Proof. Suppose $\pi : D \to C$ is in $\overline {M}_{E,g}-M_{E,g}$ . Then it must be in the image of $\widetilde {M}_{E,g}-\widetilde {M}^{\circ }_{E,g}$ . Consider the forgetful morphism $\widetilde {M}_{E,g}\to \overline {\mathcal {M}}_{1, 2g-2}$ that records the target curve with the $(2g-2)$ smooth branch points. Since $(C,R_1, \ldots , R_{2g - 2})$ is not smooth, it is represented by a point in $\overline {\mathcal {M}}_{1, 2g-2}-\mathcal {M}_{1, 2g-2}$ , with one of its irreducible components being E. We know that the boundary of $\overline {\mathcal {M}}_{1, 2g-2}$ lies in the union of the image of the following clutching morphisms:

$$ \begin{align*} \kappa_{1,i;0,j}: \overline{\mathcal{M}}_{1, i+1} \times \overline{\mathcal{M}}_{0, j+1} &\to \overline{\mathcal{M}}_{1, i+j} = \overline{\mathcal{M}}_{1,2g-2}, \\ \kappa_{0}: \overline{\mathcal{M}}_{0, 2g } &\to \overline{\mathcal{M}}_{1, 2g-2}. \end{align*} $$

Here, $(i,j)$ runs through the pairs $i \geq 0$ , $j \geq 2$ , and $i+j=2g-2$ . The map $\kappa _{1,i;0,j}$ clutches two curves together at an ordinary double point, while $\kappa _{0}$ creates a node on one curve by identifying two marked points on it.

Notice that $(C,R_1, \ldots , R_{2g - 2})$ cannot be in the image of $\kappa _{0}$ since C should have E as one of its irreducible components. Hence, there must exist $i \geq 0, j \geq 2$ such that $i+j=2g-2$ and $C= (C_1, R_1', \ldots , R_i', R_{i + 1}')\cup _{R_{i + 1}' = R_{j + 1}"}(C_2, R_1", \ldots , R_j", R_{j + 1}")$ for some $(i + 1)$ - and $(j + 1)$ -pointed stable curves $(C_1, R_1', \ldots , R_i', R_{i + 1}')$ and $(C_2, R_1", \ldots , R_j", R_{j + 1}")$ of genera $1$ and $0$ , respectively. We can further choose $C_1$ and $C_2$ such that $C_2$ is a smooth genus- $0$ curve, since $C_1$ is allowed to be degenerate. Let $D_1=\pi ^{-1}(C_1)$ and $D_2=\pi ^{-1}(C_2)$ be the preimages of $C_1$ and $C_2$ under $\pi : D \to C$ , and let Q be the node on C resulting from clutching. Note that $C_2$ has $j+1$ marked points, one of them is Q. We have the following possibilities for the configuration of $\pi : D \to C$ .

1. 1. If j is even: Write $j=2m$ . Since $C_2$ is smooth of genus $0$ , any $\mathbb {Z}/2\mathbb {Z}$ -cover of $C_2$ must have an even number of the branching points. Therefore, Q is not a branching point. In this case, $\pi : D_2 \to C_2$ is a stable $\mathbb {Z}/2\mathbb {Z}$ -cover branched at j points, so this gives an element of $\widetilde {{\mathcal {H}}}_{m-1;1}$ . Moreover, E is a component of $C_1$ and $D_1 \to C_1$ is a stable $\mathbb {Z}/2\mathbb {Z}$ -cover, branched at $i=2g-2-j=2(g-m-1)$ points, hence this gives an element of $\widetilde {M}_{E,g-m;1}$ . Therefore, $D \to C$ is in $\Xi _{g-m,m-1}$ . The stability of $D_2$ implies that $j\geq 2$ , so $m-1\geq 0$ .

2. 2. If j is odd: Write $j=2m+1$ . Then Q must be a branching point. In this case, $\pi : D_2 \to C_2$ is a stable $\mathbb {Z}/2\mathbb {Z}$ -cover branched at $j+1=2(m+1)$ points, so it is in $\widetilde {{\mathcal {H}}}_{m}$ . Moreover, E is a component of $C_1$ and $D_1 \to C_1$ is a stable $\mathbb {Z}/2\mathbb {Z}$ -cover branched at $i+1=2(g-m-1)$ points, hence it is in $\widetilde {M}_{E,g-m}$ . Therefore, $D \to C$ is in $\Delta _{g-m,m}$ . Since i is odd, we have $i \geq 1$ , implying $g-m \geq 2$ . The stability of $C_2$ implies that $j\geq 2$ , and it is odd, so $j\geq 3$ . Thus, we conclude $m \geq 1$ .

Therefore, $\pi : D \to C$ lies in the union of image of clutching maps.

Proof. Let $g_1, g_2 \geq 1$ be integers such that $g_1 + g_2 = g$ . We compute

$$ \begin{align*} \dim(\Xi_{g_1, g_2 - 1}) = \dim(\widetilde{M}_{E,g_1;1})+\dim(\widetilde{{\mathcal{H}}}_{g_2 - 1;1}) = (2g_1-2)+1+(2g_2 - 3)+1=\dim(\overline{M}_{E,g})-1, \end{align*} $$

while for $g_1 \geq 2$ , we find that

$$ \begin{align*} \dim(\Delta_{g_1, g_2}) = \dim(\widetilde{M}_{E,g_1})+\dim(\widetilde{{\mathcal{H}}}_{g_2}) =(2g_1-2)+(2g_2-1) =\dim(\overline{M}_{E,g})-1. \end{align*} $$

Therefore, all loci $\Xi _{g_1, g_2 - 1}$ and $\Delta _{g_1, g_2}$ have codimension $1$ in $\overline {M}_{E, g}$ , and Lemma 2.4 implies that the boundary of $\overline {M}_{E, g}$ has codimension $1$ in $\overline {M}_{E, g}$ .

Suppose that $M_{E,g}$ is not dense in $\overline {M}_{E,g}$ . Then there is a non-empty, open set $U \subseteq \overline {M}_{E,g}$ that does not intersect $M_{E,g}$ so that $U \subseteq \overline {M}_{E,g}-M_{E,g}$ , and hence $\dim (U)\leq 2g-3$ . On the other hand, Theorem A.11 says that $\overline {M}_{E,g}$ is irreducible of dimension $2g-2$ . This implies $\dim (U)=2g-2$ , which is a contradiction. To conclude, $M_{E,g}$ is dense in $\overline {M}_{E,g}$ .

Proof. The claim that $\overline {M}_{E,g;n}^{\mathrm {red}}$ is proper and Deligne–Mumford follows from the fact that $\overline {M}_{E,g;n}$ is proper and Deligne–Mumford. To compute its dimension, we observe that all the fibers of $\pi : \overline {M}_{E,g;n} \to E$ are isomorphic. To get an isomorphism between $\pi ^{-1}(O_E) = \overline {M}_{E,g;n}^{\mathrm {red}}$ and $\pi ^{-1}(x)$ , take a point $y \in E$ such that $(2g - 2) \cdot y = x$ and use translation by y to get an isomorphism between $\pi ^{-1}(O_E)$ and $\pi ^{-1}(x)$ . Because $\overline {M}_{E,g;n}$ has dimension $2g - 2 + n$ (Theorem A.11) and all the fibers of $\pi $ are isomorphic, it follows that $\pi ^{-1}(O_E) = \overline {M}_{E,g;n}^{\mathrm {red}}$ has dimension $2g - 3 + n$ , as desired.

Proof. This can be proved using a careful case-by-case analysis, as is done in the appendix, Proposition A.15.

Proof. Let E be an elliptic curve in characteristic $p>2$ with p-rank $f_E$ . Recall that ${\overline {\mathcal {B}}}_{E, 2}$ is irreducible and proper by Lemma 3.1, while $\mathcal {B}_{E, 2}$ cannot contain complete subvarieties by Lemma 3.2. Therefore, $\overline {\mathcal {B}}_{E, 2}-{\mathcal {B}}_{E, 2} \neq \varnothing $ . One boundary point of $\overline {\mathcal {B}}_{E, 2}-{\mathcal {B}}_{E, 2}$ corresponds to a curve ${D' = (E', q_1, q_2)/\{q_1 = q_2\}}$ as in Example 3.3 with $f_{D'} = f_E + 1$ (computed using [Reference Bosch, Lütkebohmert and Raynaud5, Section 9.2, Example 8]). Therefore, by the purity result (Lemma 3.4), the generic p-rank of ${\overline {\mathcal {B}}}_{E, 2}$ equals $f_E + 1$ .

It remains to show that $V_{f_E}(\overline {\mathcal {B}}_{E, 2})$ is non-empty. By Example 3.3, $D = (E, O_E)\cup _{O_E = O_E}(E, O_E)$ , with $f_D = 2f_E$ , represents a curve lying in the boundary of ${\overline {\mathcal {B}}}_{E, 2}$ . If $f_E = 0$ , then $f_D = 0$ . Otherwise, assume $f_E = 1$ . The locus of ordinary curves in ${\overline {\mathcal {B}}}_{E, 2}$ is quasi-affine; this follows from [Reference Oort21, Section 6.1]. Therefore, ${\overline {\mathcal {B}}}_{E, 2}$ contains an element that corresponds to a curve D with lower p-rank, $f_D = f_E = 1$ . Moreover, D is a smooth curve. This follows from Example 3.3, using that both $D_1$ and $D_2$ have $f_{D_1} = f_{D_2} = 2$ when $f_E = 1$ .

Proof. It is enough to consider the case $f_E = 0, f_D = 0$ since the proof of the preceding lemma implies the result in other situations. Write $E: y^2 = x(x - 1)(x - \lambda )$ . For $p> 3$ , there is $\lambda ' \neq \lambda $ such that $E': y^2 = x(x - 1)(x - \lambda ')$ is supersingular (see [Reference Silverman27, Section V.4]). Consider the double covers $E \to \mathbb {P}^1$ and $E' \to \mathbb {P}^1$ given by the inclusion $k(x) \subset k(x, y)$ of function fields, and the fiber product $E \times _{\mathbb {P}^1} E'$ taken with respect to these maps. The normalization of $E\times _{\mathbb {P}^1}E'$ is a supersingular genus $2$ curve with a map of degree $2$ to E and the result follows.

Proof. For $g\geq 4$ , elements of $V_f(\Xi _{1, g - 2})$ correspond to double covers of $(E, O_E)\cup _{O_E, \infty } (\mathbb {P}^1, \infty )$ , ramified above distinct points $R_i\in \mathbb {P}^1 - \{\infty \}, 1\leq i \leq 2g - 2$ and unramified otherwise, with p-rank at most f. Each such element corresponds to one of the following two types of clutches:

1. 1. $(E, O_E)\cup _{O_E = q_1} (H, q_1, q_2)\cup _{q_2 = O_E} (E, O_E)$ representing an element of $V_{f}(\delta _{1, g-2})$ , where H represents an element of $V_{f - 2f_E}(\overline {\mathcal {H}}_{g - 2})$ , and $q_1, q_2 \in H$ are two distinct points that are in the same orbit under the involution of H.

2. 2. $(E', q_1', q_2')\cup _{q_1' = q_1, q_2' = q_2} (H, q_1, q_2)$ representing an element of $V_{f}(\xi _{1, g - 2})$ , where H is as in the previous case, and $E' \to E$ an étale double cover with $q_1', q_2' \in E'$ two distinct points in the same orbit under the induced bielliptic involution.

The dimension of the locus of curves of the first form is $\dim V_{f - 2f_E}(\overline {\mathcal {H}}_{g - 2, 1}) = g - 2 + f - 2f_E$ , while for the second form, it is $\dim V_{f - f_E - 1}(\overline {\mathcal {H}}_{g - 2, 1}) = g - 3 + f - f_E$ . For $g = 2$ , the result follows from the proof of Lemma 3.6, while for $g = 3$ , it follows similarly as for $g \geq 4$ . The only difference is that $H = (H, O_H)$ is an elliptic curve, and $q_1, q_2 \in H - \{O_H\}$ are two distinct points mapping to the same point under the double cover $H \to \mathbb {P}^1$ induced by the divisor $2 \cdot O_H$ . Thus, instead of $\dim V_{f - 2f_E}(\overline {\mathcal {H}}_{g - 2, 1})$ and $\dim V_{f - f_E - 1}(\overline {\mathcal {H}}_{g - 2, 1})$ , we consider $\dim V_{f - 2f_E}(\overline {\mathcal {M}}_{1, 2})$ and $\dim V_{f - f_E - 1}(\overline {\mathcal {M}}_{1, 2})$ in the dimension computations.

Proof. Let us fix $f_E \leq f \leq g - 1 + f_E$ , $g_1 \geq 2,$ and $g_2 \geq 1$ such that $g_1 + g_2 = g$ . (Note that the outcomes of the computations below are also valid in the cases $g_1 = g - 1$ and $g_1 = g - 2$ since $\dim V_0(\overline {\mathcal {M}}_{0, 3}) = 0$ and $\dim V_{f_2}(\overline {\mathcal {M}}_{1, 1}) = f_2$ , and using the identifications $\overline {\mathcal {H}}_{0, 1} \cong \overline {\mathcal {M}}_{0, 3}$ and $\overline {\mathcal {H}}_{1} \cong \overline {\mathcal {M}}_{1, 1}$ similarly as in Section 2.1 and the proof of Lemma 3.10.)

We first compute the dimension of $V_{f}(\Delta _{g_1, g_2})$ . Let $f_1$ and $f_2$ be numbers that satisfy $f_E\leq f_1 \leq g_1$ , $0\leq f_2 \leq g_2$ , and $f_1 + f_2 = f$ . Every element of $V_f(\Delta _{g_1, g_2})$ corresponds to a clutch of a curve D whose isomorphism class lies in $V_{f_1}(\overline {\mathcal {B}}_{E, g_1})$ and a curve H from $V_{f_2}(\overline {\mathcal {H}}_{g_2})$ , identified at points $r \in D$ and $r' \in H$ , the ramification points of the structural morphisms $D \to E$ and $H \to \mathbb {P}^1$ , for some choice of $(f_1, f_2)$ as above. (Note that there are only finitely many choices for r and $r'$ as above given D and H.) By the assumption, for every such a choice of $(f_1, f_2),$ it holds that

$$ \begin{align*}\dim V_{f_1}(\overline{\mathcal{B}}_{E, g_1}) + \dim V_{f_2}(\overline{\mathcal{H}}_{g_2}) = (g_1 - 2 + f_1 - f_E) + (g_2 - 1 + f_2) = g - 3 + f - f_E,\end{align*} $$

and therefore, $\dim V_{f}(\Delta _{g_1, g_2}) = g - 3 + f - f_E$ , for any $2\leq g_1 \leq g - 1$ .

Now, we compute the dimension of $V_{f}(\Xi _{g_1, g_2 - 1})$ . Let $f_1$ and $f_2$ be numbers that satisfy $f_E\leq f_1 \leq g_1$ , $0\leq f_2 \leq g_2 - 1$ , and $f_1 + f_2 = f - 1$ . Let D be a curve whose isomorphism class lies in $V_{f_1}(\overline {\mathcal {B}}_{E, g_1})$ , let H be a hyperelliptic curve from $V_{f_2}(\overline {\mathcal {H}}_{g_2 - 1})$ , and $q_1 \in D$ and $q_1' \in H$ their points that are not ramification points of the structural morphisms $D \to E$ and $H \to \mathbb {P}^1$ . Denote by $q_2 \in D$ (resp. $q_2' \in H$ ) the image of $q_1$ (resp. $q_1'$ ) under the structural bielliptic (resp. hyperelliptic) involution. Every element of $V_f(\Xi _{g_1, g_2 - 1})$ corresponds to a clutch of such curves D and H with $q_1$ and $q_1'$ being identified, as well as $q_2$ and $q_2'$ . By the assumption, we find that

$$ \begin{align*}\dim V_{f_1}(\overline{\mathcal{B}}_{E, g_1, 1}) + \dim V_{f_2}(\overline{\mathcal{H}}_{g_2 - 1, 1}) = (g_1 - 2 + f_1 - f_E) + (g_2 - 2 + f_2) + 2 = g - 3 + f - f_E,\end{align*} $$

which implies $\dim V_{f}(\Xi _{g_1, g_2 - 1}) = g - 3 + f - f_E$ , for any $2\leq g_1 \leq g - 1$ .

Proof. First, observe that $V_f(\overline {\mathcal {B}}_{E, g})$ is non-empty for any $f_E \leq f \leq g - 1 + f_E$ . For $g = 2$ , this follows from Lemma 3.6, and for $g \geq 3$ , it follows from Lemmas 3.5 and 3.6, which establish that $V_f(\Delta _{2, g - 2}) \subset V_f(\overline {\mathcal {B}}_{E, g})$ is non-empty.

We follow the strategy of the proof of theorem [Reference Faber and van der Geer11, Theorem 2.3]. Let $W = W_f$ be a component of the p-rank $\leq f$ locus $V_f(\overline {\mathcal {B}}_{E, g})$ for a fixed f with $f_E \leq f \leq g - 1 + f_E$ . Let $W_{f + 1}$ be a component of $V_{f+ 1}(\overline {\mathcal {B}}_{E, g})$ such that $W_f \subseteq W_{f + 1}$ . This gives us a sequence

$$ \begin{align*}W_f \subseteq W_{f + 1} \subseteq \dots \subseteq W_{g - 2 + f_E} \subseteq W_{g - 1 + f_E}.\end{align*} $$

Note that $W_{g-1+f_E} = \overline {\mathcal {B}}_{E,g}$ and $\dim W_{g - 1 + f_E} = 2g - 3$ by Lemma 3.1. By Lemma 3.4, if $W_{i} \neq W_{i + 1}$ , for $f \leq i \leq g - 2 + f_{E}$ , then $W_{i}$ has codimension $1$ in $W_{i + 1}$ . This implies that the codimension of $W = W_f$ in $\overline {\mathcal {B}}_{E,g} = W_{g - 1 + f_E}$ is at most $g - 1 + f_E - f$ , or equivalently, that

$$ \begin{align*}\dim W \geq g - 2 + f - f_E.\end{align*} $$

To establish an upper bound on the dimension of W, we use induction on g with the base case $g = 2$ covered by Lemma 3.6. The inductive hypothesis assumes that for all $2\leq g' < g$ and $1\leq f' \leq f$ , $V_{f'}(\overline {\mathcal {M}}_{E, g'})$ is pure of dimension $g' - 2 + f' - f_E$ . Since $\mathcal {B}_{E, g}$ cannot contain complete subvarieties (see Lemma 3.2), it follows that $W\cap \Delta \neq \varnothing $ , where $\Delta = \Delta _0 \cup \Delta _1 \cup \dots \cup \Delta _{\lfloor g/2 \rfloor }$ is the boundary divisor of $\overline {\mathcal {M}}_g$ .

In the case $f_E = 1$ , we find that the underlying curve of the generic point of W is smooth by comparing dimensions. Therefore, $W \cap \Delta _i \neq \varnothing $ for some $0 \leq i \leq \lfloor \frac {g}{2} \rfloor $ . In Lemmas 3.10 and 3.11, we compute the dimensions of the loci of curves in $\overline {\mathcal {B}}_{E, g} - \mathcal {B}_{E, g}$ , under the assumption given by the inductive hypothesis, resulting in $\dim (W\cap \Delta _i) \leq g - 4 + f$ for any $0 \leq i \leq \lfloor \frac {g}{2} \rfloor $ . Since $\Delta _i$ has codimension $1$ in $\overline {\mathcal {M}}_g$ and $\overline {\mathcal {M}}_g$ is a smooth stack, every component of $W\cap \Delta _i$ has codimension at most $1$ in W (see [Reference Vistoli29, p. 614]), and we get the following inequality:

(3) $$ \begin{align} \dim W \leq \dim (W \cap \Delta_i) + 1 = g - 3 + f. \end{align} $$

Consequently, we conclude that $\dim W \leq g - 3 + f = g - 2 + f - f_E$ .

For $g\geq 3$ and $f_E = 0$ , the situation requires more careful consideration. The inductive hypothesis gives $\dim V_f(\delta _{1, g-2}) = g - 2 + f$ , making it impossible to apply the previous argument directly. A stronger inequality than (3) is needed. If W consists entirely of isomorphism classes of singular curves, by comparing dimensions, it has to be a component of $V_f(\delta _{1, g-2})$ , yielding

$$ \begin{align*}\dim(W) = g-2+f=g-2+f-2f_E.\end{align*} $$

In the remaining case, the generic point of W is represented by a smooth curve. That means that there is a component $\Gamma $ of $V_f(\mathcal {B}_{E, g}) \subseteq \mathcal {M}_g$ such that $W = \overline {\Gamma } \subseteq \overline {\mathcal {M}}_g$ . In particular, note that $\dim \Gamma = \dim W \geq g - 2 + f\geq 1$ , and that $W \cap \Delta \neq \varnothing $ because of Lemma 3.2. Moreover, since $\Gamma \subseteq \mathcal {M}_g$ , it follows that $\dim \Gamma = \dim j(\Gamma )$ , which consequently implies that $\dim W = \dim j(W)$ , while $W \cap \Delta \neq \varnothing $ implies that $\dim j(W \cap \Delta )\geq 0$ . Let us prove the following inequality:

(4) $$ \begin{align} \dim W \leq \dim j(W \cap \Delta) + 1, \end{align} $$

which will give us the desired upper bound on W; we do this using a similar argument as in [Reference Oort19, Section 1]. Assume that $\dim j(W) \geq \dim j(W\cap \Delta ) + 2$ . In a projective embedding of $\tilde {\mathcal {A}}_g$ , we could then intersect $j(W)$ with $(\dim j(W) - 1)$ -hyperplanes in general position to get a complete one-dimensional subvariety X that does not meet $j(W \cap \Delta )$ . This X would be a subvariety of $j(W) - j(W\cap \Delta ) = j(\Gamma )$ , whose preimage in $\overline {\mathcal {M}}_g$ would thus be a complete one-dimensional subvariety of $\Gamma $ . This is impossible since $\mathcal {B}_{E, g}$ and $\Gamma $ cannot contain such subvarieties by Lemma 3.2. Therefore, the inequality (4) holds. Note that $\dim j(V_f(\delta _{1, g-2}) ) = g - 3 + f$ since the Torelli morphism “forgets” the pair $q_1, q_2 \in H$ of points in the same orbit of hyperelliptic involution as in (1) of the proof of Lemma 3.10. Combining this with Lemmas 3.10 and 3.11, we conclude that $\dim j(W \cap \Delta ) \leq g - 3 + f$ , which implies $\dim W \leq g - 2 + f$ . This finishes the proof.

Proof. The third part follows from the proof of Theorem 3.12. Moreover, we can explicitly describe these components. Given a supersingular elliptic curve $(E, O_E)$ in characteristic $p>0$ , a component of $V_f(\overline {\mathcal {H}}_{g - 2})$ , for $g\geq 4$ , corresponding to a family of stable hyperelliptic curves H, and two distinct points $q, q' \in H$ in the same orbit under the hyperelliptic involution, a component of $V_f(\overline {\mathcal {B}}_{E,g})$ consists of isomorphism classes of stable curves of the form $(E, O_E)\cup _{q = O_E} (H, q, q') \cup _{q' = O_E} (E, O_E)$ ; note that the dimension of this family equals $\dim V_f(\overline {\mathcal {H}}_{g - 2, 1}) = g - 2 + f = g - 2 + f - f_E$ , as desired. For $g = 2$ , this follows from the proof of Lemma 3.6, while for $g = 3$ , it follows similarly as in the proof of Lemma 3.10 using the identification $\overline {\mathcal {H}}_{1}\cong \overline {\mathcal {M}}_{1, 1}$ .

To establish the existence of smooth bielliptic curves with a prescribed p-rank, we employ a technique similar to the one from the proof of [Reference Pries23, Theorem 6.4]. Let $p>3$ be a prime number and E be an elliptic curve in characteristic p whose p-rank equals $f_E$ . By Lemma 3.6, for any $f_E\leq f'\leq 1 + f_E$ , there is a smooth bielliptic curve D of genus $2$ with $f_D = f'$ . By Lemma 3.5 and by comparing dimensions, there is a smooth hyperelliptic curve $H \to \mathbb {P}^1$ of genus $g - 2$ whose p-rank equals $f - f'$ . Given ramification points $r \in D$ and $r' \in H$ of the structure bielliptic and hyperelliptic morphisms of D and H, respectively, consider the curve $(D, r)\cup _{r = r'}(H, r')$ . Any irreducible family of such curves has dimension $(2 - 2 + f' - f_E) + (g - 3 + f - f') = g - 3 + f - f_E$ , and is therefore contained in a component of $V_f(\overline {\mathcal {B}}_{E, g})$ that cannot consist entirely of isomorphism classes of singular curves, due to the description provided earlier and Theorem 3.12. Consequently, a generic member of this component is represented by a smooth bielliptic curve of genus g with p-rank f, which is a double cover of E. A similar argument applies for $p = 3$ and $f \neq 0$ . In the remaining case, $p = 3$ , $f_E = 0$ , and $f = 0$ , by Remark 3.9, there are no smooth curves of genus $2$ that are double covers of E. However, by Lemma 5.1 below, there is a smooth curve $D'$ of genus $3$ that is a double cover of E and whose $3$ -rank equals $0$ . To show the existence of a smooth curve of genus $g\geq 4$ with $3$ -rank $0$ that is a double cover of E, we now use the existence of $D'$ and a smooth (hyper)elliptic curve of genus $g - 3$ with $3$ -rank $0$ and a similar argument to the one we just presented for $p>3$ .

Proof. Let W be a component of $V_f(\overline {\mathcal {B}}_g)$ . As before, we find that $\dim W \geq g - 2 + f$ , using the purity of the p-rank stratification of $\overline {\mathcal {M}}_g$ . Consider the morphism $\Phi : \overline {\mathcal {B}}_g^{\mathrm {adm}} \to {\overline {\mathcal {M}}}_{1, 2g - 2} \to {\overline {\mathcal {M}}}_{1, 1}$ induced by $(D \to (E', P_1, \ldots , P_{2g - 2})) \mapsto ({E}, P_1)$ , where $E = ({E}, P_1)$ denotes the elliptic curve obtained by stabilizing $(E', P_1)$ . Let $W'$ denote the preimage of W under the map $\Psi : \overline {\mathcal {B}}_g^{\mathrm {adm}} \to \overline {\mathcal {M}}_g$ . We separate the proof into different cases depending on whether the generic point of W is represented by a smooth curve.

1. 1. Suppose that the generic point of W is represented by a smooth curve; note that $\dim W = \dim W'$ . There are two possibilities:

   1. (a) $\Phi (W')$ is a single point in $\mathcal {M}_{1,1} \subseteq \overline {\mathcal {M}}_{1,1}$ corresponding to a smooth elliptic curve E. In this case, W is an irreducible locus consisting of isomorphism classes of smooth curves D with $f_D \leq f$ , each admitting a double cover $D \to E$ . In other words, $W \subseteq V_f(\overline {\mathcal {B}}_{E, g})$ . Therefore,

      $$ \begin{align*}\dim W \leq \dim V_f(\overline{\mathcal{B}}_{E, g}) = g - 2 + f - f_E \leq g - 2 + f\end{align*} $$

      by Theorem 3.12.

   2. (b) Otherwise, $\Phi (W')$ consists of at least two points in $\overline {\mathcal {M}}_{1,1}$ . Moreover, $\Phi (W') \subseteq \overline {\mathcal {M}}_{1,1}$ is irreducible, being the image of an irreducible locus under a morphism. These two properties imply that $\Phi (W') = \overline {\mathcal {M}}_{1,1}$ . In particular, there exists an element of $W'$ that maps to the singular elliptic curve $E_0 = (\mathbb {P}^1, \infty , 0, 1)/\{0 = 1\}$ under $\Phi $ , and hence whose image under $\Psi $ lies in $W \cap \Delta _0$ . Therefore, $W \cap \Delta _0 \neq \varnothing $ . Since $\overline {\mathcal {M}}_g$ is a smooth stack, we have $\dim W \leq \dim (W \cap \Delta _0) + 1$ . Thus, to conclude that $\dim W \leq g - 2 + f$ , it suffices to show that $\dim (W \cap \Delta _0) \leq g - 3 + f$ .

      Let $\Gamma _{ns} = \Gamma _{ns}' \cap W$ and $\Gamma _s = \Gamma _s' \cap W$ , where $\Gamma _{ns}'$ and $\Gamma _s'$ are the loci described in Remark 3.15. It suffices to show that $\dim \Gamma _{ns} \leq g - 3 + f$ and $\dim \Gamma _s \leq g - 3 + f$ . Recall that $W \subseteq V_f(\overline {\mathcal {B}}_g)$ .

      • • To show $\dim \Gamma _{ns} \leq g-3+f$ , note that $\Gamma _{ns}$ is the union of the loci $V_f(\overline {\mathcal {B}}_{E, g}) \cap \Delta _0$ over all isomorphism classes $[E]$ of (smooth) elliptic curves E. By Theorem 3.12 and Lemmas 3.10 and 3.11, no component of $V_f(\overline {\mathcal {B}}_{E, g})$ is contained in $\Delta _0$ , and $\dim (V_f(\overline {\mathcal {B}}_{E, g}) \cap \Delta _0) \leq g - 3 + f - f_E$ . Moreover, the locus of curves in $\mathcal {M}_{1,1}$ with p-rank $f_E$ has dimension $f_E$ for $0 \leq f_E \leq 1$ . Hence, we conclude that

        $$ \begin{align*}\dim \Gamma_{ns} \leq (g - 3 + f - f_E) + f_E = g - 3 + f.\end{align*} $$

      • • To show $\dim \Gamma _{s} \leq g-3+f$ , let $V_{f - 1}(\overline {\mathcal {H}}_{g - 1, 2})$ be the locus of stable $2$ -pointed hyperelliptic curves $(H, q_1, q_2)$ of genus $g - 1$ whose p-rank is $\leq f - 1$ , and let $\Gamma " \subseteq V_{f - 1}(\overline {\mathcal {H}}_{g - 1, 2})$ be the sublocus where the marked points $q_1, q_2$ of $(H, q_1, q_2)$ are chosen to be two ramification points of the double cover $H \to C$ , with C a rational curve. Since there are only finitely many ramification points, we have $\dim \Gamma " = \dim V_{f - 1}(\overline {\mathcal {H}}_{g - 1})$ . Using the definition of $\Gamma _s'$ and the inclusion $W \subseteq V_f(\overline {\mathcal {B}}_{g})$ , we deduce that $\Gamma _s = \Gamma _s' \cap W$ is contained in the image of $\Gamma "$ under the clutching morphism $\overline {\mathcal {M}}_{g - 1, 2} \to \overline {\mathcal {M}}_{g}$ . Lemma 3.5 then implies that

        $$ \begin{align*}\dim \Gamma_s = \dim V_f(\overline{\mathcal{H}}_{g - 1}) \leq g - 3 + f.\end{align*} $$

2. 2. If W is not generically represented by a smooth curve, then W consists entirely of isomorphism classes of singular curves. If the image of $W'$ under $\Phi $ contains the isomorphism class of $E_0 = (\mathbb {P}^1, \infty , 0, 1)/\{0 = 1\}$ , the same discussion as above implies that $W \cap \Delta _0 \neq \varnothing $ and $\dim W \leq g - 2 + f$ ; this covers the cases $\Phi (W') = \overline {\mathcal {M}}_{1,1}$ or $\Phi (W') = [E_0]$ . Otherwise, $\Phi (W') = [E]$ , where $E = (E, O_E)$ is a smooth elliptic curve. As before, Theorem 3.12 implies that $\dim W \leq \dim V_f(\overline {\mathcal {B}}_{E, g}) = g - 2 + f - f_E \leq g - 2 + f$ . (More precisely, $\dim W \geq g - 2 + f$ , and Lemmas 3.10 and 3.11 imply that $f_E = 0$ in this case and that W could only possibly be the closure of the locus consisting of isomorphism classes of double covers of curves of the form $(E, O_E) \cup _{O_E = \infty } (\mathbb {P}^1, \infty )$ , ramified above $2g - 2$ distinct points $P_i \in \mathbb {P}^1 - \{\infty \}$ , $1 \leq i \leq 2g - 2$ , and unramified otherwise, as in Corollary 3.14.)

In both cases, we conclude that $\dim W \leq g - 2 + f$ , which implies $\dim W = g - 2 + f$ , as desired. Consequently, $V_f(\overline {\mathcal {B}}_g)$ is pure of dimension $g - 2 + f$ .

Proof. These observations follow immediately from the construction in (7), combined with [Reference Kani and Rosen13, Theorem C] and the fact that $\mathrm {Jac}(D) \sim \mathrm {Jac}(D)_{\text {new}} \oplus \mathrm {Jac}(E)$ .

Proof of Proposition 5.1.

We use the fact that in characteristic $3$ , $y^2=x^4+ax^3+bx^2+cx+d$ is a supersingular elliptic curve if and only if $b=0$ and $x^4+ax^3+cx+d=0$ has four distinct roots. This is because we need four distinct roots to ensure that the curve is smooth and then the Hasse-invariant is b.

We write $i\in \mathbb {F}_9$ for any of the roots of $x^2=-1$ and consider

$$\begin{align*}E:y^2=x(x-i-2)(x-2i-2)(x-1)=x^4+x^3+x.\end{align*}$$

Note that over an algebraically closed field of characteristic $3$ , there is up to isomorphism a unique supersingular elliptic curve, so, without loss of generality, we can assume that E is given by the above equation. Let D be the normalization of $H \times _{\mathbb {P}^1} E$ , where

$$ \begin{align*} H: z^2&=x(x-i-2)(x-2i)(x-i-1)=x^4+2ix^3+ix. \end{align*} $$

From (8), we see that $\mathrm {Jac}(D) \sim \mathrm {Jac}(E) \oplus \mathrm {Jac}(H) \oplus \mathrm { Jac}(H')$ , where

$$ \begin{align*} &H: &z^2=x(x-i-2)(x-2i)(x-i-1)&=x^4+2ix^3+ix,\\ &H': &w^2=(x-2i-2)(x-1)(x-2i)(x-i-1)&=x^4+(i+2)x^3+(2i+2)x+1. \end{align*} $$

We see that E, $H,$ and $H'$ are supersingular and $g_{D}=3$ . Thus, D is a genus- $3$ double cover of E and the $3$ -rank of D is $0$ .

Proof. For $H:z^2=x^n-t^n$ , its genus is $g_H=\lceil \frac {n}{2}\rceil -1$ . Let M be the Hasse–Witt matrix of H. Note that H is ordinary if and only if the p-rank of D is $g_H$ , which is if and only if the rank of M is $g_H$ .

It is known that $H^1(H,\mathcal {O}_{H})$ has basis given by $\frac {z}{x^k}$ for $1\leq k\leq g_H$ . Note that $\mathbb {Z}/n\mathbb {Z}$ acts on H, where $1 \in \mathbb {Z}/n\mathbb {Z}$ acts by sending $(x,z)$ to $(\zeta _n x,z)$ . Since $p \nmid n$ , this action leads to the decomposition

$$ \begin{align*}H^1(H,\mathcal{O}_{H})=\bigoplus_{i=0}^{n-1}H^1(H,\mathcal{O}_{H})_i.\end{align*} $$

Here, $H^1(H,\mathcal {O}_{H})_i$ is the eigenspace where $1 \in \mathbb {Z}/n\mathbb {Z}$ acts via scalar multiplication by $\zeta _n^i$ .

By the computation in [Reference Bouw6, Section 5],

$$ \begin{align*}H^1(H,\mathcal{O}_{H})_i=\begin{cases} \left\langle \frac{z}{x^{n-i}} \right\rangle & \text{if } n-g_H\leq i\leq n-1 \\ 0 & \text{if } 0\leq i\leq n-g_H-1. \end{cases}\end{align*} $$

Moreover, M maps $H^1(H,\mathcal {O}_{H})_i$ to $H^1(H,\mathcal {O}_{H})_{pi\pmod {n}}$ . This means the rank of M equals $\#S_1$ , where $S_1$ is the following set:

$$\begin{align*}S_1 =\{n-g_H\leq i\leq n-1 \mid n-g_H\leq pi\quad\pmod{n}\leq n-1\}.\end{align*}$$

Clearly, it has the same size as

$$\begin{align*}S_2=\{1\leq i\leq g_H \mid 1\leq pi\quad\pmod{n}\leq g_H\}.\end{align*}$$

Note that H is ordinary if and only if $S_2=\{1,2,\dots ,g_H\}$ . Let a be the congruence class of p modulo n. The assumption $p \not \equiv 1 \pmod {n}$ implies $2\leq a\leq n-1$ . Depending on the value of a, we have the following cases:

1. 1. If $g_H+1\leq a\leq n-1$ : then $1\notin S_2$ , so H is not ordinary.

2. 2. If $2\leq a\leq g_H$ : let $i=\lceil \frac {g_H+1}{a}\rceil $ , so $1\leq i\leq g_H$ . Moreover, observe

   $$ \begin{align*}g_H+1\leq ai< g_H+1+a\leq 2g_H+1\leq n.\end{align*} $$

   This implies that $i\notin S_2$ , so H is not ordinary.

In both cases, H is not ordinary. In addition, if $p \equiv -1 \pmod {n}$ , then $S_2=\varnothing $ . Therefore, M has rank $0$ and hence the p-rank of H is also $0$ .

Proof. We first show that $\mathrm {Jac}(D_t)_{\text {new}}$ is not ordinary. Observe that $D_t$ has two hyperelliptic quotients:

$$\begin{align*}H_t: z^2=x^n-t^n \text{ and } H^{\prime}_t: w^2=x(x-1)(x-\lambda)(x^n-t^n).\end{align*}$$

Since $\mathrm {Jac}(D_t)_{\text {new}} \sim \mathrm {Jac}(H_t) \oplus \mathrm {Jac}(H^{\prime }_t)$ , it suffices to show that at least one of the factors is not ordinary. By Lemma 5.2, we have that $f_{H_t} \leq g_{H_t}-1$ , so $H_t$ is not ordinary, and hence $\mathrm {Jac}(D)_{\text {new}}$ is not ordinary. Furthermore, since $\mathrm {Jac}(D_t)\sim \mathrm { Jac}(D_t)_{\text {new}} \oplus \mathrm {Jac}(E)$ , we see that $f_{D_t} \leq f_E+g-2$ . Moreover, when $p \equiv -1 \pmod {n}$ , we obtain $f_{H_t}=0$ , so $f_{\mathrm {Jac}(D_t)_{\text {new}}} = f_{H_t'} \leq g_{H_t'}=\lceil \frac {n+1}{2} \rceil =\lceil \frac {g}{2} \rceil $ . Hence, $f_{D_t} \leq \lceil \frac {g}{2} \rceil +f_E \leq \lceil \frac {g}{2} \rceil +1$ .