\mathcal {M}}}_g$ the map forgetting the Prym structure and by $p_{g,2}$ the forgetful map from $\overline{{\mathcal {R}}}_{g,2}$ to $\overline{{\mathcal {M}}}_g$.

To obtain effective divisors on $\overline{{\mathcal {R}}}_{g,2}$ we will consider pullbacks of effective divisors from other spaces. We will use the descriptions of the pullbacks $\chi ^*_{g,2}$ and $i_{g,2}^*$ appearing in Sect. 3, as well as some pullbacks of forgetful maps, that are immediate to describe. For example, the map $p_{g,2}$ is the composition

$$\begin{aligned} \overline{{\mathcal {R}}}_{g,2}\xrightarrow {\pi _{g,2}} \overline{{\mathcal {M}}}_{g,2/{\mathbb {Z}}_2} \rightarrow \overline{{\mathcal {M}}}_g \end{aligned}$$

The map $\pi _{g,2}$ is ramified only along the divisor $\Delta _0^{\text {ram}}$. Knowing all divisors above which the map is ramified is sufficient to describe the pullbacks of all boundary divisors. The pullback of the other map is well-known in the literature, see [3, Formula (6)]. Lastly, the map $\pi ^*_g$ at the level of Picard groups appears in [26, Sect. 0].

We will mainly use the class of Gieseker–Petri (see [24, Theorem 1.6]) and Brill–Noether divisors (see [23]). We recall here their classes (up to multiplication with a constant).

Let $d = rs+r$ and $g = rs+s$. In this situation, the class of the Gieseker–Petri divisor in $\text {Pic}({\overline{M}}_g)$ is

$$\begin{aligned} {[}\overline{\mathcal{G}\mathcal{P}}^r_{g,d}] = a\lambda - b_0\delta _0 - \cdots \end{aligned}$$

where

$$\begin{aligned} \frac{a}{b_0} = 6 + \frac{12}{g+1} + \frac{6(s+r+1)(g-2)(g-1)}{(s+1)(r+2)g(g+1)} \end{aligned}$$

Let g, r, d satisfying $g+1 = (r+1)(g-d+r)$ then the class of the Brill–Noether divisor in $\text {Pic}({\overline{M}}_g)$ is

$$\begin{aligned} {[}\overline{{\mathcal {M}}}^r_{g,d}] = 6(g+3)\lambda - (g+1)\delta _0 - \cdots \end{aligned}$$

$\bullet$ For $g = 13$ we consider the following classes of divisors (up to multiplication by some positive constant):

• 1. The pullback by $\chi _{13,2}$ of the Brill–Noether divisor $\overline{{\mathcal {M}}}^2_{26, 19}$ on $\overline{{\mathcal {M}}}_{26}$:

  $$\begin{aligned} \chi _{13,2}^*[\overline{{\mathcal {M}}}^2_{26, 19}] = 29\psi + 464\lambda - 94\delta _0^{\textrm{ram}} - 72\delta _0' - 72\delta _{0:13,\left\{ \eta \right\} }-\cdots \end{aligned}$$

• 2. The pullback by $p_{13,2}$ of the Brill–Noether divisor $\overline{{\mathcal {M}}}^1_{13,7}$ on $\overline{{\mathcal {M}}}_{13}$:

  $$\begin{aligned} p_{13,2}^{*}[\overline{{\mathcal {M}}}^1_{13,7}] = 48\lambda - 14\delta _0^{\textrm{ram}} - 7\delta _0'-\cdots \end{aligned}$$

• 3. The pullback by $i_{13,2}$ of the Prym–Koszul divisor ${\mathcal {U}}_{14,4}$ on $\overline{{\mathcal {R}}}_{14}$ (see [26, Theorem 0.6]):

  $$\begin{aligned} i^*_{13,2}[{\mathcal {U}}_{14,4}] = 21\psi + 180\lambda -42\delta ^{\textrm{ram}}_0 - 28 \delta _0' - \alpha \delta _{0:13,\left\{ \eta \right\} }- \cdots \end{aligned}$$

  where we get from Remark 3.1 and [26, Proposition 1.9] that $\alpha \ge 100$.

We consider the sum

$$\begin{aligned} \frac{1}{92}\chi ^*_{13,2}[\overline{{\mathcal {M}}}^2_{26, 19}] + \frac{1}{23}p_{13,2}^{*}[\overline{{\mathcal {M}}}^1_{13,7}] + \frac{3}{92}i^*_{13,2}[{\mathcal {U}}_{14,4}] = \psi + 13\lambda - 2\delta _0' - 3\delta _0^{\textrm{ram}}- \cdots \end{aligned}$$

We still need to check that this divisor satisfies the slope requirements for the other boundary divisors. The coefficient of the divisor $\delta _{0:13,\left\{ \eta \right\} }$ is greater than or equal to $\frac{1}{92}\cdot 72 + 0 + \frac{3}{92}\cdot 100 > 3$. For the divisor $\delta _{0:13,\left\{ {\mathcal {O}}\right\} }$ the coefficient is greater than or equal to $2\cdot \frac{\text {coefficient}_{\delta _1}[\overline{{\mathcal {M}}}^2_{26, 19}]}{92} = 2\cdot \frac{200}{92} > 3$.

It is easy to check that all other slope requirements are respected. Consequently $K_{\overline{{\mathcal {R}}}_{13,2}}$ is in the effective cone and $\overline{{\mathcal {R}}}_{13,2}$ has non-negative Kodaira dimension.

$\bullet$ For the space $\overline{{\mathcal {R}}}_{16,2}$ we consider the following divisors (up to multiplication by a positive constant):

• 1. The pullback by $\chi _{16,2}$ of the Brill–Noether divisor $\overline{{\mathcal {M}}}^2_{32,23}$ on $\overline{{\mathcal {M}}}_{32}$:

  $$\begin{aligned} \chi _{16,2}^*[\overline{{\mathcal {M}}}^2_{32,23}] = 35\psi + 560\lambda -114\delta _0^{\textrm{ram}}-88\delta _0'- 88\delta _{0:16,\left\{ \eta \right\} } - \cdots \end{aligned}$$

• 2. The pullback by $p_{16,2}$ of the Koszul divisor $\overline{{\mathcal {Z}}}_{16,1}$ (see [24, Theorem 1.1]) on $\overline{{\mathcal {M}}}_{16}$:

  $$\begin{aligned} p_{16,2}^*[\overline{{\mathcal {Z}}}_{16,1}] = 407\lambda - 122\delta _0^{\textrm{ram}} - 61\delta _0' -\cdots \end{aligned}$$

• 3. The pullback by $\pi _{17}\circ i_{16,2}$ of the Brill–Noether divisor $\overline{{\mathcal {M}}}^1_{17,9}$ on $\overline{{\mathcal {M}}}_{17}$:

  $$\begin{aligned} i_{16,2}^*\circ \pi _{17}^*[\overline{{\mathcal {M}}}^1_{17,9}] = 3\psi + 20\lambda - 6\delta _0^{\textrm{ram}} - 3\delta _0' - 16\delta _{0:16,\left\{ \eta \right\} } - \cdots \end{aligned}$$

For small enough $\epsilon$, we consider the sum

$$\begin{aligned}{} & {} \frac{62+\epsilon }{4933}\cdot \chi _{16,2}^*[\overline{{\mathcal {M}}}^2_{32,23}] + \frac{27+80\epsilon }{4993}\cdot p_{16,2}^*[\overline{{\mathcal {Z}}}_{16,1}] \\{} & {} \quad + \frac{921-1656\epsilon }{4933}\cdot i_{16,2}^*\circ \pi _{17}^*[\overline{{\mathcal {M}}}^1_{17,9}] \end{aligned}$$

which is equal to

$$\begin{aligned}{} & {} (1-\epsilon )\psi + 13\lambda - 2\delta _0' - \left( \frac{15888}{4933}-\frac{62}{4933}\epsilon \right) \delta ^{\textrm{ram}}_0\\{} & {} \quad - \left( \frac{20192}{4933}-\frac{26408}{4933}\epsilon \right) \delta _{0:16,\left\{ \eta \right\} }-\cdots \end{aligned}$$

Checking that all other slope requirements are satisfied is immediate. The conclusion follows because $\psi$ is big and nef.

$\bullet$ For the space $\overline{{\mathcal {R}}}_{17,2}$ we consider the following divisors (up to multiplication by a positive constant):

• 1. The pullback by $\chi _{17,2}$ of the Brill–Noether divisor $\overline{{\mathcal {M}}}^4_{34, 31}$ on $\overline{{\mathcal {M}}}_{34}$:

  $$\begin{aligned} \chi _{17,2}^*[\overline{{\mathcal {M}}}^4_{34, 31}] = 111\psi + 1776\lambda - 362\delta _0^{\textrm{ram}} - 280\delta _0' - 280\delta _{0:17,\left\{ \eta \right\} } -\cdots \end{aligned}$$

• 2. The pullback by $p_{17,2}$ of the Brill–Noether divisor $\overline{{\mathcal {M}}}^1_{17,9}$ on $\overline{{\mathcal {M}}}_{17}$:

  $$\begin{aligned} p^*_{17,2}[\overline{{\mathcal {M}}}^1_{17,9}] = 20\lambda - 6\delta _0^{\textrm{ram}} - 3\delta _0' - \cdots \end{aligned}$$

• 3. The pullback by $\pi _{18}\circ i_{17,2}$ of the Gieseker–Petri divisor $\overline{\mathcal{G}\mathcal{P}}^5_{18,20}$ on $\overline{{\mathcal {M}}}_{18}$:

  $$\begin{aligned} i_{17,2}^*\circ \pi _{18}^*[\overline{\mathcal{G}\mathcal{P}}^5_{18,20}] = 77\psi + 516 \lambda - 154\delta _0^{\textrm{ram}} - 77\delta _0' - 408\delta _{0:17,\left\{ \eta \right\} } -\cdots \end{aligned}$$

For $\epsilon$ small enough, we consider the sum

$$\begin{aligned}{} & {} \frac{85-8\epsilon }{21832}\cdot \chi _{17,2}^*[\overline{{\mathcal {M}}}^4_{34, 31}] + \frac{2489+7728\epsilon }{21832}\cdot p^*_{17,2}[\overline{{\mathcal {M}}}^1_{17,9}] \\{} & {} \quad + \frac{161-272\epsilon }{21832}\cdot i_{17,2}^*\circ \pi _{18}^*[\overline{\mathcal{G}\mathcal{P}}^5_{18,20}] \end{aligned}$$

which is equal to

$$\begin{aligned} (1-\epsilon )\psi + 13\lambda - 2\delta _0' - \left( \frac{70498}{21832}+\frac{198}{2729}\epsilon \right) \delta _0^{\textrm{ram}}- \cdots \end{aligned}$$

and it is immediately checked that it respects all the slope requirements. The divisor $i_{17,2}^*\circ \pi _{18}^*[\overline{\mathcal{G}\mathcal{P}}^5_{18,20}]$ is necessary here for the coefficient of $\delta _{0:17,\left\{ \eta \right\} }$.

As the situation is entirely similar for all the other cases, we will simply state which divisors are used, skipping the numerical details.

• For the space $\overline{{\mathcal {R}}}_{18,2}$ we will use the divisors $\chi ^*_{18,2}[\overline{\mathcal{G}\mathcal{P}}^5_{36,35}]$, $p_{18,2}^*[\overline{\mathcal{G}\mathcal{P}}^5_{18,20}]$ and $i_{18,2}^*\circ \pi _{19}^*[\overline{{\mathcal {M}}}^1_{19,10}]$.

• For the space $\overline{{\mathcal {R}}}_{19,2}$ we will use the divisors $\chi _{19,2}^*[\overline{{\mathcal {M}}}^2_{38,27}]$, $p_{19,2}^*[\overline{{\mathcal {M}}}^1_{19,10}]$ and $i_{19,2}^*\circ \pi _{20}^*[\overline{{\mathcal {M}}}^2_{20,15}]$.

• For the space $\overline{{\mathcal {R}}}_{20,2}$ we will use the divisors $\chi _{20,2}^*[\overline{\mathcal{G}\mathcal{P}}^7_{40,42}]$, $p_{20,2}^*[\overline{{\mathcal {M}}}^2_{20,15}]$ and $i_{20,2}^*\circ \pi _{21}^*[\overline{{\mathcal {M}}}^1_{21,11}]$.

• For the space $\overline{{\mathcal {R}}}_{21,2}$ we will use the divisors $\chi _{21,2}^*[\overline{\mathcal{G}\mathcal{P}}^6_{42,42}]$, $p_{21,2}^*[\overline{{\mathcal {M}}}^1_{21,11}]$ and $i_{21,2}^*\circ \pi _{22}^*[\overline{\mathcal{G}\mathcal{P}}^{11}_{22,12}]$.

Using the outlined divisors, we conclude that $K_{\overline{{\mathcal {R}}}_{g,2}}$ is big for $g\ge 16$. The proof follows as a consequence of Theorem 5.1. $\square$

Next, we will study the birational geometry of ${\mathcal {R}}_{g,2}$ when g is small.

Proof of Theorem 1.2

For the partition $\mu = (1,1,2,\ldots ,2,2g-4)$ of length $g+2$, we consider the irreducible stratum ${\mathcal {Q}}_g(\mu )$ defined as the locus:

$$\begin{aligned} \left\{ [C,x,y, z_1,\ldots ,z_g]\in {\mathcal {M}}_{g,g+2} \ | \ {\mathcal {O}}_C\left( x+y+2\sum _{i=1}^{g-1}z_i + (2g-4)z_g\right) \cong \omega _C^{\otimes 2}\right\} \end{aligned}$$

and the map ${\mathcal {Q}}_g(\mu ) \rightarrow {\mathcal {R}}_{g,2}$ defined as:

$$\begin{aligned} {[}C,x,y,z_1,\ldots ,z_g]\mapsto \left[ C,x+y, \omega _C\left( -x-y-\sum _{i=1}^{g-1}z_i-(g-2)z_g\right) \right] . \end{aligned}$$

This map is dominant and ${\mathcal {Q}}_g(\mu )$ is uniruled for $3\le g\le 6$, see [7, Theorem 0.3]. This concludes the proof. $\square$

It is important to note that the Prym moduli spaces provide an interesting geometric property for the divisors of "small" slope on $\overline{{\mathcal {M}}}_g$.

Proposition 4.3

Let D be an effective divisor on ${\mathcal {M}}_g$ of slope $s(D) < 10$ and ${\overline{D}}$ its closure in $\overline{{\mathcal {M}}}_g$. Depending on the parity of g we have the following:

• 1. If $g = 2i+1$, then ${\overline{D}}$ contains the locus $\chi _{i+1}(\Delta _{1:i})$,

• 2. If $g =2i$, then ${\overline{D}}$ contains the locus $\chi _{i,2}(\Delta _{i-1:1,\left\{ \eta \right\} })$.

Proof

We start with the case $g = 2i+1$. If $\chi _{i+1}(\overline{{\mathcal {R}}}_{i+1})$ is contained in ${\overline{D}}$, the conclusion is clear; hence we can assume the contrary.

We consider a generic pencil of elliptic curves. We attach a base point to a generic point of a generic curve of genus i and obtain in this way a test curve A in $\overline{{\mathcal {M}}}_{i+1}$. We denote by $A_{1:i}$ the pullback of this test curve to $\Delta _{1:i}\subseteq \overline{{\mathcal {R}}}_{i+1}$. We know from [12, Sect. 3.1], [37, Example 1.16] that

$$\begin{aligned} A_{1:i} \cdot \lambda = 3, \ A_{1:i} \cdot \delta _0' = 12, \ A_{1: i} \cdot \delta ^{\textrm{ram}}_0 = 12, \ A_{1:i} \cdot \delta _{1:i} = -3 \end{aligned}$$

while the intersection with all other boundary divisors is 0.

We have that $\chi _{i+1}^*[{\overline{D}}]$ is an effective divisor in $\overline{{\mathcal {R}}}_{i+1}$. If this divisor does not contain $\Delta _{1:i}$ in its support, it follows that $\chi _{i+1}^*[{\overline{D}}]\cdot A_{1:i} \ge 0$. If we write (up to multiplication by a constant) the class of ${\overline{D}}$ as $s\lambda - \delta _0 -\cdots$ the inequality $\chi _{i+1}^*[{\overline{D}}]\cdot A_{1:i} \ge 0$ becomes:

$$\begin{aligned} 2s - 4\left( 2 + 1+\frac{s}{4}\right) + 2 \ge 0 \end{aligned}$$

that is $s\ge 10$. Hence our assumption was wrong and we get the conclusion for the odd case.

For the case $g = 2i$ we can define a test curve $A_{i-1:1,\left\{ \eta \right\} }$ on $\overline{{\mathcal {R}}}_{i,2}$ by considering two points on the genus $i-1$ component of the test curve A in $\overline{{\mathcal {M}}}_i$ and pulling it back to $\Delta _{i-1:1,\left\{ \eta \right\} }$. We have the intersection numbers:

$$\begin{aligned} A_{i-1:1,\left\{ \eta \right\} }\cdot \lambda = 3, \ \ \ A_{i-1:1,\left\{ \eta \right\} }\cdot \delta _0' = 12, \ \ \ A_{i-1:1,\left\{ \eta \right\} } \cdot \delta ^{\textrm{ram}}_0 = 12 \end{aligned}$$

and $A_{i-1:1,\left\{ \eta \right\} } \cdot \delta _{i-1:1,\left\{ \eta \right\} } = -3$, while the intersection of $A_{i-1:1,\left\{ \eta \right\} }$ with $\psi$ and all other boundary classes is 0.

To prove that $A_{i-1:1,\left\{ \eta \right\} }\cdot \psi = 0$, we consider the map $\pi _{i,2}:\overline{{\mathcal {R}}}_{i,2} \rightarrow \overline{{\mathcal {M}}}_{i,2/{\mathbb {Z}}_2}$. The relation

$$\begin{aligned} \pi _{i,2*}A_{i-1:1,\left\{ \eta \right\} }\cdot \psi = 0 \end{aligned}$$

is a consequence of [40, Test curve E]. The push-pull formula implies the equality $A_{i-1:1,\left\{ \eta \right\} }\cdot \psi = 0$.

For the non-zero intersection numbers, we look at the image of $A_{i-1:1,\left\{ \eta \right\} }$ through the map

$$\begin{aligned} i_{i,2}:\overline{{\mathcal {R}}}_{i,2} \rightarrow \Delta _0^{\textrm{ram}}\subseteq \overline{{\mathcal {R}}}_{i+1}. \end{aligned}$$

The image is a curve that has the same intersection numbers as $A_{1:i}$. We can use again the push-pull formula to compute the intersection numbers we wanted. To see that the intersection with all other boundary divisors is 0 is immediate, as the test curve $A_{i-1:1,\left\{ \eta \right\} }$ does not intersect these divisors.

By considering the pullback of $\chi _{i,2}:\overline{{\mathcal {R}}}_{i,2} \rightarrow \overline{{\mathcal {M}}}_{2i}$, the proof follows analogously to the case $g= 2i+1$. $\square$

5 The singularities of $\overline{{\mathcal {R}}}_{g,2}$

In order to conclude Theorem 1.1 we still need to prove that any pluricanonical form on the smooth locus of $\overline{{\mathcal {R}}}_{g,2}$ can be holomorphically extended to any desingularisation. The goal of this section is to conclude:

Theorem 5.1

We fix $g\ge 4$ and let ${\widehat{{\mathcal {R}}}}_{g,2} \rightarrow \overline{{\mathcal {R}}}_{g,2}$ be any desingularisation. Then every pluricanonical form defined on the smooth locus $\overline{{\mathcal {R}}}^{\text {reg}}_{g,2}$ extends holomorphically to the space ${\widehat{{\mathcal {R}}}}_{g,2}$.

Because $\overline{{\mathcal {R}}}_{g,2}$ is a normal variety with finite quotient singularities, our goal is to prove Theorem 5.1 using the Reid–Shepherd-Barron–Tai criterion. We describe the locus of non-canonical singularities and show that every pluricanonical form extends holomorphically over it. This is Theorem 5.5, as well as an analysis similar to [31, pages 41–44].

To prove Theorem 5.1, we follow closely the approaches in [36] and [26]. As our proofs are very similar to those in the aforementioned papers, our main objective is to point out the distinctions in the statements.

Next, we want to describe the smooth locus of $\overline{{\mathcal {R}}}_{g,2}$ and for this we give the following definitions.

Definition 5.2

An irreducible component $C_j$ of a quasistable curve $[X,x+y]$ is called a rational tail if the arithmetic genus $p_a(C_j)$ is 0 and $C_j \cap \overline{X\setminus C_j} = \left\{ p\right\}$. The node p is then called a rational tail node. A non-trivial automorphism $\sigma$ of $[X, x+y]$ is called a rational tail automorphism (with respect to $C_j$) if $\sigma _{X\setminus C_j}$ is the identity.

It is clear from the definition that if $C_j$ is a rational tail then the two points x and y are on $C_j$. With the obvious modifications, we can define what it means for a morphism $\sigma$ to be an elliptic tail automorphism (with respect to an elliptic tail $C_j$), with the remark that in the definition we add the extra condition $x, y \notin C_j$.

Definition 5.3

An exceptional component E of a quasistable curve $[X, x+y]$ is called a disconnecting exceptional component if $\overline{X\setminus E}$ consists of two disjoint connected components, which we denote $X_1$ and $X_2$.

Let $[X, x+y, \eta , \beta ]$ a 2-branched Prym curve and E a disconnecting exceptional component of $[X, x+y]$. We denote by $\gamma _E \in \text {Aut}_0(X,x+y,\eta , \beta )$ the inessential automorphism that is the multiplication with 1 and respectively $-1$ in every fiber of $\eta$ over $X_1$ and respectively $X_2$.

Similarly to Theorem 6.5 in [26] and Proposition 2.15 in [36] we get the following theorem:

Theorem 5.4

Let $(X, x+y, \eta , \beta )$ be a 2-branched Prym curve of genus $g\ge 4$. Then the point $[X, x+y, \eta , \beta ]$ in $\overline{{\mathcal {R}}}_{g,2}$ is smooth if and only if $\textrm{Aut}(X, x+y, \eta , \beta )$ is generated by rational tail involutions, elliptic tail involutions and automorphisms of the form $\gamma _E$ for some disconnecting exceptional component E.

Proof

Denote by $(C, x+y)$ the stable model of $(X, x+y)$. We can partition the nodes of C into the set $N\subseteq \text {Sing}(C)$ of exceptional nodes (i.e. an exceptional component of X was collapsed to that node) and $\Delta {:=}\text {Sing}(C){\setminus } N$. Then, the versal deformation space of $(X,x+y,\eta ,\beta )$ can be described as:

$$\begin{aligned} {\mathbb {C}}^{3g-1}_\tau = \bigoplus _{p_i\in N}{\mathbb {C}}_{\tau _i}\oplus \bigoplus _{p_i\in \Delta }{\mathbb {C}}_{\tau _i} \oplus \bigoplus _{C_j\subseteq C} H^1\left( C_j^\nu , T_{C_j^\nu }(-D_j)\right) \end{aligned}$$

where $C_j^\nu$ is the normalization of the irreducible component $C_j$ and $D_j$ is the divisor of special points on $C_j^\nu$ (preimages of nodes and of $x+y$). Moreover, if ${\mathbb {C}}^{3g-1}_t$ is the versal deformation space of $(C,x+y)$ then the map ${\mathbb {C}}_\tau ^{3g-1}\rightarrow {\mathbb {C}}_t^{3g-1}$ is given by $t_i = \tau _i^2$ if $p_i \in N$ and $t_i = \tau _i$ otherwise.

Locally at $[X,x+y,\eta , \beta ]$, the space $\overline{{\mathcal {R}}}_{g,2}$ is given as the quotient of the versal deformation space ${\mathbb {C}}_\tau ^{3g-1}$ by the automorphism group $\text {Aut}(X,x+y,\eta , \beta )$. Because of [13, Lemme 1], we can assume that $\text {Aut}(X,x+y,\eta , \beta )$ acts linearly on the coordinates of ${\mathbb {C}}_\tau ^{3g-1}$.

In this situation, we know from [46] that $[X, x+y, \eta , \beta ]$ is smooth in $\overline{{\mathcal {R}}}_{g,2}$ if and only if the group $\textrm{Aut}(X, x+y, \eta , \beta )$ is generated by quasi-reflections (i.e. automorphisms having 1 as an eigenvalue of multiplicity exactly $3g-2$). Arguing as in [26, Proposition 6.6] we conclude that the only automorphisms acting as quasi-reflections are those appearing in the statement. $\square$

The non-canonical singularities of the space $\overline{{\mathcal {R}}}_{g,2}$ can be easily described.

Theorem 5.5

Let $g\ge 4$. Then a point $[X, x+y, \eta , \beta ]\in \overline{{\mathcal {R}}}_{g,2}$ is a non-canonical singularity if and only if X has an elliptic tail $C_j$ with j-invariant 0 and $\eta$ is trivial on $C_j$.

This is equivalent to showing that all the other singularities of $\overline{{\mathcal {R}}}_{g,2}$ are canonical. Let $\varphi \in \text {Aut}(X,x+y,\eta , \beta )$ be an order n automorphism and $\xi _n$ a fixed n-th root of unity. If $\varphi$ has eigenvalues $\xi _n^{a_1},\ldots , \xi _n^{a_{3\,g-1}}$ with $0\le a_i < n$ for all $1\le i \le 3\,g-1$, we define the age of $\varphi$ to be

$$\begin{aligned} \text {age}(\varphi , \xi _n){:=}\frac{1}{n}\sum _{i=1}^{3g-1} a_i. \end{aligned}$$

The Reid–Shepherd-Barron–Tai criterion states that ${\mathbb {C}}^{3g-1}_\tau /\text {Aut}(X,x+y,\eta , \beta )$ has canonical singularities if and only if for every non-trivial $\varphi \in \text {Aut}(X,x+y,\eta , \beta )$ we have $\text {age}(\varphi ,\xi _n) \ge 1$. Because $\text {Aut}(X,x+y,\eta , \beta )$ can be assumed to act linearly on ${\mathbb {C}}_\tau ^{3g-1}$, we reduce Theorem 5.5 to a combinatorial study of the action of an element $\varphi \in \text {Aut}(X,x+y,\eta , \beta )$ on the nodes of X, as well as on its irreducible components.

The description of the locus of non-canonical singularities is very similar to other cases in the literature, see [26, 31, 35, 36] and [27]. The approach in proving Theorem 5.5 will be similar to the one in [26] and [36] and we will continue by pointing out the differences in our case. First, we will define what it means for a pair $\left( (X,x+y,\eta , \beta ), \sigma \right)$, where $\sigma \in \text {Aut}(X, x+y, \eta , \beta )$, to be singularity reduced.

For such a pair $\left( (X,x+y,\eta , \beta ), \sigma \right)$ we take again $(C,x+y)$ to be the stable model of $(X,x+y)$ and $\sigma _C$ to be the automorphism of C induced by $\sigma$. We consider the definition:

Definition 5.6

Let $p_{i_0}, p_{i_1} = \sigma _C(p_{i_0}), \ldots , p_{i_{m-1}} = \sigma ^{m-1}_C(p_{i_0})$ be distinct nodes of C, cyclically permuted by $\sigma _C$ and $p_{i_0}$ is not a disconnecting exceptional node, a rational tail node or an elliptic tail node. Then $\sigma$ acts on the subspace $\bigoplus _{j=0}^{m-1}{\mathbb {C}}_{\tau _{i_j}}$ of ${\mathbb {C}}_\tau ^{3g-1}$ as $\sigma \cdot \tau _{i_{j-1}} = c_j\tau _{i_j} \ \forall j =\overline{0,m-1}$ for some constants $c_j$. We say the pair $\left( (X,x+y, \eta , \beta ), \sigma \right)$ is singularity reduced if for every cycle as above we have $\prod _{j=1}^{m}c_j \ne 1$.

Arguing as in [26, Proposition 6.8] and [36, Proposition 3.6] it is sufficient to prove that if the pair $\left( (X,x+y, \eta , \beta ), \sigma \right)$ is singularity reduced and satisfies $\text {age}(\sigma , \xi _n) <1$ then X has an elliptic tail as in Theorem 5.5. As in [26] and [36] we denote by $(*)$ the assumption that $\left( (X,x+y, \eta , \beta ), \sigma \right)$ is singularity reduced and satisfies $\text {age}(\sigma , \xi _n) <1$. We then have that:

Proposition 5.7

If $(*)$ holds, then $\sigma _C$ fixes all nodes and all components of the stable model $(C, x+y)$ of $(X, x+y)$.

Proof

The proof follows exactly as in [36, Proposition 3.7 and Proposition 3.8]. We will sketch here some of the details.

We assume there exist a point $p_{i_0}$ that is not fixed by $\sigma _C$. Let m be the length of the orbit of $p_{i_0}$ and denote $p_{i_k} {:=}\sigma _C(p_{i_k-1})$ for $1\le k \le m-1$. We consider the subspace

$$\begin{aligned} W {:=}\bigoplus _{j=0}^{m-1}{\mathbb {C}}_{\tau _{i_j}} \end{aligned}$$

of the versal deformation space ${\mathbb {C}}^{3g-1}_\tau$. The action of $\sigma$ on W is given by a matrix of type

$$\begin{aligned} B={\small \begin{pmatrix} 0&{}c_1&{}&{}\\ \vdots &{}&{}\ddots &{}\\ 0&{}&{}&{}c_{m-1}\\ c_m&{}0&{}\cdots &{}0 \end{pmatrix}} \end{aligned}$$

with $\prod _{j=1}^{m}c_j \ne 1$.

Because $B^m = \left( \prod _{j=1}^m c_j\right) \cdot {\mathbb {I}}_m$ and $B^n = {\mathbb {I}}_m$ for n the order of $\sigma$, it follows that $\prod _{j=1}^m c_j$ is a $\frac{n}{m}-$th root of unity, say $\xi _n^{m\cdot l}$ for some $1\le l < \frac{n}{m}$.

The eigenvalues of B are $\xi _n^{l+j\cdot \frac{n}{m}}$ for $0\le j \le m-1$ and hence

$$\begin{aligned} \text {age}(\sigma , \xi _n) \ge \frac{1}{n}\sum _{j=0}^{m-1}\left( l+j\cdot \frac{n}{m}\right) > 1 \end{aligned}$$

when $m \ge 3$. We are left with the case $m = 2$, where we get there exists at most one pair of nodes that are interchanged by $\sigma _C$. To treat the case $m = 2$ we need first to obtain more restrictions for $\sigma _C$ (for example that it fixes all components of C).

The proof that it fixes all components of C is similar to the proof above and to the computation in [31, page 35]. If there is an irreducible component $C_j$ that is not fixed by $\sigma _C$, we will obtain either that the component has genus less than 2, or that it is in an orbit with 2 elements (i.e. $\sigma _C$ interchanges $C_j$ and another component). These cases can be treated on an ad-hoc basis.

Once we have that all components are fixed by $\sigma _C$, we can go back to show that all nodes are fixed by $\sigma _C$. $\square$

Next we will look at how $\sigma$ acts on the components of C and describe all possible situations where the age contribution of the respective component is less than 1. This description is similar to that in [26, Proposition 6.12] with some extra cases coming from the existence of the two marked points.

Proposition 5.8

Assume $(*)$ holds and let $C_j$ a component of C with normalization denoted $C_j^\nu$. We denote by $D_j$ the divisor of marked points on $C_j^\nu$ and $\varphi _j {:=}\sigma ^\nu _{|C^\nu _j}$. Then $(C^\nu _j, D_j, \varphi _j)$ is of one of the following types and the contribution to $\text {age}(\sigma , \xi _n)$ coming from $H^1(C^\nu _j, T_{C_j^\nu }(-D_j)) \subseteq {\mathbb {C}}_v^{3\,g-1}$ is at least the quantity $w_j$:

• i. Identity component $\varphi _j = \text {Id}_{C^\nu _j}$, the pair $(C^\nu _j, D_j)$ is arbitrary and $w_j = 0$.

• ii. Elliptic tail: $C^\nu _j$ is elliptic, $D_j = p_1$ and $p_1$ is fixed by $\varphi _j$. Depending on the order of $\varphi _j$ we distinguish the subcases:

  • a. $\textrm{ord}(\varphi _j) = 2$ and $w_j = 0$

  • b. $\textrm{ord}(\varphi _j) = 4$, $C^\nu _j$ has j-invariant 1728 and $w_j = \frac{1}{2}$

  • c. $\textrm{ord}(\varphi _j) = 3$ or 6, $C^\nu _j$ has j-invariant 0 and $w_j = \frac{1}{3}$

• iii. Elliptic ladder: $C_j^\nu$ is elliptic, $D_j = p_1 + p_2$ with both markings coming from nodes of C and $\varphi _j$ fixes $p_1$ and $p_2$. We distinguish three subcases depending on the order of $\varphi _j$:

  • a. $\textrm{ord}(\varphi _j) = 2$ and $w_j =\frac{1}{2}$

  • b. $\textrm{ord}(\varphi _j) = 4$, $C_j^\nu$ has j-invariant 1728 and $w_j =\frac{3}{4}$

  • c. $\textrm{ord}(\varphi _j) = 3$, $C_j^\nu$ has j-invariant 0 and $w_j =\frac{2}{3}$

• iv. Hyperelliptic tail: $C^\nu _j$ has genus 2, $\varphi _j$ is the hyperelliptic involution, $D_j$ is of the form $D_j = p_1$ with $p_1$ fixed by $\varphi _j$ and $w_j = \frac{1}{2}$

• v. Rational tail: $C^\nu _j$ is rational, $D_j = p_1 + x+y$, $\textrm{ord}(\varphi _j) = 2$, the point $p_1$ is fixed by $\varphi _j$ while x and y are permuted, and $w_ j =0$.

• vi. Rational ladder: $C^\nu _j$ is rational, $D_j = p_1 + p_2+ x+y$, $\textrm{ord}(\varphi _j) = 2$, the points $p_1, p_2$ are fixed by $\varphi _j$ while x and y are permuted, and $w_ j =\frac{1}{2}$.

• vii. 1-pointed elliptic tail: $C_j^\nu$ is elliptic, $D_j = p_1 + p_2$ where $p_1$ comes from a node of C and $p_2$ comes from one of the markings x, y. Both $p_1$ and $p_2$ are fixed by $\varphi _j$. We distinguish three subcases depending on the order of $\varphi _j$:

  • a. $\textrm{ord}(\varphi _j) = 2$ and $w_j = \frac{1}{2}$

  • b. $\textrm{ord}(\varphi _j) = 4$, $C_j^\nu$ has j-invariant 1728 and $w_j = \frac{3}{4}$

  • c. $\textrm{ord}(\varphi _j) = 3$ or 6, $C_j^\nu$ has j-invariant 0 and $w_j = \frac{2}{3}$ viii. 2-pointed elliptic tail: $C_j^\nu$ is elliptic, $D_j = p_1 +x+y$ with x and y permuted by $\varphi _j$. Again, we distinguish two subcases depending on the order of $\varphi _j$:

  • a. $\textrm{ord}(\varphi _j) = 2$ and $w_j = \frac{1}{2}$

  • b. $\textrm{ord}(\varphi _j) = 6$, $C_j^\nu$ has j-invariant 0 and $w_j = \frac{1}{3} + \frac{1}{3} +\frac{1}{6} = \frac{5}{6}$.

If $(*)$ holds, the cases where $w_j > \frac{1}{3}$ cannot appear, while if for every irreducible component of C we have $w_j$ = 0, we get that $\sigma$ is a composition of quasi-reflections. This implies Theorem 5.5. In order to extend the pluricanonical forms over the locus of non-canonical singularities, the method outlined in [31, pages $41-44$] works in our situation, hence Theorem 5.1 follows.

6 The Prym-canonical divisorial strata

We consider the moduli space ${\mathcal {C}}^n{\mathcal {R}}_g$ parametrizing pairs $(X, x_1,\ldots ,x_n, \eta )$ where $(X,x_1,\ldots ,x_n)$ is an n-pointed smooth curve of genus g and $\eta$ is a non-trivial element of $\textrm{Pic}^0(X)$.

For a partition ${\underline{d}} = (d_1,\ldots ,d_n)$ of $g-1$, we consider the Prym-canonical divisorial stratum $PD_{{\underline{d}}}$, defined as the locus

$$\begin{aligned} PD_{{\underline{d}}} {:=}\left\{ [X,x_1,\ldots , x_n,\eta ] \in {\mathcal {C}}^n{\mathcal {R}}_g \ | \ h^0\left( X, \omega _X\otimes \eta \left( -\sum _{i=1}^nd_ix_i\right) \right) \ge 1 \right\} \end{aligned}$$

Our goal is to provide a suitable compactification $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ of ${\mathcal {C}}^n{\mathcal {R}}_g$ and compute the class $[{\overline{PD}}_{{\underline{d}}}]$ in $\text {Pic}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$.

Using the same approach as in [15], we compactify ${\mathcal {C}}^n{\mathcal {R}}_g$ to a moduli space $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ parametrizing isomorphism classes of pairs $(X, x_1,\ldots ,x_n, \eta , \beta )$ where the n-pointed curve $(X, x_1,\ldots ,x_n)$ is quasistable of genus g and $\beta :\eta ^{\otimes 2} \rightarrow {\mathcal {O}}_X$ is a homomorphism of invertible sheaves that satisfies the properties:

• 1. The line bundle $\eta$ has total degree 0 on X and degree 1 on every exceptional component,

• 2. The morphism $\beta$ is generically non-zero away from the exceptional components.

The notion of isomorphism is simply the pointed generalization of the one considered in [6].

Next, we consider the irreducible boundary divisors of $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ and describe a generic element for each one of them. They are as follows:

• The divisors $\Delta _0', \Delta _0''$ and $\Delta _0^{\textrm{ram}}$ whose generic point corresponds to an element $[X, x_1,\ldots ,x_n, \eta , \beta ]$ where $[X,\eta , \beta ] \in \overline{{\mathcal {R}}}_g$ is generic in $\Delta _0', \Delta _0''$ and $\Delta _0^{\textrm{ram}}$, respectively, and $x_1,\ldots ,x_n\in X$ are generic points of the non-exceptional component of X.

• The divisors $\Delta _{g,S}$ for $S\subseteq \left\{ 1,\ldots ,n\right\}$ of cardinality $|S|\le n-2$. A generic element $[X,x_1,\ldots ,x_n,\eta ,\beta ]$ satisfies that $[X,x_1,\ldots ,x_n]\in \Delta _{0,S^c}\subseteq \overline{{\mathcal {M}}}_{g,n}$, the line bundle $\eta$ is trivial on the rational component and is a non-trivial 2-torsion on the genus g component.

• The divisors $\Delta _{i,S:g-i}$ for $1\le i\le g-1$ and $S\subseteq \left\{ 1,\ldots ,n\right\}$, whose generic element $[X, x_1,\ldots ,x_n, \eta , \beta ]$ satisfies that $[X, \eta , \beta ]$ is in $\Delta _{i:g-i}\subseteq \overline{{\mathcal {R}}}_g$ and $[X,x_1,\ldots ,x_n]$ is in $\Delta _{i,S}\subseteq \overline{{\mathcal {M}}}_{g,n}$. We remark that the notations $\Delta _{i, S:g-i}$ and $\Delta _{g-i,S^c:i}$ refer to the same divisor of $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$.

• The divisors $\Delta _{i,S}$ for $1\le i\le g-1$ and $S\subseteq \left\{ 1,\ldots ,n\right\}$, whose generic element $[X, x_1,\ldots ,x_n, \eta , \beta ]$ satisfies that $[X, \eta , \beta ]$ is in $\Delta _i\subseteq \overline{{\mathcal {R}}}_g$ and $\left\{ x_i\right\} _{i\in S}$ are generic points of the component of X on which $\eta$ is non-trivial.

Next, we consider maps between moduli spaces of pointed (Prym) curves and we describe the action of the pullback at the level of Picard groups. This will allow us to compute the classes of the Prym-canonical divisorial strata.

We consider the map $\pi :\overline{{\mathcal {C}}^n{\mathcal {R}}}_g \rightarrow \overline{{\mathcal {M}}}_{g,n}$ forgetting the Prym structure and stabilizing the underlying n-pointed curve. We denote $\psi _j {:=}\pi ^{*}\psi _j$, $\lambda {:=}\pi ^{*}\lambda$ in $\textrm{Pic}(\mathrm {\overline{{\mathcal {C}}^n{\mathcal {R}}}_g})$ and we want to describe the pullback of the classes $\lambda , \psi _j, \delta _0', \delta _0'', \delta _0^{\textrm{ram}}$, $\delta _{i,S:g-i}$ and $\delta _{i,S}$ with respect to different maps.

Proposition 6.1

Consider the map $\pi _1:\overline{{\mathcal {M}}}_{g-i,n+1-s} \rightarrow \overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ defined as

$$\begin{aligned} {[}C,x_1,\ldots ,x_{n-s},x] \mapsto [C\cup _{x\sim y}Y,x_1,\ldots ,x_n, {\mathcal {O}}_C, \eta _Y] \end{aligned}$$

where $[Y,y,x_{n-s+1},\ldots ,x_n,\eta _Y]$ is a generic element in ${\mathcal {C}}^{s+1}{\mathcal {R}}_i$. The pullback at the level of Picard groups $\pi _1^*:\textrm{Pic}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g) \rightarrow \textrm{Pic}(\overline{{\mathcal {M}}}_{g-i,n+1-s})$ satisfies:

$$\begin{aligned} \pi _1^{*}\lambda= & {} \lambda , \ \pi _1^*\delta _0'' = \pi _1^*\delta _0^{\textrm{ram}} = 0, \ \pi _1^*\delta _0' = \delta _0 \\ \pi _1^*\psi _j= & {} \psi _j \ \textrm{for} \ 1\le j\le n-s, \ \pi _1^*\psi _j =0 \ \textrm{for} \ j\ge n-s+1\\ \pi _1^*\delta _{j,S:g-j}= & {} 0 \ \text {for every} \ 1\le j\le g-1 \ \text {and every} \ S\subseteq \left\{ 1,\ldots , n\right\} \end{aligned}$$

For $T = \left\{ n-s+1,\ldots , n\right\}$ we have:

$$\begin{aligned} \pi _1^*\delta _{j,S} = {\left\{ \begin{array}{ll} \delta _{j-i, (S\setminus T)\cup \left\{ n-s+1\right\} } &{} \textrm{when} \ i\le j\le g, \ T\subseteq S \ \textrm{and} \ (j,S)\ne (i,T) \\ -\psi _{n-s+1} &{} \textrm{when} \ j=i \ \textrm{and} \ T=S \\ 0 &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

Proof

The equalities with right term 0 follow because $\textrm{Im}(\pi _1)$ does not intersect the respective divisors. For the other equalities we look at the composition map $\pi \circ \pi _1:\overline{{\mathcal {M}}}_{g-i,n+1-s} \rightarrow \overline{{\mathcal {M}}}_{g,n}$ and we use that $\pi _1^*\circ \pi ^* = (\pi \circ \pi _1)^*$.

The map $\pi$ is a finite map, with ramification divisor $\Delta _0^{\text {ram}}$. Using this, we can describe the map $\pi ^*$ at the level of Picard groups. The map $(\pi \circ \pi _1)^*$ is well understood and is described in [4, Lemma 1.4]. Using these, we can conclude that $\pi _1^*$ is as described in the proposition, implying the conclusion. $\square$

Similarly we get:

Proposition 6.2

Let $\pi _2:\overline{{\mathcal {C}}^{n-s+1}{\mathcal {R}}}_{g-i}\rightarrow \overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ be given as

$$\begin{aligned} {[}C,x_1,\ldots ,x_{n-s},x, \eta _C] \mapsto [C\cup _{x\sim y}Y,x_1,\ldots ,x_n, \eta _C, \eta _Y] \end{aligned}$$

where $[Y,y,x_{n-s+1},\ldots ,x_n,\eta _Y]$ is a generic point of ${\mathcal {C}}^{s+1}{\mathcal {R}}_i$. When $s\ne 0$, the pullback at the level of Picard groups $\pi _2^*:\textrm{Pic}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g) \rightarrow \textrm{Pic}(\overline{{\mathcal {C}}^{n-s+1}{\mathcal {R}}}_{g-i})$ satisfies:

$$\begin{aligned} \pi _2^{*}\lambda= & {} \lambda , \ \pi _2^*\delta _0'' = 0, \ \pi _2^*\delta _0' = \delta _0'+\delta _0'', \ \pi _2^*\delta _0^{\textrm{ram}} = \delta _0^{\textrm{ram}}\\ \pi _2^*\psi _j= & {} \psi _j \ \textrm{for} \ 1\le j\le n-s, \ \pi _2^*\psi _j =0 \ \textrm{for} \ j\ge n-s+1 \end{aligned}$$

For $T = \left\{ n-s+1,\ldots , n\right\}$ we have:

$$\begin{aligned} \pi _2^*\delta _{j,S} = {\left\{ \begin{array}{ll} \delta _{j-i, (S\setminus T)\cup \left\{ n-s+1\right\} } &{} \textrm{when} \ i+1\le j\le g \ \textrm{and} \ T\subseteq S \\ 0 &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

We recall that the divisor $\Delta _{j,S:g-j}$ admits the alternative notation $\Delta _{g-j,S^c:j}$. We choose the one where the set contains $\left\{ n-s+1\right\}$ and we have:

$$\begin{aligned} \pi _2^*\delta _{j,S:g-j} = \delta _{j-i, (S\setminus T)\cup \left\{ n-s+1\right\} :g-j} + \delta _{g-j,((S\setminus T)\cup \left\{ n-s+1\right\} )^c} \end{aligned}$$

when $i\le j\le g-1, \ T\subseteq S$ and $(j,S)\ne (i,T)$,

$$\begin{aligned} \pi _2^*\delta _{j,S:g-j} = -\psi _{n-s+1} \end{aligned}$$

if $i = j$ and $S= T$, and is 0 otherwise.

When $s=0$ we have $T = \emptyset$ and

$$\begin{aligned} \pi _2^*\delta _{j,S:g-j} = \delta _{j-i, S\cup \left\{ n+1\right\} :g-j} + \delta _{g-j,(S\cup \left\{ n+1\right\} )^c} + \delta _{j, S:g-j-i} + \delta _{j,S} \end{aligned}$$

when $(j,S) \ne (i,\emptyset )$ and

$$\begin{aligned} \pi _2^*\delta _{i,\emptyset :g-i} = -\psi _{n+1} + \delta _{i,\emptyset :g-2i} + \delta _{g-2i,\left\{ 1,2,\ldots , n+1\right\} } \end{aligned}$$

All the other pullbacks are as in the case $s\ne 0$.

Notice that when $j \le i$ or $j \ge g-i$ some classes in the expression of $\pi _2^*\delta _{j,S:g-j}$ do not make sense. The convention is that these classes are 0 in the formula.

Proof

We can check using test curves that the boundary classes are linearly independent in $\textrm{Pic}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$. Using this and the obvious commutative diagram

the conclusion follows. $\square$

Lastly, we have:

Proposition 6.3

Let $\pi _3:\overline{{\mathcal {C}}^{n-s+1}{\mathcal {R}}}_{g-i}\rightarrow \overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ be given as

$$\begin{aligned} {[}C,x_1,\ldots ,x_{n-s},x, \eta _C] \mapsto [C\cup _{x\sim y}Y,x_1,\ldots ,x_n, \eta _C, {\mathcal {O}}_Y] \end{aligned}$$

where $[Y,y,x_{n-s+1},\ldots ,x_n]$ is a generic element of ${\mathcal {M}}_{i,s+1}$. When $s\ne 0$, the pullback at the level of Picard groups $\pi _3^*:\textrm{Pic}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g) \rightarrow \textrm{Pic}(\overline{{\mathcal {C}}^{n-s+1}{\mathcal {R}}}_{g-i})$ satisfies:

$$\begin{aligned} \pi _3^{*}\lambda= & {} \lambda , \ \pi _3^*\delta _0'' = \delta _0'', \ \pi _3^*\delta _0' = \delta _0', \ \pi _3^*\delta _0^{\textrm{ram}} = \delta _0^{\textrm{ram}}\\ \pi _3^*\psi _j= & {} \psi _j \ \textrm{for} \ 1\le j\le n-s, \ \pi _3^*\psi _j =0 \ \textrm{for} \ j\ge n-s+1 \end{aligned}$$

We denote again $T = \left\{ n-s+1,\ldots , n\right\}$ and we have:

$$\begin{aligned} \pi _3^*\delta _{j,S}= & {} {\left\{ \begin{array}{ll} \delta _{j-i, (S\setminus T)\cup \left\{ n-s+1\right\} } &{} \textrm{when} \ i+1\le j\le g \ \textrm{and} \ T\subseteq S \\ \delta _{j,S} &{} \textrm{when} \ i+j\le g, \ S\subseteq T^c \ \textrm{and} \ (j,S)\ne (g-i, T^c) \\ -\psi _{n-s+1} &{} \textrm{when} \ j=g-i \ \textrm{and} \ S=T^c \\ 0 &{} \text {otherwise} . \end{array}\right. }\\ \pi _3^*\delta _{j,S:g-j}= & {} {\left\{ \begin{array}{ll} \delta _{j-i, (S\setminus T)\cup \left\{ n-s+1\right\} :g-j} &{} \textrm{when} \ i+1\le j\le g-1 \ \textrm{and} \ T\subseteq S \\ \delta _{j,S:g-j-i} &{} \textrm{when} \ i+j\le g-1 \ \textrm{and} \ S\cap T =\emptyset \\ 0 &{} \text {otherwise} . \end{array}\right. } \end{aligned}$$

When $s = 0$, we have $T = \emptyset$ and

$$\begin{aligned} \pi _3^*\delta _{j,S} = \delta _{j,S} + \delta _{j-i, S\cup \left\{ n+1\right\} } \end{aligned}$$

for $(j,S) \ne (g-i,\left\{ 1,2,\ldots , n\right\} )$,

$$\begin{aligned}{} & {} \pi _3^*\delta _{g-i,\left\{ 1,2,\ldots , n\right\} } = -\psi _{n+1} + \delta _{g-2i, \left\{ 1,2,\ldots , n+1\right\} }\\{} & {} \pi _3^*\delta _{j,S:g-j} = \delta _{j-i, S\cup \left\{ n+1\right\} :g-j-i} + \delta _{j,S:g-j}. \end{aligned}$$

All the other pullbacks are the same as in the case $s \ne 0$.

As in Proposition 6.2, the convention is that whenever a boundary class does not make sense, it is 0 in the corresponding formula.

Proof

This follows analogously to Propositions 6.1 and 6.2. $\square$

As we do not know if $\psi _1,\ldots , \psi _n$ and $\lambda$ generate $\textrm{Pic}({\mathcal {C}}^n{\mathcal {R}}_g)$ we start by proving that the class $[PD_{{\underline{d}}}]$ is a linear combination of these classes in $\textrm{Pic}({\mathcal {C}}^n{\mathcal {R}}_g)$.

Proposition 6.4

Let ${\underline{d}} = (d_1,\ldots , d_n)$ be a partition of $g-1$ with all entries positive. We have the following equality in $\textrm{Pic}({\mathcal {C}}^n{\mathcal {R}}_g)$:

$$\begin{aligned} {[}PD_{{\underline{d}}}] = \sum _{j=1}^n \frac{d_j(d_j+1)}{2}\psi _j - \lambda \end{aligned}$$

Proof

We work over the locus ${\mathcal {R}}^0_g$ of smooth Prym curves of genus g without automorphisms. We consider the Cartesian diagram

Because the Prym curves have no automorphisms, it follows that there exists a line bundle ${\mathcal {P}}$ on ${\mathcal {C}}^1{\mathcal {R}}_g^0$ restricting to $\eta$ over each fiber $p^{-1}([X,\eta ])$. We denote by $\Delta _i$ the diagonal of ${\mathcal {C}}^{n+1}$ parametrizing points $[C,x,x_1,\ldots , x_n]$ satisfying $x=x_i$, and consider the short exact sequence

$$\begin{aligned} 0 \rightarrow p_2^*{\mathcal {P}} \rightarrow p_2^*{\mathcal {P}}\otimes {\mathcal {O}}_{{\mathcal {C}}^{n+1}}\left( \sum _{i=1}^{n}d_i\Delta _i\right) \rightarrow p_2^*{\mathcal {P}} \otimes \left( {\mathcal {O}}_{{\mathcal {C}}^{n+1}}\left( \sum _{i=1}^{n}d_i\Delta _i \right) /{\mathcal {O}}_{{\mathcal {C}}^{n+1}}\right) \rightarrow 0 \end{aligned}$$

We pushforward this by $p_{1*}$ and obtain the exact sequence:

$$\begin{aligned}{} & {} 0 \rightarrow p_{1*}\left( p_2^*{\mathcal {P}}\otimes {\mathcal {O}}_{{\mathcal {C}}^{n+1}}\left( \sum _{i=1}^{n}d_i\Delta _i\right) \right) \\{} & {} \quad \rightarrow p_{1*}\left( p_2^*{\mathcal {P}} \otimes \left( {\mathcal {O}}_{{\mathcal {C}}^{n+1}}\left( \sum _{i=1}^{n}d_i\Delta _i\right) /{\mathcal {O}}_{{\mathcal {C}}^{n+1}}\right) \right) \xrightarrow {\alpha } \cdots \\{} & {} \qquad \cdots \xrightarrow {\alpha } R^1p_{1*}p_2^*{\mathcal {P}}\rightarrow R^1p_{1*}\left( p_2^*{\mathcal {P}}\otimes {\mathcal {O}}_{{\mathcal {C}}^{n+1}}\left( \sum _{i=1}^{n}d_i\Delta _i\right) \right) \rightarrow 0 \end{aligned}$$

The pushforward $p_{1*}\left( p_2^*{\mathcal {P}} \otimes \left( {\mathcal {O}}_{{\mathcal {C}}^{n+1}}(\sum _{i=1}^{n}d_i\Delta _i)/{\mathcal {O}}_{{\mathcal {C}}^{n+1}}\right) \right)$ is a vector bundle of rank $g-1$ on ${\mathcal {C}}^n{\mathcal {R}}^0_g$ with the fiber over a point $[X,x_1,\ldots ,x_n,\eta ]$ identified with $H^0(X, \eta \otimes \left( {\mathcal {O}}_X(\sum _{i=1}^nd_ix_i)/{\mathcal {O}}_X\right) )$. Similarly $R^1p_{1*}p_2^*{\mathcal {P}}$ is a vector bundle of rank $g-1$ with fiber $H^1(X, \eta )$ over the point $[X,x_1,\ldots ,x_n,\eta ]$. The map $\alpha$ restricted to the fiber over $[X,x_1,\ldots ,x_n, \eta ]$ is the one induced by the exact sequence:

$$\begin{aligned} 0\rightarrow \eta \rightarrow \eta \left( \sum _{i=1}^nd_ix_i\right) \rightarrow \eta \otimes \left( {\mathcal {O}}_X\left( \sum _{i=1}^nd_ix_i\right) /{\mathcal {O}}_X)\right) \rightarrow 0 \end{aligned}$$

The Riemann-Roch Theorem implies that $[PD_{{\underline{d}}}]$ is the degeneration locus of the map $\alpha$ and consequently

$$\begin{aligned} {[}PD_{{\underline{d}}}] = -c_1\left( p_{1!}\left( p_2^*{\mathcal {P}}\otimes {\mathcal {O}}_{{\mathcal {C}}^{n+1}}\left( \sum _{i=1}^{n}d_i\Delta _i\right) \right) \right) \end{aligned}$$

We apply the Grothendieck-Riemann-Roch formula and obtain that $[PD_{{\underline{d}}}]$ is equal to

$$\begin{aligned} -p_{1*}\left( \frac{\left( c_1(p_2^*{\mathcal {P}}) +\sum _{i=1}^{n}d_i\Delta _i\right) ^2}{2} - \frac{c_1(\omega _{p_1})\cdot \left( c_1(p_2^*{\mathcal {P}}) +\sum _{i=1}^{n}d_i\Delta _i\right) }{2} + \frac{c_1(\omega _{p_1})^2}{12} \right) \end{aligned}$$

But ${\mathcal {P}}^{\otimes 2} \cong {\mathcal {O}}_{{\mathcal {C}}^1{\mathcal {R}}^0_g}$ and we conclude that $2c_1({\mathcal {P}}) = 0$. Since the torsion terms disappear in the rational Picard group, we have the equality

$$\begin{aligned} {[}PD_{{\underline{d}}}] = -p_{1*}\left( \frac{(\sum _{i=1}^{n}d_i\Delta _i)^2}{2} - \frac{(\sum _{i=1}^{n}d_i\Delta _i)\cdot c_1(\omega _{p_1})}{2}+ \frac{c_1(\omega _{p_1})^2}{12} \right) \end{aligned}$$

hence

$$\begin{aligned} {[}PD_{{\underline{d}}}] = \sum _{i=1}^n\frac{d_i^2}{2}\psi _i + \sum _{i=1}^n\frac{d_i}{2}\psi _i - \lambda \end{aligned}$$

$\square$

We are now ready to compute the class $[{\overline{PD}}_{{\underline{d}}}]$ in $\textrm{Pic}(\overline{{\mathcal {C}}^n{\mathcal {R}}}_g)$.

Proof

We will denote the Prym-canonical class $[{\overline{PD}}_{{\underline{d}}}]$ by

$$\begin{aligned}{}[{\overline{PD}}_{{\underline{d}}}]= & {} \sum _{j=1}^n \frac{d_j(d_j+1)}{2}\psi _j -\lambda - b_0'\delta _0'-b_0''\delta _0'' - b_0^{\textrm{ram}}\delta _0^{\textrm{ram}} - \cdots \\{} & {} \cdots -\sum _{\begin{array}{c} 1\le i \le g \\ S\subseteq \left\{ 1,\ldots ,n\right\} \end{array}} b_{i,S} \delta _{i,S} -\sum _{\begin{array}{c} 1\le i \le g-i \\ S\subseteq \left\{ 1,\ldots ,n\right\} \end{array}} b_{i,S:g-i} \delta _{i,S:g-i}. \end{aligned}$$

Our goal is to compute the coefficients of the boundary divisors.

We define ${\mathcal {H}}^2_g(2{\underline{d}}, 2^{g-1})$ as the locus

$$\begin{aligned} \left\{ [C,x_1,\ldots ,x_{g+n-1}]\in {\mathcal {M}}_{g,g+n-1} \ | \ {\mathcal {O}}_C\left( \sum _{i=1}^{n}2d_i x_i + \sum _{i=n+1}^{g+n-1}2x_i\right) \cong \omega _C^{\otimes 2}\right\} \end{aligned}$$

and denote by ${\mathcal {Q}}_g(2{\underline{d}}, 2^{g-1})$ the component of ${\mathcal {H}}^2_g(2{\underline{d}}, 2^{g-1})$ parametrizing divisors that are not twice the divisor of a holomorphic differential.

We consider the morphism ${\mathcal {Q}}_g(2{\underline{d}}, 2^{g-1})\rightarrow {\mathcal {C}}^n{\mathcal {R}}_g$ defined as

$$\begin{aligned} {[}C,x_1,\ldots ,x_{g+n-1}] \mapsto \left[ C,x_1,\ldots , x_n, \omega _C\otimes {\mathcal {O}}_C\left( -\sum _{i=1}^{n}d_i x_i -\sum _{i=n+1}^{g+n-1}x_i\right) \right] \end{aligned}$$

and we immediately observe that the image of this map is the divisor $PD_{{\underline{d}}}$. Consequently, the closure ${\overline{PD}}_{{\underline{d}}}$ in $\overline{{\mathcal {C}}^n{\mathcal {R}}}_g$ is well understood, see [5], and we can use the method of [41, Proposition 1.4] to compute its class.

In order to conclude the proof, we first set some notations. For a partition ${\underline{m}} = (m_1,\ldots ,m_n)$ of g with all entries positive, we consider the stratum

$$\begin{aligned} {\mathcal {H}}_g({\underline{m}}, 1^{g-2}) = \left\{ [C,x_1,\ldots ,x_{g+n-2}]\in {\mathcal {M}}_{g,g+n-2} | \ {\mathcal {O}}_C\left( \sum _{i=1}^{n}m_i x_i + \sum _{i=n+1}^{g+n-2}x_i\right) \cong \omega _C\right\} \end{aligned}$$

We consider the map ${\mathcal {H}}_g({\underline{m}}, 1^{g-2})\rightarrow {\mathcal {M}}_{g,n}$ forgetting the points $x_{n+1}, \ldots , x_{g+n-2}$ and denote $D^g_{{\underline{m}}}$ its image. We have a similar approach in the case when ${\underline{m}}$ is a length n partition of $g-1$ with at least one negative entry. We denote

$$\begin{aligned} {\mathcal {H}}_g({\underline{m}}, 1^{g-1}) = \left\{ [C,x_1,\ldots ,x_{g+n-1}]\in {\mathcal {M}}_{g,g+n-1} | \ {\mathcal {O}}_C\left( \sum _{i=1}^{n}m_i x_i + \sum _{i=n+1}^{g+n-1}x_i\right) \cong \omega _C\right\} \end{aligned}$$

and we consider the map to ${\mathcal {M}}_{g,n}$ forgetting the last $g-1$ entries. We denote its image by $D^g_{{\underline{m}}}$.

In the notations of Propositions 6.1, 6.2 and 6.3, we have

$$\begin{aligned}{} & {} \pi _2^*[{\overline{PD}}_{{\underline{d}}}] = PD_{\underline{d'}} + \text {Boundary divisors} ,\\{} & {} \pi _3^*[{\overline{PD}}_{{\underline{d}}}] = PD_{\underline{d'}} + \text {Boundary divisors, and} \\{} & {} \pi _1^*[{\overline{PD}}_{{\underline{d}}}] = {\left\{ \begin{array}{ll} D^{n+1-s}_{{\underline{d}}''} + \text {Boundary terms} &{} \textrm{when} \ d_T\ge 2i-2 \\ D^{n+1-s}_{{\underline{d}}'} + \text {Boundary terms} &{} \textrm{when} \ d_T \le 2i-4 \end{array}\right. } \end{aligned}$$

where ${\underline{d}}' = (d_1,\ldots , d_{n-s}, d_T-i)$ and ${\underline{d}}'' = (d_1,\ldots , d_{n-s}, d_T+1-i)$. These equalities are satisfied as a consequence of the proof of [41, Proposition 1.4]. Because we can consider a multitude of variations of the maps $\pi _1, \pi _2$ and $\pi _3$, and because we computed the coefficients of the $\psi _j$’s in Proposition 6.4, we conclude that:

$$\begin{aligned} b_{i,S:g-i}= & {} \frac{(d_S-i)(d_S-i+1)}{2} \ \text {and} \\ b_{i,S}= & {} {\left\{ \begin{array}{ll} \frac{(d_S-i+1)(d_S-i+2)}{2} &{} \textrm{when} \ d_S\ge i-1 \\ \frac{(d_S-i)(d_S-i+1)}{2} &{} \textrm{when} \ d_S \le i-2 \end{array}\right. } \end{aligned}$$

Moreover, because $\delta _0$ is not one of the boundary terms appearing in the pullback $\pi _1^*$, as remarked in [41, Proposition 1.4], we deduce that $b_0' = 0$. A similar argument for the map $\pi _2^*$ implies further that $b_0'' = b_0' = 0$. The pushforward of the class $[{\overline{PD}}_{{\underline{d}}}]$ was computed in [41, Proposition 1.4]. We use this to conclude $b_0^{\textrm{ram}} = \frac{1}{4}$, thus completing the proof. $\square$

Remark 6.5

All the Prym-canonical divisorial strata $PD_{{\underline{d}}}$ are irreducible, see [34].

We consider the moduli space ${\mathcal {Q}}_g(\mu )$ parametrizing divisors of quadratic differentials with zero multiplicities given by the partition $\mu$. It is clear that if $g\ge 22$ and $l(\mu )\ge g$, the stratum ${\mathcal {Q}}_g(\mu )$ is of general type, because it maps with finite fibers to ${\mathcal {M}}_{g,l(\mu )-g}$. Mapping to ${\mathcal {C}}^{l(\mu )-g}{\mathcal {R}}_g$ instead allows us to find examples of strata of general type in genus as low as 13.

Remark 6.6

Let ${\mathcal {Q}}_g(\mu )$ a stratum with all entries of $\mu$ even and $l(\mu )\ge g$. If ${\mathcal {R}}_g$ is of general type, then ${\mathcal {Q}}_g(\mu )$ is also of general type. Similarly, if we allow $\mu$ to have two odd entries and assume $l(\mu )\ge g+2$ we get that ${\mathcal {Q}}_g(\mu )$ is of general type when ${\mathcal {R}}_{g,2}$ is.