The Torelli map restricted to the hyperelliptic locus

Abstract

Let $g \geq 2$ and let the Torelli map denote the map sending a genus $g$ curve to its principally polarized Jacobian. We show that the restriction of the Torelli map to the hyperelliptic locus is an immersion in characteristic not $2$. In characteristic $2$, we show the Torelli map restricted to the hyperelliptic locus fails to be an immersion because it is generically inseparable; moreover, the induced map on tangent spaces has kernel of dimension $g-2$ at every point.

1. Introduction

Let $\mathscr{H}_{g}$ over $\operatorname {Spec}\mathbb{Z}$ denote the moduli stack of smooth hyperelliptic curves of genus $g$, $\mathscr{M}_{g}$ over $\operatorname {Spec}\mathbb{Z}$ denote the moduli stack of smooth curves of genus $g$, and $\mathscr{A}_{g}$ over $\operatorname {Spec}\mathbb{Z}$ denote the moduli stack of principally polarized abelian varieties of dimension $g$. Throughout, for $S$ a scheme, by a curve of genus $g$ over $S$, we mean a smooth proper morphism of schemes $f: C \to S$ of relative dimension $1$ whose fibers are geometrically connected $1$-dimensional schemes of arithmetic genus $g$. For $R$ a ring, we use $\mathscr{H}_{g,R}, \mathscr{M}_{g,R}, \mathscr{A}_{g,R}$ to denote the base changes of $\mathscr{H}_{g}, \mathscr{M}_{g}$, and $\mathscr{A}_{g}$ over $\operatorname {Spec}\mathbb{Z}$ along $\operatorname {Spec}R \rightarrow \operatorname {Spec}\mathbb{Z}$. Throughout, we assume $g \geq 2$.

The main goal of this paper is to understand whether the composition $\mathscr{H}_{g} \xrightarrow {\iota _g} \mathscr{M}_{g} \xrightarrow {\tau _g} \mathscr{A}_{g}$ is an immersion, (i.e., a locally closed immersion,) for $\tau _g$ the Torelli map sending a curve to its principally polarized Jacobian. Let $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ denote this composition. We use $\tau _{g,R}$ and $\phi _{g,R}$ for the base changes of $\tau _g$ and $\phi _g$ along a map $\operatorname {Spec}R \rightarrow \operatorname {Spec}\mathbb{Z}$. To start, we recall the classical characterization of when $\tau _g$ is injective on tangent vectors. This follows from OS79, Theorems 2.6 and 2.7 together with the converse of OS79, Theorem 2.7, which is easy to verify directly, see Proposition 3.1. Also, see Lan19, Theorem 4.3.

Theorem 1.1 (OS79, Theorems 2.6 and 2.7).

Let $k$ be a field. For $g \geq 3$, and $[C] \in \mathscr{M}_{g,k}$ a field-valued point, the Torelli map $\tau _{g,k}: \mathscr{M}_{g,k} \rightarrow \mathscr{A}_{g,k}$ is injective on tangent vectors at $[C]$ if and only if $[C] \in \mathscr{M}_{g,k} - \mathscr{H}_{g,k} \subset \mathscr{M}_{g,k}$. When $g = 2$, the map $\tau _{g,k}$ is injective on tangent vectors at all points $[C] \in \mathscr{M}_{g,k}$.

Moreover, away from characteristic $2$, the precise scheme theoretic fiber of $\tau _g$ over geometric points corresponding to hyperelliptic Jacobians is computed in Ric20, p. 7. As a consequence, Theorem 1.1 shows that $\tau _g$ is not even a monomorphism at points of $\mathscr{H}_{g}$ when $g \geq 3$. It is therefore natural to ask whether the restriction $\tau _g|_{\mathscr{H}_{g}} = \phi _g : \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ is a monomorphism. Our main theorem answers this question. We say a morphism of algebraic stacks is a radimmersion if it factors as the composition of a finite radicial morphism and an open immersion, see Definition A.1.

Theorem 1.2.

For $g \geq 2$, the map $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ over $\operatorname {Spec}\mathbb{Z}$ is a radimmersion. Additionally, $\phi _2$ is an immersion and $\phi _{g, \mathbb{Z}[1/2]}: \mathscr{H}_{g,\mathbb{Z}[1/2]} \rightarrow \mathscr{A}_{g,\mathbb{Z}[1/2]}$ is an immersion. However, when $g > 2$, for $k$ a field of characteristic $2$, $\phi _{g, k}$ is not an immersion; instead, $\phi _{g, k}$ is generically inseparable and the induced map on tangent spaces at any geometric point of $\mathscr{H}_{g,k}$ has kernel of dimension $g-2$.

We carry out the proof of Theorem 1.2 at the end of § 6. To paraphrase the statement, Theorem 1.1 says, loosely speaking, that there are many tangent vectors to a hyperelliptic point in $\mathscr{M}_{g}$ that are killed under $\tau _g$. We wish to understand whether those tangent vectors which are killed can lie inside $\mathscr{H}_{g}$, or whether they correspond to deformations to non-hyperelliptic curves. The answer, provided by Theorem 1.2, is that they do all correspond to deformations to non-hyperelliptic curves when the characteristic is not $2$, but this fails quite badly in characteristic $2$.

Remark 1.3.

The statement that $\phi _{g,k}$ is an immersion for a field $k$ of characteristic not $2$ appears in OS79, p. 176, though some details are omitted. Our guess is that the authors verified the map $\phi _{g,k}$ is injective on tangent spaces via explicitly calculating the Kodaira Spencer map sending a differential to a corresponding deformation, in order to understand the image of the map $T_{[C]} \mathscr{H}_{g,k} \rightarrow T_{[C]} \mathscr{M}_{g,k} \simeq H^1(C, T_C)$ on tangent spaces at a point $[C] \in \mathscr{H}_{g,k}$. However, no indication is given there as to how to use this to show the map of stacks is an immersion. In this article we take a different approach, which also applies in characteristic $2$; we do not see how to carry out the approach of OS79 in characteristic $2$.

Remark 1.4.

Our initial interest in this problem was motivated by the computation of the essential dimension of level structure covers of $\mathscr{H}_{g}$. Using Theorem 1.2, it is shown in FKW19, Example 2.3.6 that for $g \geq 2$, the cover of $\mathscr{H}_{g}$ parameterizing pairs of a hyperelliptic curve and an $n$-torsion point of the Jacobian of that curve (over a field of characteristic $0$) has essential dimension $2g-1$ when $2 \nmid n$ but only has essential dimension $g + 1$ when $n = 2$.

There are two main components to the proof of Theorem 1.2. The first component of the proof is to describe the map $\phi _g$ induces on tangent vectors. This is done by analyzing the deformation theory of hyperelliptic curves, which is possible by means of their relatively simple equations. The key tool to analyzing the induced map on tangent spaces is Proposition 3.1, which relies on a nonstandard definition of $\mathscr{H}_{g}$ given in § 2.2. The second component of the proof is to very that $\phi _g$ is a radimmersion. This will imply $\phi _g$ is an immersion when it is a monomorphism, i.e., away from characteristic $2$. For checking $\phi _g$ is a radimmersion, we use a valuative criterion, which roughly says that a map of stacks $f: X \rightarrow Y$ is a radimmersion when, given a map from the spectrum a discrete valuation ring to $Y$ with its two points factoring through $X$, the map from the spectrum of the discrete valuation ring factors uniquely through $X$. We verify this valuative criterion using that $\phi _g$ factors as the composition of an immersion into the moduli stack of compact type curves $\mathscr{H}_{g} \rightarrow \mathscr{M}_{g}^{c}$, and a proper “compactified Torelli map” $\mathscr{M}_{g}^{c} \rightarrow \mathscr{A}_{g}$.

The outline of the paper is as follows: In § 2 we recall background on the infinitesimal Torelli theorem and the moduli stack of hyperelliptic curves. In § 3 we describe the map $\phi _g$ induces on tangent spaces. We then compute this map on tangent spaces when the characteristic is not $2$ in § 4 and when the characteristic is $2$ in § 5. After some preliminaries on the compactified Torelli map, we prove Theorem 1.2 in § 6. Finally, in Appendix A, we prove a valuative criterion for immersions of stacks, which is used in the proof of Theorem 1.2.

2. Background

In this section, we review relevant background notation we will need from Theorem 1.1 in § 2.1 and also a nonstandard construction of the moduli stack of hyperelliptic curves which will be crucial to the ensuing proof. We define the stack $\mathscr{H}_{g}$ of genus $g$ hyperelliptic curves in § 2.2. We show $\mathscr{H}_{g}$ is a smooth algebraic stack in § 2.3. Finally, in § 2.4, we show $\mathscr{H}_{g}$ has a closed immersion into $\mathscr{M}_{g}$.

2.1. Key inputs in the proof of the infinitesimal Torelli theorem

We next review the key inputs in the proof of Theorem 1.1, as we will rely on understanding explicitly the map on tangent spaces associated to $\tau _g: \mathscr{M}_{g} \rightarrow \mathscr{A}_{g}$ in our ensuing analysis of the map $\mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$.

The statement regarding injectivity on geometric points is the classical Torelli theorem Cor86, Chapter VII, Theorem 12.1(a), see also the original proof by Torelli Tor13 and Andreotti’s beautiful proof And58. Thus, we just address the statement on tangent vectors. This too is classical, and boils down to Noether’s theorem, regarding the map 2.2 below, though is perhaps less well known.

Let $k$ be a field and let $[C] \in \mathscr{M}_{g,k}$ be a field valued point of $\mathscr{M}_{g,k}$ corresponding to the curve $C$. We’d like to understand whether the map

$$\begin{align*} T_{[C]} \mathscr{M}_{g,k} \rightarrow T_{[\tau _{g,k}(C)]} \mathscr{A}_{g,k} \end{align*}$$

is injective, for $T_{[C]} \mathscr{M}_{g,k}$ denoting the tangent space to $\mathscr{M}_{g,k}$ at $[C]$. By deformation theory, $T_{[C]} \mathscr{M}_{g,k} \simeq H^1(C, T_C)$. For $V$ a vector space, define $\operatorname {Sym_2}V$ as the kernel of $V^{\otimes 2} \rightarrow \wedge ^2 V$, where $\wedge ^2 V = V^{\otimes 2}/ \operatorname {Span}(v\otimes v : v \in V)$. (Note that $\operatorname {Sym_2}V \simeq \operatorname {Sym}^2 V$ in characteristic $p \neq 2$, but differs in characteristic $2$. Here $\operatorname {Sym}^2 V$ denotes the natural quotient of $V \otimes V$ by the span of $v \otimes w - w \otimes v$ for $v, w \in V$.) Further,

$$\begin{align*} T_{[\tau _{g}(C)]}\mathscr{A}_{g,k} &= \ker (H^1(\tau _g(C), \mathscr{O}_{\tau _g(C)})^{\otimes 2} \rightarrow \wedge ^2 H^1(\tau _g(C), \mathscr{O}_{\tau _g(C)}) ) \\ &\simeq \operatorname {Sym_2}H^1(C, \mathscr{O}_C). \end{align*}$$

as described in the proof of OS79, Theorem 2.6. Also see Ser06, Theorem 3.3.11(iii) for this identification.

Therefore, we wish to understand whether the natural map

$$\begin{align} H^1(C, T_C) \rightarrow \operatorname {Sym_2}H^1(C, \mathscr{O}_C), {\tag{2.1}} \end{align}$$

induced by $T_{[C]}\tau _{g,k}: T_{[C]} \mathscr{M}_{g,k} \rightarrow T_{[\tau _g(C)]}\mathscr{A}_{g,k}$, is injective.

Applying Serre duality, since $H^1(C, T_C)$ is dual to $H^0(C, \omega _C^{\otimes 2})$, it is equivalent to understand surjectivity of the corresponding map

$$\begin{align} H^0(C, \omega _C^{\otimes 2}) \leftarrow \operatorname {Sym}^2 H^0(C, \omega _C). {\tag{2.2}} \end{align}$$

This duality, valid even in characteristic $2$, uses that, for $V$ a finite dimensional vector space, $\operatorname {Sym_2}V^\vee$ can be viewed as the second graded piece of the algebra $\operatorname {Sym}_\bullet V^\vee$ with its divided power structure that is naturally dual to $\operatorname {Sym}^\bullet V$. The map 2.2 is explicitly the map given by multiplying two sections, see Lan19, Theorem 4.3 and OS79, Theorem 2.6. By Noether’s theorem SD73, Theorem 2.10 the map 2.2 is surjective when $C$ is not hyperelliptic and fails to be surjective when $C$ is hyperelliptic. See § 4.1 and § 5.2 for an explicit description of the image of 2.2 in the hyperelliptic case.

2.2. Definition of $\mathscr{H}_{g}$

There are several different definitions of $\mathscr{H}_{g}$, the moduli stack of hyperelliptic curves of genus $g$, in the literature. For the purposes of this paper, we will be especially concerned with the more delicate case when $2$ is not invertible on the base, so let us now expand a bit on the definition of $\mathscr{H}_{g}$ over $\operatorname {Spec}\mathbb{Z}$ we employ. We will essentially define $\mathscr{H}_{g}$ as the Hurwitz stack of degree $2$ covers of a genus $0$ curve. We assume $g \geq 2$. For the next definition, recall that a map $\phi : X \to Y$ is locally free of degree $d$ if $\phi _* \mathscr{O}_X$ is a locally free rank $d$ sheaf on $Y$, or equivalently $\phi$ is a degree $2$ finite map which is flat and of finite presentation Sta, Tag 02KB.

Definition 2.1.

Suppose $g \in \mathbb{Z}$ and $g \geq 2$. Define the $\mathscr{H}_{g}$, the stack of hyperelliptic genus $g$ curves as the category fibered in groupoids over schemes, whose fiber over a scheme $B$ corresponds tuples $(B,C,f,P,h)$ where $C$ is a smooth proper curve of genus $g$ over $B$ with geometrically connected fibers, $h: P \to B$ is a smooth proper curve of genus $0$ over $B$ with geometrically connected fibers, and $f: C \to P$ is a degree $2$ locally free morphism. Morphisms $(B,C,f,P,h) \to (B',C',f',P',h')$ are morphisms $t: B \to B', r: C \to C', s: P \to P'$ making all squares in the diagram

$$\begin{equation} \vcenter{\img[][57pt][78pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} C \ar{r}{r} \ar{d}{f} & C' \ar{d}{f'} \\ P \ar{r}{s} \ar{d}{h} & P' \ar{d}{h'} \\ B \ar{r}{t} & B' \end{tikzcd}}]{Images/img101a856b9441d9302c1bec2802fee77f.svg}}\tag{2.3} \end{equation}$$

fiber squares.

Note that $\mathscr{H}_{g}$ as defined above is indeed a stack because the $\operatorname {isom}$ presheaf between any two points is a sheaf and descent data for $(B,C,f,P,h)$ is effective; the effectivity of descent data holds because descent data for $C$ and $(P,h)$ are separately effective, as is descent data for the morphism $f$.

Remark 2.2.

We note that hyperelliptic curves over fields as defined in Definition 2.1 may fail to have a map to $\mathbb{P}^1$, as they may have a map to a genus $0$ curve which is not isomorphic to $\mathbb{P}^1$. However, over an algebraically closed field, any hyperelliptic curve has a map to $\mathbb{P}^1$.

2.3. Showing $\mathscr{H}_{g}$ is an algebraic stack

To show $\mathscr{H}_{g}$ is an algebraic stack, we will construct a smooth cover by a scheme. This scheme will be an open subscheme of a certain linear system on a Hirzebruch surface, which we now define.

Notation 2.3.

For $n \in \mathbb{Z}_{\geq 0}$, let $\mathbb{F}_n \coloneq \mathbb{P}\left( \mathscr{O}_{\mathbb{P}^1_\mathbb{Z}}(-n) \oplus \mathscr{O}_{\mathbb{P}^1_\mathbb{Z}} \right)$ denote the $n$th Hirzebruch surface over $\operatorname {Spec}\mathbb{Z}$. Let $E \subset \mathbb{F}_n$ denote the “directrix” section of $\mathbb{F}_n \to \mathbb{P}^1$ corresponding to the surjection $\mathscr{O}_{\mathbb{P}^1_\mathbb{Z}}(-n) \oplus \mathscr{O}_{\mathbb{P}^1_\mathbb{Z}} \to \mathscr{O}_{\mathbb{P}^1_\mathbb{Z}}(-n)$ and let $F$ denote a fiber of the map $\mathbb{F}_n \to \mathbb{P}^1$. Letting $\underline{G}$ denote the constant group scheme associated to a group $G$, we claim there is an isomorphism $\underline{\mathbb{Z} \oplus \mathbb{Z}} \to \operatorname {Pic}_{\mathbb{F}_n/\mathbb{Z}}$ sending $(1,0) \mapsto \mathscr{O}(F)$ and $(0,1) \mapsto \mathscr{O}(E)$. Indeed, this is an isomorphism by the fibral isomorphism criterion Gro67, 17.9.5 because it induces an isomorphism over every geometric point of $\operatorname {Spec}\mathbb{Z}$. The intersection pairing on $\mathbb{F}_n$ satisfies $E \cdot E = -n, E \cdot F = 1, F \cdot F = 0$, see Bea96, Proposition IV.1.

To start, we show that every hyperelliptic curve of genus $g$ has a canonical immersion into a Hirzebruch surface, and lies in a particular linear system. This will allow us to check that the schematic cover of $\mathscr{H}_{g}$ we construct maps surjectively to $\mathscr{H}_{g}$.

Lemma 2.4.

Any hyperelliptic curve $C$ of genus $g \geq 2$ over an algebraically closed field $k$ is a closed subscheme of $\mathbb{F}_{g+1}$ in the linear system $\mathscr{O}(2E + (2g+2)F)$. Further, any curve in the linear system associated to $\mathscr{O}(2E + (2g+2)F)$ has genus $g$.

Proof.

Given a map $f: C \to \mathbb{P}^1$ over $k$ the surjective adjunction map $f^* f_* \mathscr{O}_C \to \mathscr{O}_C$ induces an map $C \to \mathbb{P} \left( f_* \mathscr{O}_C \right)$. This map is an immersion, as can be verified on fibers over $\mathbb{P}^1$. We claim $f_* \mathscr{O}_C \simeq \mathscr{O}_{\mathbb{P}^1}(-g-1) \oplus \mathscr{O}_{\mathbb{P}^1}$. We know $f_* \mathscr{O}_C$ splits as a direct sum of line bundles because is a locally free sheaf on $\mathbb{P}^1$. Because $1 = h^0(C, \mathscr{O}_C) = h^0(\mathbb{P}^1, f_* \mathscr{O}_C)$, one of the summands of $f_* \mathscr{O}_C$ must be $\mathscr{O}_{\mathbb{P}^1}$. To compute the other summand, we find the degree of $f_* \mathscr{O}_C$. For sufficiently large $m$, we have

$$\begin{align*} h^0(\mathbb{P}^1, (f_* \mathscr{O}_C)(m)) = h^0(\mathscr{O}_C \otimes f^* \mathscr{O}_{\mathbb{P}^1}(m)) = h^0(f^* \mathscr{O}_{\mathbb{P}^1}(m)) = 2m-g+1. \end{align*}$$

If $f_* \mathscr{O}_C \simeq \mathscr{O}_{\mathbb{P}^1}(-n) \oplus \mathscr{O}_{\mathbb{P}^1}$ then we find

$$\begin{align*} h^0(\mathbb{P}^1, (f_* \mathscr{O}_C)(m)) = (m+1) + (m+1-n) = 2m+2-n \end{align*}$$

and therefore $2-n = -g + 1$ and $n = g + 1$ as claimed.

To conclude, it remains to check $C$ lies in the linear system $\mathscr{O}(2E + (g+1)F)$. Write $C = \alpha E + \beta F$. Since $C \to \mathbb{P}^1$ has degree $2$, the intersection $C \cdot E$ has degree $2$, which implies $\alpha = 2$. Since $C$ has genus $g$ and the canonical divisor of $\mathbb{F}_{g+1}$ is $-2E - (g+1+2)F$ adjunction implies $2g-2 = ((-2E - (g+1+2)F) + (2E + \beta F)) \cdot (2E + \beta F)$. Solving for $\beta$ yields $\beta = 2g + 2$, as we wanted. The final statement that any curve of class $\mathscr{O}(2E + (2g+2)F)$ has genus $g$ can be deduced directly from Riemann-Roch. Alternatively, one may deduce it from the fact that hyperelliptic curves of genus $g$ exist and that the genus is constant in flat connected families.

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The above lemma will allow us to show that a certain linear space of sections on a Hirzebruch surface is a cover of the stack $\mathscr{H}_{g}$. Let $G \coloneq \operatorname {Aut}_{\mathbb{F}_n/\mathbb{Z}}$ denote the automorphisms group scheme of the Hirzebruch surface $\mathbb{F}_n$ over $\operatorname {Spec}\mathbb{Z}$. The cover of $\mathscr{H}_{g}$ we construct will be a $G$ torsor, and so to show $\mathscr{H}_{g}$ is an algebraic stack (so that it has a smooth cover by a scheme) we will need to know $G$ is smooth.

Lemma 2.5.

For any $n \geq 1$ $\operatorname {Aut}_{\mathbb{F}_n/\mathbb{Z}}$ is isomorphic to a certain semi-direct product $\mathbb{G}_a^{n+1} \rtimes (\operatorname {GL}_2/\mu _n)$. In particular, it is smooth and connected of relative dimension $n+5$.

Proof.

The plan is to construct a map from the smooth group scheme $\mathbb{G}_a^{n+1} \rtimes (\operatorname {GL}_2/\mu _n)$ to $\operatorname {Aut}_{\mathbb{F}_n/\mathbb{Z}}$, which we will verify to be an isomorphism. We now describe the semidirect product structure in terms of a group action on $H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n))$. Choose $x,y$ as a basis for $H^0(\mathbb{P}^1, \mathscr{O}(1))$ and represent elements of $H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n))$ by pairs $(r, z(x,y))$ for $z$ a degree $n$ polynomial. Then $g \in \operatorname {GL}_2$ sends $(r,z(x,y)) \mapsto (r, z(gx,gy))$ and $(\alpha _0, \ldots , \alpha _n) \!\in \! \mathbb{G}_a^{n+1}$ sends $(r,z(x,y)) \!\mapsto \! (r, z(x,y) + \sum _{i=0}^n \alpha _i x^iy^{n-i})$. The action of $\operatorname {GL}_2$ has kernel $\mu _n \subset \operatorname {GL}_2$. Now, because the pushforward of $\mathscr{O}_{\mathbb{F}_{n}}(E + nF)$ along $\mathbb{F}_{n} \to \mathbb{P}^1$ is $\mathscr{O} \oplus \mathscr{O}(n)$ the complete linear system associated to $nF + E$ induces a map $\mathbb{F}_n \to \mathbb{P} H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n))$. This contracts the directrix $E$ but is an immersion on the open complement of $E$. We claim the action of $\mathbb{G}_a^{n+1} \rtimes (\operatorname {GL}_2/\mu _n)$ fixes the image $Y$ of $\mathbb{F}_{n}$ in $\mathbb{P} H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n))$. First we check the action of $\operatorname {GL}_2$ fixes the image. The action fixes both the point $p$ corresponding to the quotient $H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n)) \to H^0(\mathbb{P}^1, \mathscr{O})$ and the rational normal curve $R$ lying in the hyperplane associated to the quotient $H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n)) \to H^0(\mathbb{P}^1, \mathscr{O}(n))$. As $Y$ is the cone over $R$ with cone point $p$, any such automorphism must preserve $Y$. The action of $\mathbb{G}_a^{n+1}$ fixes $p$ and sends $R$ to another curve on the cone $Y$ not meeting $p$. It therefore preserves $Y$ as well. We obtain the induced map $\theta : \mathbb{G}_a^{n+1} \rtimes (\operatorname {GL}_2/\mu _n) \to \operatorname {Aut}_{\mathbb{F}_n/\mathbb{Z}}$ via the identification $\operatorname {Aut}_{Y/\mathbb{Z}} \simeq \operatorname {Aut}_{\mathbb{F}_n/\mathbb{Z}}$ from blowing up $p$.

It remains to check this map is an isomorphism. By the fibral isomorphism criterion Gro67, 17.9.5, since $\mathbb{G}_a^{n+1} \rtimes (\operatorname {GL}_2/\mu _n)$ is flat over $\operatorname {Spec}\mathbb{Z}$, in order to check $\theta : \mathbb{G}_a^{n+1} \rtimes (\operatorname {GL}_2/\mu _n) \to \operatorname {Aut}_{\mathbb{F}_{n}/\mathbb{Z}}$ is an isomorphism, it suffices to do so on geometric fibers. We now check the restriction of $\theta$ to $\operatorname {Spec}k$, for $k$ an algebraically closed field, is an isomorphism. This may be checked using the above description of $G$ acting on $Y$, the image of $\mathbb{F}_n \to \mathbb{P} H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n))$ induced by $\mathscr{O}_{\mathbb{F}_{n}}(nF + E)$. As mentioned above, we can identify $\operatorname {Aut}_{Y/k} \simeq \operatorname {Aut}_{\mathbb{F}_n/k}$. Every automorphism of $Y$ must preserve the point $p$. Therefore, every automorphism of $\mathbb{F}_n$ preserves $E$ and we obtain a map $\operatorname {Aut}_{Y/k} \simeq \operatorname {Aut}_{\mathbb{F}_n/k} \to \operatorname {Aut}_{E/k} = \operatorname {PGL}_2$. Using that every automorphism of $Y$ is induced by one of $\mathbb{P} H^0(\mathbb{P}^1, \mathscr{O} \oplus \mathscr{O}(n))$ fixing $Y$, this map is surjective, and the kernel can be identified with $\mathbb{G}_a^{n+2} \times \mathbb{G}_m$ by direct calculation. The latter factor corresponds to the central torus in $\operatorname {GL}_2/\mu _n$. This verifies that $\theta$ is an isomorphism.

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To state the next proposition proving that $\mathscr{H}_{g}$ is a smooth algebraic stack, we now introduce a smooth scheme with a map to $\mathscr{H}_{g}$. Let $\pi : \mathbb{F}_{g+1} \to \operatorname {Spec}\mathbb{Z}$ denote the projection and define $U \subset \mathbb{P} \left(\pi _*\mathscr{O}_{\mathbb{F}_{g+1}}(2E+(2g+2)F) \right)$ over $\operatorname {Spec}\mathbb{Z}$ as the open subscheme parameterizing smooth curves in the linear system $\pi _*\mathscr{O}_{\mathbb{F}_{g+1}}(2E+(2g+2)F$ with universal family $C \to \mathbb{P}^1_U \to U$. The family $C \to \mathbb{P}^1_U \to U$ is equivariant for the action of $G \coloneq \operatorname {Aut}_{\mathbb{F}_{g+1}/\mathbb{Z}}$ and descends to a map of stacks $[C/G] \to [\mathbb{P}^1_U/G] \to [U/G]$ inducing a map $[U/G] \to \mathscr{H}_{g}$.

Proposition 2.6.

For $g \geq 2$, the above constructed map $[U/G] \to \mathscr{H}_{g}$ is an equivalence of stacks. Further, $U \to \mathscr{H}_{g}$ is a smooth cover and hence $\mathscr{H}_{g}$ is a smooth integral algebraic stack of relative dimension $2g - 1$ over $\operatorname {Spec}\mathbb{Z}$.

Proof.

Note that the final statement follows from the first by Lemma 2.5 because $U \to [U/G]$ is a smooth cover, and $[U/G]$ is an algebraic stack.

Hence, it suffices to show $[U/G] \to \mathscr{H}_{g}$ is an equivalence of stacks. In the statement of Proposition 2.6 we have constructed a map $[U/G] \to \mathscr{H}_{g}$ and we now construct an inverse map. In order to construct a map $\mathscr{H}_{g} \to [U/G]$, given any $B \to \mathscr{H}_{g}$, we wish to construct a $G$ torsor over $B$ with a map to $U$. The map $B \to \mathscr{H}_{g}$ yields a family $C \xrightarrow {f} P \xrightarrow {h} B$ and we have an immersion $C \to \mathbb{P}(f_*\mathscr{O}_C)$ induced by the surjective adjunction map $f^* f_* \mathscr{O}_C \to \mathscr{O}_C$. Then, consider the scheme $I \coloneq \operatorname {isom}(\mathbb{F}_{g+1}, \mathbb{P}(f_* \mathscr{O}_C)) \to B$. To construct a map $I \to U$, it is enough to produce a family of smooth curves in the linear system $\mathscr{O}(2E + (2g+2)F)$ in $(\mathbb{F}_{g+1})_I$. This is provided by the pullback of $C \to \mathbb{P}(f_* \mathscr{O}_C) \to B$ along $I \to B$, using the isomorphism $(\mathbb{F}_{g+1})_I \simeq (\mathbb{P}(f_* \mathscr{O}_C))_I$ coming from the universal property of $I$. We next claim $I$ is a $G$-torsor over $B$. It certainly has a $G$ action via the action of $G$ on $\mathbb{F}_{g+1}$, and it is straightforward to verify the map $G \times _{\operatorname {Spec}\mathbb{Z}} I \to I \times _B I, (g,x) \mapsto (x,gx)$ is an isomorphism. The cover $I \to B$ is therefore smooth by Lemma 2.5, and it is surjective by Lemma 2.4.

To conclude, it is enough to show that this map $\mathscr{H}_{g} \to [U/G]$ defines an inverse equivalence to the map $[U/G] \to \mathscr{H}_{g}$ from the statement. On objects, the composition $\mathscr{H}_{g} \to [U/G] \to \mathscr{H}_{g}$ constructs the immersion into a relative Hirzebruch surface, and then forgets it, so is equivalent to the identity. Further, the composition $[U/G] \to \mathscr{H}_{g} \to [U/G]$ is also equivalent to the identity because the immersion into a Hirzebruch surface is uniquely determined by the hyperelliptic curve, and any automorphism of the hyperelliptic curve induces a unique automorphism of the relative Hirzebruch surface.

The final statement about the dimension holds because $G$ has relative dimension $g + 6$, using Lemma 2.5 and $U$ has relative dimension $3g + 5$ by Lemma 2.7 below. Therefore, $\dim \mathscr{H}_{g} = (3g+5) - (g+6) = 2g - 1$.

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The following standard calculation on the dimension of a linear system was needed above to compute the dimension of $\mathscr{H}_{g}$.

Lemma 2.7.

For $k$ a field and $g \geq 0$, $\mathbb{P} H^0( \mathbb{F}_{g+1}, \mathscr{O}_{\mathbb{F}_{g+1}}(2E + (2g+2)F))$ on $\mathbb{F}_{g+1}$ over $k$ has dimension $3g + 5$.

Proof.

Let $C$ be a smooth curve in the linear system $2E + (2g + 2)F$. We obtain an ideal sheaf exact sequence

$$\begin{equation} \vcenter{\img[][234pt][13pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} 0 \ar{r} & \mathscr{O}_{\mathbb F_{g+1}} \ar{r} & \mathscr{O}_{\mathbb F_{g+1}}(C) \ar{r} & \mathscr{O}_C(C) \ar{r} & 0. \end{tikzcd}}]{Images/imge3ce6f0873e68424a71c99f658d98029.svg}}{\tag{2.4}} \end{equation}$$

Note that $H^1(\mathbb{F}_{g+1}, \mathscr{O}_{\mathbb{F}_{g+1}})= 0$ as can be computed by the Leray spectral sequence associated to the composition $\mathbb{F}_{g+1} \xrightarrow {f} \mathbb{P}^1 \to \operatorname {Spec}k$ because $R^1 f_* \mathscr{O}_{\mathbb{F}_{g+1}} = 0$ and $H^1(\mathbb{P}^1, \mathscr{O}_{\mathbb{P}^1}) = 0$. Therefore, upon taking cohomology of 2.4, we get an exact sequence

$$\begin{equation} \vcenter{\img[][364pt][14pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} 0 \ar{r} & H^0(\mathbb F_{g+1}, \mathscr{O}_{\mathbb F_{g+1}}) \ar{r} & H^0(\mathbb F_{g+1}, \mathscr{O}_{\mathbb F_{g+1}}(C)) \ar{r} & H^0(\mathbb F_{g+1}, \mathscr{O}_C(C)) \ar{r} & 0. \end{tikzcd}}]{Images/img0a1eac729713436abc4850f49562fd4c.svg}}\tag{2.5} \end{equation}$$

We are aiming to show $h^0(\mathbb{F}_{g+1}, \mathscr{O}_{\mathbb{F}_{g+1}}(C)) = 3g + 6$, which has dimension $1$ more than the projectivization in the statement of the lemma. Since $h^0(\mathbb{F}_{g+1}, \mathscr{O}_{\mathbb{F}_{g+1}}) = 1$, it is enough to show $h^0(\mathbb{F}_{g+1}, \mathscr{O}_C(C)) = 3g +5$. We have

$$\begin{align*} \deg \mathscr{O}_C(C) \!=\! (2E \!+\! (2g+2)F) \cdot (2E + (2g+2)F) = -4(g+1) \!+\! 4(2g+2) \!=\! 4g + 4. \end{align*}$$

Since $C$ has genus $g$ by Lemma 2.4, it follows from Riemann Roch that $h^0(\mathbb{F}_{g+1}, \mathscr{O}_C(C)) = 4g + 4 - g + 1 = 3g + 5$.

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2.4. Showing $\mathscr{H}_g$ has a closed immersion into $\mathscr{M}_g$

We next check the natural map $\mathscr{H}_g \rightarrow \mathscr{M}_g$ is a closed immersion. The following general lemma will be useful.

Lemma 2.8.

Suppose $f: X \to Y$ is a proper morphism of algebraic stacks with diagonals that are separated and of finite type. Assume $f$ induces a bijection on isotropy subgroups at every geometric point of $X$, is injective on geometric points, and injective on tangent vectors at every geometric point. Then $f$ is a closed immersion.

Proof.

Let $Y' \to Y$ be a smooth cover by a scheme, let $X' \coloneq X \times _Y Y'$, and let $f' : X' \to Y'$ denote the base change of $f$. It is enough to check $f'$ is a closed immersion. Because $f$ is an injection on isotropy groups, and $X$ and $Y$ have separated finite type diagonals, $f$ is representable by Con07, Theorem 2.2.5(1). Thus, $X'$ is an algebraic space. Note that $X'$ is in fact a scheme because $f'$ is quasi-affine by a version of Zariski’s main theorem Sta, Tag 082J. Further, because $f$ is a surjection on isotropy groups and $f$ is injective on geometric points, $f'$ is also injective on geometric points. We find that $f'$ is proper, injective on geometric points, and injective on tangent vectors, hence a closed immersion Gro67, 18.12.6.

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Lemma 2.9.

For $g \geq 2$, the natural map $\mathscr{H}_g \to \mathscr{M}_g$ sending $(B,C,f,P,h)$ over a scheme $B$ to the curve $C$ over $B$ is a closed immersion.

Proof.

The idea is to verify the conditions of Lemma 2.8. To check that $\mathscr{H}_g \rightarrow \mathscr{M}_g$ is a closed immersion, we first verify that the map induces bijections on isotropy group spaces at each geometric point. In particular, this will imply the map is representable. As a preliminary step, we claim given a hyperelliptic curve $C$ of genus $g \geq 2$ over an algebraically closed field $k$ and an automorphism $r: C \to C$, there is a unique map $f: C \to \mathbb{P}^1$, up to automorphism of $\mathbb{P}^1$. To see this, it is enough to verify that for any degree $2$ map $C \to \mathbb{P}^1$, the composition with the $(g-1)$-Veronese $\mathbb{P}^1 \to \mathbb{P}^{g-1}$ realizes the map $C \to \mathbb{P}(H^0(C, \omega _C)) \simeq \mathbb{P}^{g-1}$ associated to the invertible sheaf $\omega _C$. This holds because $\omega _C$ is the unique invertible sheaf on $C$ of degree $2g-2$ with a $g$-dimensional space of global sections. Fixing such a map $C \to \mathbb{P}^1$, we find that $\mathbb{P}^1$ is the quotient of $C$ by the hyperelliptic involution as a scheme. Therefore, by the universal property of quotients, there is a unique automorphism $s: \mathbb{P}^1 \to \mathbb{P}^1$ over $k$ so that $f \circ r = s \circ f$. This shows the map on isotropy groups is surjective, and has kernel supported on a single geometric point. To show the kernel is trivial, it is enough to show the kernel is reduced over $k$. The same argument as above applied over the dual numbers in place of $k$ shows the kernel is indeed reduced.

Next, we check $\mathscr{H}_g \to \mathscr{M}_g$ is injective on geometric points and tangent vectors at each geometric point. The fiber over an map $\operatorname {Spec}k \to \mathscr{M}_g$ for $k$ an algebraically closed field, corresponding to a curve $C$ over $\operatorname {Spec}k$ is identified with the scheme $W$ parameterizing degree $2$ line bundles on $C$ which have a $2$-dimensional space of global sections, up to isomorphism. First, $W$ is either empty or set theoretically a single point, because given any curve $C$ of genus at least $2$ over an algebraically closed field, there is at most one degree $2$ map to $\mathbb{P}^1$, up to automorphisms of $\mathbb{P}^1$. It remains to check $W$ is reduced in the case it is nonempty. The tangent space to the unique point of $W$ corresponding to the line bundle $\mathscr{L}$ on $C$ can be identified with the cokernel of the multiplication map $\mu _0 : H^0(C, \mathscr{L}) \otimes H^0(C, \omega _C \otimes \mathscr{L}^\vee ) \to H^0(C, \omega )$ ACGH85, Proposition 4.2(i), so it is enough to verify $\mu _0$ is surjective. We verify this standard calculation below in Proposition 3.1.

To conclude, by Lemma 2.8, it suffices to check $\mathscr{H}_g \rightarrow \mathscr{M}_g$ is proper. For this, we use the valuative criterion for properness. Let $R$ be a discrete valuation ring, let $s$ denote the closed point of $\operatorname {Spec}R$ and let $\eta$ denote the generic point. We begin with a curve $h: C \to \operatorname {Spec}R$ so that the generic fiber has a degree $2$ map to a genus $0$ curve $P$. It is enough to show there is some extension of $R$ on which $C$ factors through a degree $2$ map to $\mathbb{P}^1_R$. By replacing $R$ with a suitable extension, we may assume we have a factorization $C_\eta \to \mathbb{P}^1_\eta \to \eta$. Therefore, it is enough to show that given $C \to \operatorname {Spec}R$ with $C_\eta \to \mathbb{P}^1_\eta$ there is a unique extension of this map to a map $C \to \mathbb{P}^1_R \to \operatorname {Spec}R$. The pullback of $\mathscr{O}_{\mathbb{P}^1_\eta }(1)$ to $C_\eta$ gives a degree $2$ invertible sheaf on $C_\eta$ which extends uniquely to a degree $2$ invertible sheaf $\mathscr{L}$ on $H$, for example by properness of $\operatorname {Pic}_{C/R}$. Riemann Roch (using $g \geq 2)$ and upper semicontinuity of cohomology together imply $h^0(C_s, \mathscr{L}_s) = 2$. Therefore, Grauert’s theorem tells us $h_* \mathscr{L}$ is locally free, hence free, of rank $2$. Then, up to elements of $R^\times$, there is then a unique choice of basis for $h_* \mathscr{L}$ inducing a map $C \to \mathbb{P}^1_R$ compatible with the given map $C_\eta \to \mathbb{P}^1_\eta$, as we wished to show.

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3. The general setup for checking injectivity on tangent vectors

Let $k$ be a field and let $C$ be a hyperelliptic curve over $k$. To understand whether the map $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ is injective on tangent vectors, we want to understand the composition

$$\begin{align*} T_{[C]} \mathscr{H}_{g,k} \xrightarrow {T_{[C]} \iota _{g,k}} T_{[C]} \mathscr{M}_{g,k} \xrightarrow {T_{[C]}\tau _{g,k}} T_{[\tau _g(C)]} \mathscr{A}_{g,k}. \end{align*}$$

We have already explicitly described $T_{[C]}\tau _{g,k}$ by identifying it as dual to 2.2 (see also § 4.1 and § 5.2 below for explicit descriptions of 2.2 in terms of differentials) so we next want to understand the image of $T_{[C]}\iota _{g,k}$. Following ACG11, Chapter 21, §5-§6 we can identify $T_{[C]}\iota _{g,k}$ as follows.

Let $C$ be a hyperelliptic curve as above and $\mathscr{L}$ the unique isomorphism class of invertible sheaf giving rise to a hyperelliptic map $C \rightarrow \mathbb{P}^1$. This $\mathscr{L}$ has nontrivial automorphisms, but the automorphisms will be irrelevant for the ensuing computations. Let

$$\begin{align*} \mu _0 : H^0(C, \mathscr{L}) \otimes H^0(C, \omega _C \otimes \mathscr{L}^\vee ) \rightarrow H^0(C, \omega _C) \end{align*}$$

denote the multiplication map. Then, as in ACG11, Chapter 21, (6.1), there is a canonical map

$$\begin{align} \mu _1 : \ker \mu _0 \rightarrow H^0(C, \omega _C^{\otimes 2}). {\tag{3.1}} \end{align}$$

To describe this map explicitly, recall that the differential is a map $\mathscr{O}_C \to \Omega _C = \omega _C$. This induces a map $\mathscr{L} \to \omega _C \otimes \mathscr{L}$, and hence a map $H^0(C, \mathscr{L}) \to H^0(C, \omega _C \otimes \mathscr{L})$ which sends $r \mapsto dr$. One can describe 3.1 explicitly as the map sending $r \otimes s \mapsto dr \cdot s$. It is not obvious this is well-defined, but the well definedness along with this description is verified in ACG11, Chapter 21, p. 810. (The language used there makes it seem like they are working over $\mathbb{C}$, but their proof works equally well over any field.)

Proposition 3.1.

For $C$ a hyperelliptic curve, the composition

$$\begin{align} T_{[C]} \mathscr{H}_{g,k} \rightarrow T_{[C]} \mathscr{M}_{g,k} \rightarrow T_{[\tau _{g}(C)]} \mathscr{A}_{g,k} {\tag{3.2}} \end{align}$$

is dual to the composition

$$\begin{align} H^0(C, \omega _C^{\otimes 2})/\operatorname {im}\mu _1 \xleftarrow {(T_{[C]}\iota _{g,k})^\vee } H^0(C, \omega _C^{\otimes 2}) \xleftarrow {(T_{[C]}\tau _{g,k})^\vee } \operatorname {Sym}^2 H^0(C, \omega _C). {\tag{3.3}} \end{align}$$

Hence, the dimension of the kernel of 3.2 agrees with the dimension of the cokernel of 3.3. In particular, 3.2 is injective if and only if 3.3 is surjective. Further, $\operatorname {coker}\mu _0 = 0, \dim \operatorname {im}\mu _1 = g - 2,$ and $\dim \operatorname {im}(T_{[C]} \tau _{g,k})^\vee = 2g-1$.

Proof.

Let $C$ a smooth projective geometrically connected curve and $L$ an invertible sheaf on $C$. Then, $C$ has a rank $2$ locally free sheaf, $\Sigma _L$, as defined in ACG11, p. 804, such that $H^1(C, \Sigma _L)$ parameterizes first order deformations of the pair $(C,L)$ ACG11, Chapter 21, Proposition 5.15. Further, as described in ACG11, Chapter 21, (5.24), there is a natural map $\mu : H^0(C, L) \otimes H^0(C, \omega _C \otimes L^\vee ) \rightarrow H^0(C, \omega _C \otimes \Sigma _L^\vee )$. The key property of $\mu$ is that if $\alpha \in H^1(C,\Sigma _L)$ corresponds to a first order deformation $(\mathscr{C}, \mathscr{L})$ of $(C,L)$, then all sections of $L$ extend to sections of $\mathscr{L}$ (meaning that, if $\pi : \mathscr{C} \rightarrow \operatorname {Spec}k[\varepsilon ]/(\varepsilon ^2)$ is the structure map, $\pi _* \mathscr{L}$ is locally free,) if and only if $\alpha \in (\operatorname {im}\mu )^\perp$ ACG11, Chapter 21, Proposition 5.26. Here $(\operatorname {im}\mu )^\perp$ denotes the orthogonal subspace to $\operatorname {im}\mu$ under the Serre duality pairing $H^0(C, \omega _C \otimes \Sigma _L^\vee ) \otimes H^1(C, \Sigma _L) \rightarrow k$.

Suppose $C$ is a hyperelliptic curve with corresponding line bundle $L$ defining the hyperelliptic map $C \rightarrow \mathbb{P}^1$. Given $\beta \in H^1(C, T_C)$, we obtain a deformation of $C$, corresponding to a curve $\pi : \mathscr{C} \rightarrow \operatorname {Spec}k[\varepsilon ]/(\varepsilon ^2)$. Our main goal is to show that $\beta$ corresponds to a hyperelliptic curve precisely when $\beta \in (\operatorname {im}\mu _1)^\perp$. By definition, if $[\mathscr{C}]$ is hyperelliptic, there is an invertible sheaf $\mathscr{L}$ on $\mathscr{C}$ with $\pi _* \mathscr{L}$ locally free of rank $2$. As described above, using ACG11, Chapter 21, Proposition 5.15, we obtain that the pair $(\mathscr{C}, \mathscr{L})$ corresponds to some $\alpha \in H^1(C, \Sigma _L)$ with $\alpha \in (\operatorname {im}\mu )^\perp$.

We next show that if $\beta$ corresponds to a hyperelliptic curve, then $\beta \in (\operatorname {im}\mu _1)^\perp$. In other words, we will show $T_{[C]} \mathscr{H}_{g,k} \subset \left( \operatorname {im}\mu _1 \right)^\perp$. By ACG11, Chapter 21, Proposition 5.15, there is a natural map of sheaves $\Sigma _L \rightarrow T_L$ inducing the map $\sigma : H^1(C, \Sigma _L) \rightarrow H^1(C, T_L)$ which sends the deformation $[(\mathscr{C}, \mathscr{L})]$ to the deformation $[\mathscr{C}]$. In particular, $\sigma (\alpha ) = \beta$. Let $(T_{[C]}\tau _{g,k})^\vee : H^0(C, \omega _C^{\otimes 2}) \rightarrow H^0(C, \omega _C \otimes \Sigma _L^\vee )$ denote the map which is Serre dual to $\sigma$. By definition of $\mu _1$ (see ACG11, Chapter 21, (6.1)) we have a commutative diagram

$$\begin{equation} \vcenter{\img[][216pt][52pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} H^0(C, L)\otimes H^0(C, \omega_C \otimes L^\vee) \ar{r}{\mu} & H^0(C, \omega_C \otimes\Sigma_L^\vee) \\ \ker\mu_0 \ar{r}{\mu_1}\ar{u} & H^0(C, \omega_C^{\otimes2}) \ar{u}{\sigma^\vee}. \end{tikzcd}}]{Images/imgdba563e0e63595d4f763a71fec7354f0.svg}}{\tag{3.4}} \end{equation}$$

For any $a \in \operatorname {im}\mu$, we have $\langle a, \alpha \rangle = 0$, where $\langle , \rangle$ denotes the pairing from Serre duality. Therefore, for $b \in \operatorname {im}\mu _1$, commutativity of 3.4 implies $\langle \sigma ^\vee (b), \alpha \rangle = 0$. Functoriality of Serre duality then implies $0= \langle \sigma ^\vee (b), \alpha \rangle = \langle b, \sigma (\alpha ) \rangle = \langle b,\beta \rangle$, and hence $\beta \in \left( \operatorname {im}\mu _1 \right)^\perp$.

We claim that $\dim \ker \mu _0 = g-2, \dim \operatorname {im}(T_{[C]} \tau _{g,k})^\vee = 2g-1,$ and $\mu _1$ is injective. We now explain why verifying these three claims finishes the proof. First, if $\dim \ker \mu _0 = g - 2$ then $\operatorname {coker}\mu _0 = 0$ because $\mu _0$ is a map from a $2g - 2$ dimensional vector space to a $g$ dimensional vector space. We next explain why these claims imply $\dim \operatorname {im}\mu _1 = g-2$. We know $T_{[C]} \mathscr{H}_{g,k}$ is $2g-1$ dimensional (as $\mathscr{H}_{g,k}$ is smooth of dimension $2g-1$ Proposition 2.6) and $T_{[C]}\mathscr{H}_{g,k} \subset (\operatorname {im}\mu _1)^\perp$ inside the $3g-3$ dimensional vector space $H^0(C, T_C)$, as shown above. Since $\dim \operatorname {im}\mu _1 = g-2$, by dimensional considerations, the containment $T_{[C]} \mathscr{H}_{g,k} \subset \left( \operatorname {im}\mu _1 \right)^\perp$ must be an equality. Therefore, the inclusion $T_{[C]} \mathscr{H}_{g,k} \hookrightarrow H^0(C, T_C)$ is dual to the surjection $H^0(C, \omega _C^{\otimes 2}) \twoheadrightarrow H^0(C, \omega _C^{\otimes 2})/\operatorname {im}\mu _1$.

It remains to check $\dim \ker \mu _0 = g-2, \dim \operatorname {im}(T_{[C]} \tau _{g,k})^\vee = 2g-1,$ and $\mu _1$ is injective. First, let us check $\dim \ker \mu _0 = g-2$. Because $C$ is hyperelliptic, we have $\omega _C = L^{\otimes (g-1)}$. In particular, $\omega _C \otimes L^\vee \simeq L^{\otimes (g-2)}$ and so $\mu _0$ is given by the map $H^0(C, L) \otimes H^0(C, L^{\otimes (g-2)}) \rightarrow H^0(C, L^{\otimes (g-1)})$. Letting $\left\{ 1,x \right\}$ denote a basis for $H^0(C, L)$, we see $\left\{ 1, x, \ldots , x^i \right\}$ is a basis for $H^0(C, L^{\otimes i})$. Applying this when $i = g-2$ and $i = g-1$, we find $\mu _0$ is surjective, so $\dim \ker \mu _0 = g-2$. Explicitly, we see $\ker \mu _0$ is generated by $\left\{ 1 \otimes x^j - x \otimes x^{j-1} \right\}_{1 \leq j \leq g-2}$.

Next, we verify $\dim \operatorname {im}(T_{[C]} \tau _{g,k})^\vee = 2g-1$. As in the previous paragraph, we may choose a basis $\left\{ 1, x \right\}$ for $H^0(C, L)$ so that $1, x, \ldots , x^{g-1}$ is a basis for $H^0(C, \omega _C)$. Then, the image of the multiplication map $(T_{[C]} \tau _{g,k})^\vee$ is spanned by $1, x, \ldots , x^{2g-2}$, which has dimension $2g-1$.

To conclude, we show $\mu _1$ is injective. Recall $\left\{ 1 \otimes x^j - x \otimes x^{j-1} \right\}_{1 \leq j \leq g-2}$ spans $\ker \mu _0$, as shown above, and recall that $\mu _1$ is given by $r \otimes s \mapsto dr \otimes s$ ACG11, Chapter 21, (6.6), (alternatively, see ACG11, p. 813-p. 814). Therefore, $\operatorname {im}\mu _1$ is spanned by $dx \cdot x^{j-1}$ for $1 \leq j \leq g-2$, which are independent elements of $H^0(C, \omega _C \otimes L^{\otimes (g-1)}) \simeq H^0(C, \omega _C^{\otimes 2})$. Hence, $\mu _1$ is injective.

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4. Hyperelliptic curves in characteristic not 2

The key to analyzing the map induced by $\phi _{g,k}$ on tangent spaces for $\operatorname {char}(k) \neq 2$ is Lemma 4.1 below. Let $k$ be an algebraically closed field of characteristic $p$ (allowing $p = 0$) with $p \neq 2$. Before proving injectivity, we set up some notation.

4.1. Hyperelliptic differentials in characteristic not $2$

Every hyperelliptic curve $C$ over an algebraically closed field $k$ of characteristic not $2$ can be expressed as the proper regular model of the affine curve $y^2 = f$ for $f \in k[x]$ a polynomial of degree $2g + 1$ with no repeated roots. We can choose a basis of differentials for $C$ of the form

$$\begin{equation} \frac{dx}{y} , \frac{x \, dx}{y}, \ldots , \frac{x^{g-1} \, dx}{y}. {\tag{4.1}} \end{equation}$$

where here $x$ and $y$ are viewed as rational functions and $dx$ is viewed as a rational section of $H^0(C, \omega _C)$.

In the above basis, the multiplication map 2.2 above (which is dual to $T_{[C]} \mathscr{M}_{g,k} \rightarrow T_{[\tau _g(C)]} \mathscr{A}_{g,k}$) has image

$$\begin{align} \frac{(dx)^2}{y^2}, \ldots , \frac{x^{2g-2}\,(dx)^2}{y^2}. {\tag{4.2}} \end{align}$$

Written another way, the basis is $\frac{1}{f}\,(dx)^2, \ldots , \frac{x^{2g-2}}{f}\,(dx)^2$. This is just what one obtains by multiplying together pairs of functions from the above described basis 4.1. In particular, the image is a $2g-1$ dimensional subspace of the $3g-3$ dimensional vector space $H^0(C, \omega _C^{\otimes 2})$.

4.2. Computing the tangent map in characteristic not $2$

We now use our explicit description of the differentials to show 3.3 is surjective when the characteristic is not $2$.

Lemma 4.1.

For $C$ a hyperelliptic curve over an algebraically closed field $k$ of characteristic $p \neq 2$, the composition 3.3 is surjective.

Proof.

We have an explicit understanding of the image of 2.2 from § 4.1. If we can also explicitly describe $\operatorname {im}\mu _1$, then we will be able to determine surjectivity of the composite map 3.3. To start, let’s describe $\mu _0$, using the notation from § 4.1. Letting $\{1, x\}$ denote a basis of $\mathscr{L}$, we find that $\{\frac{dx}{y},\frac{x\,dx}{y} \ldots , \frac{x^{g-2} \,dx}{y}\}$ is a basis for $H^0(C, \omega _C \otimes \mathscr{L}^\vee )$. Then, it follows that

$$\begin{align} x \otimes \frac{dx}{y} - 1 \otimes \frac{x \, dx}{y}, x \otimes \frac{x \, dx}{y} - 1 \otimes \frac{x^2 \, dx}{y}, \ldots , x \otimes \frac{x^{g-3}\, dx}{y} - 1 \otimes \frac{x^{g-2} \, dx}{y} {\tag{4.3}} \end{align}$$

is a basis for $\ker \mu _0$.

The map $\mu _1$ is given explicitly by sending $\alpha \otimes \beta \mapsto d\alpha \cdot \beta$ ACG11, Chapter 21, (6.6). Therefore, applying $\mu _1$ to 4.3, we find $\operatorname {im}\mu _1$ is generated by

$$\begin{align} \frac{(dx)^2}{y},\frac{x(dx)^2}{y} \ldots , \frac{x^{g-3} (dx)^2}{y}. {\tag{4.4}} \end{align}$$

On the other hand, the image of the map $H^0(C, \omega _C^{\otimes 2}) \leftarrow \operatorname {Sym}^2 H^0(C, \omega _C)$ from 2.2 is given in 4.1. Since together, the union of the $2g-1$ elements of 4.1 and the $g-2$ elements of 4.4 span $H^0(C, \omega _C^{\otimes 2})$, it follows that the composite map 3.3 is surjective.

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5. Hyperelliptic curves in characteristic $2$

As in § 4, to check injectivity of $\phi _g$ on tangent vectors, we may assume our base field $k$ is algebraically closed. To conclude the proof of Theorem 1.2, we only need prove Lemma 5.3 below. We now set up notation for the proof. The key difference in characteristic $2$ is that hyperelliptic curves cannot be described in terms of an equation of the form $y^2 = f$, for $f \in k[x]$, of degree more than $1$, as any such curve would be singular at the roots of $\frac{\partial f}{\partial x}$. We now describe a general form for hyperelliptic curves in characteristic $2$.

5.1. Equations for hyperelliptic curves in characteristic 2

We start by reviewing a standard normal form for hyperelliptic curves in characteristic $2$. This is stated in EP13, Notation 1.1, and we provide some more details here.

Lemma 5.1.

Over an algebraically closed field $k$ of characteristic $2$, every hyperelliptic curve of genus at least $2$ can be written as the projective regular model of a curve of the form $y^2 - y = f$, for $f \in k(x)$. For a general such curve, $f$ can be chosen in the form

$$\begin{align} f = \alpha _0 x + \frac{\alpha _1}{x-a_1} + \cdots + \frac{\alpha _g}{x-a_g} {\tag{5.1}} \end{align}$$

with $\alpha _0, \ldots , \alpha _g, a_1, \ldots , a_g \in k$. The corresponding curve $C$ is ramified over $\mathbb{P}^1$ at the preimages of $a_1, \ldots , a_g$, and $\infty$, with ramification order $2$ at each such point.

Remark 5.2.

A hyperelliptic curve can be written in this form precisely if it is ordinary, though we will not need this fact.

Proof.

To start, we claim the curve can be written in the form $ay^2 + by + c = 0$ with $a \in k, b \in k[x], c \in k[x]$. To see this, given a hyperelliptic curve $C \to \mathbb{P}^1$, we know from Lemma 2.4 it can be described as a closed subscheme of $\mathbb{F}_{g+1}$ in the linear system $2E + (2g+2)F$. This implies the curve can be written as $\beta _0 y^2 + \beta _1 yz + \beta _2 z^2$ with $\beta _i \in H^0(\mathbb{P}^1, \mathscr{O}(i \cdot (g+1)))$. Restricting this to $\mathbb{A}^1 \subset \mathbb{P}^1$ gives the claim.

Next, we show one may change variables to put a generic such curve in the form of 5.1. By our generality assumption, we may assume $b$ has $g + 1$ distinct roots. Then, by applying a change of variables, we may send one of the roots of $b$ to $\infty$, and hence assume that $b$ has degree $g$ as a polynomial in $x$ and has $g$ distinct roots. By scaling $y$ by $1/\sqrt {a}$, we may assume $a = 1$ and the curve is given by $y^2 + byz + cz^2$, or equivalently just $y^2 + by + c$ in affine coordinates. Next, we may replace $y$ by $by$ and divide the whole equation by $b^2$ so as to assume that the equation is of the form $y^2 - y = \frac{c}{b^2}$. We next apply a partial fraction decomposition for $\frac{c}{b^2}$ so as to write $\frac{c}{b^2} = c_2 x^2 + c_1x + c_0 + \sum _i \frac{p_i}{(x-a_i)^{2r_i}}$, where $b = \prod (x-a_i)^{r_i}$ for distinct $a_i$ and $p_i$ are polynomials in $x$ of degree at most $2r_i$. Observe that the change of variables $y \mapsto \frac{y+1}{(x-a_i)^{r_i}}$ sends $y^2 - y \mapsto \left( \frac{y}{(x-a_i)^{r_i}} \right)^2 - \frac{y}{(x-a_i)^{r_i}} + \left( \frac{1}{\left( x-a_i \right)^{2r_i}} - \frac{1}{(x-a_i)^{r_i}} \right)$. Hence, renaming $\frac{y}{(x-a_i)^{r_i}}$ to $y$, we can perform such changes of variables to cancel out the highest even power of any denominator appearing in the partial fraction decomposition of $\frac{c}{b^2}$. By a similar change of variables, we may modify the quadraic and linear terms to assume $c_2 = c_0 = 0$. Therefore, in the case that $a,b,c$ are general so that $r_i = 1$ for all $i$, it follows that $f$ may be written in the form 5.1.

To conclude, it remains to compute the ramification points and their ramification orders. A general hyperelliptic curve is given by $\beta _0 y^2 + \beta _1 yz + \beta _2 z^2$ with $\beta _i \in H^0(\mathbb{P}^1, \mathscr{O}(i \cdot (g+1)))$ as above, so that $\beta _1$ has $g+1$ distinct roots. Smoothness of the curve implies $\beta _1$ and $\beta _2$ have no common roots. We can also assume that one of the roots is over $\infty \in \mathbb{P}^1$. Then, for $b \in k[x]$ the dehomogenized version of $\beta _1$, $b$ has $g$ distinct simple roots. By computing the derivative of $\beta _0 y^2 + \beta _1 yz + \beta _2 z^2$ with respect to $y$, we see $C \to \mathbb{P}^1$ is precisely ramified at the roots of $\beta _1$.

We claim that under the setup of the previous paragraph, the ramification orders at the preimages of $a_1, \ldots , a_g, \infty$ is $2$, and $C \to \mathbb{P}^1$ has no other ramification points. To see this, because we have a degree $2$ map and we are in characteristic $2$, the ramification order is at least $2$ at each such point. Observe also that the sum of the ramification orders at all ramified points is $\deg \Omega _{C/\mathbb{P}^1} = 2g + 2$ and there are $g+1$ points $a_1, \ldots , a_g$, and $\infty$. Hence, each point must have ramification order exactly $2$, and these must account for all the ramification points of $C \to \mathbb{P}^1$.

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5.2. Hyperelliptic differentials in characteristic $2$

Now let’s proceed to explicitly write down the differentials, as in the characteristic $0$ case.

Let $k$ be a field of characteristic $2$ and let $C$ be a hyperelliptic curve given as the projective regular model of the curve defined by the equation $y^2 - y = f$, for $f = \alpha _0 x + \frac{\alpha _1}{x-a_1} + \cdots + \frac{\alpha _g}{x-a_g} \in k(x).$ As described in Lemma 5.1, $\pi : C \rightarrow \mathbb{P}^1$ is ramified at the preimages of the points $a_1, \ldots , a_g$ and $\infty$ (where $\infty$ is the point in the complement of $\operatorname {Spec}k[x] \simeq \mathbb{A}^1 \subset \mathbb{P}^1$).

By the assumption that $f$ is as in 5.1, it follows from Lemma 5.1 that there are $g+1$ ramification points of $\pi$, and the differential $dx$ is ramified to order $2$ at the preimages of $V(x-a_1), \ldots , V(x-a_g)$. In other words, the relative sheaf of differentials $\Omega _\pi$ is a skyscraper sheaf at $\pi ^{-1}(V(x - a_i))$ with degree $2$ at $\pi ^{-1}(V(x - a_i))$, for $1 \leq i \leq g$. Since $\deg dx = 2g - 2$, we conclude that $dx$ has a pole of order $2$ at the unique point over $\infty \in \mathbb{P}^1$. Further, the function $x-a_i$ vanishes to order $2$ at the preimage of $\pi ^{-1}(V(x-a_i))$ and therefore has a pole of order $2$ at $\pi ^{-1}(\infty )$. It follows that the functions

$$\begin{align*} \omega _1 \coloneq \frac{dx}{x-a_1}, \ldots , \omega _g \coloneq \frac{dx}{x-a_g} \end{align*}$$

form a basis of $H^0(C, \omega _C)$.

In the above basis, the map 2.2

$$\begin{align*} H^0(C, \omega _C^{\otimes 2}) \leftarrow \operatorname {Sym}^2 H^0(C, \omega _C). \end{align*}$$

has image spanned by elements of the form

$$\begin{align} \frac{(dx)^2}{(x-a_i)(x-a_j)} {\tag{5.2}} \end{align}$$

Of course, these elements will not be independent, but they will necessarily span a $2g-1$ dimensional subspace of $H^0(C, \omega _C^{\otimes 2})$ by Proposition 3.1.

5.3. Computing the map on tangent spaces in characteristic $2$

Lemma 5.3.

For $C$ a hyperelliptic curve of genus $g \geq 2$ over an algebraically closed field $k$ of characteristic $2$, the composition 3.3 has cokernel of dimension $g -2$.

Proof.

For $C$ any hyperelliptic curve, with notation as in Proposition 3.1, $\dim \operatorname {im}\mu _1 = g-2$ and $\dim \operatorname {im}(T_{[C]}\tau _g)^\vee = 2g - 1.$ It follows that the composition 3.3 has image of dimension at least $(2g-1) - (g-2) = g + 1$. Equivalently, 3.3 has cokernel of dimension at most $g-2$, since $\dim \mathscr{H}_{g,\mathbb{F}_2} = 2g-1 = (g+1)+(g-2)$. Therefore, by upper semicontinuity of cokernel dimension, in order to show 3.3 has cokernel of dimension $g-2$ at all $[C] \in \mathscr{H}_{g,\mathbb{F}_2}$ over a field of characteristic $2$, it suffices to show this holds over a general such $[C] \in \mathscr{H}_{g,\mathbb{F}_2}$.

As described in § 5.1, a general hyperelliptic curve over $k = \overline{k}$ with $\operatorname {char}(k) = 2$ can be written in the form $y^2 - y = f$, for $f = \alpha _0 x+ \frac{\alpha _1}{x-a_1} + \cdots + \frac{\alpha _g}{x-a_g} \in k(x)$, with elements $\alpha _0, \ldots , \alpha _g, a_1, \ldots , a_g \in k$ and with $a_1, \ldots , a_g$ distinct. Following the strategy outlined in § 3, we next determine $\ker \mu _0$ and then $\operatorname {im}\mu _1$ for such curves $C$.

We can choose $1, x-a_1$ as a basis for $H^0(C, \mathscr{L})$ identifying $\mathscr{L} \simeq \mathscr{O}_C(2 \cdot \pi ^{-1}(\infty ))$, for $\pi : C \rightarrow \mathbb{P}^1$ the hyperelliptic map. For $1 < i \leq g$, the differential forms $\frac{1}{a_1 - a_i} \left( \omega _1 - \omega _i \right) = \frac{dx}{(x-a_1)(x-a_i)}$ have a zero of order $2$ at $\infty$, and therefore lie in $H^0(C, \omega _C \otimes \mathscr{L}^\vee )$. Because they are also independent, it follows that

$$\begin{align} \frac{dx}{(x-a_1)(x-a_2)}, \frac{dx}{(x-a_1)(x-a_3)}, \ldots , \frac{dx}{(x-a_1)(x-a_g)} {\tag{5.3}} \end{align}$$

form a basis for $H^0(C, \omega _C \otimes \mathscr{L}^\vee )$. The kernel of $\mu _0$ is then spanned by elements of the form

$$\begin{align*} (x-a_1) \otimes r_i + 1 \otimes t_i \end{align*}$$

for $1 \leq i \leq g-2$ with $r_i, t_i$ linear combinations of the elements in 5.3.

Therefore, $\operatorname {im}\mu _1$ is generated by $d(x-a_1) \otimes r_i + d(1) \otimes t_i = dx \otimes r_i$ for $1 \leq i \leq g-2$. In particular, the $r_i$ are some linear combination of elements appearing in 5.3. It follows that $dx \otimes r_i$ are also linear combinations of the elements appearing in 5.2. Therefore, $\operatorname {im}\mu _1 \subset \operatorname {im}(H^0(C, \omega _C^{\otimes 2}) \leftarrow \operatorname {Sym}^2 H^0(C, \omega _C)).$ By Proposition 3.1, $\dim \operatorname {im}\mu _1 = g-2$ and $\dim \operatorname {im}(T_{[C]} \tau _g)^\vee = 2g-1$. Hence, the composition 3.3 has image of dimension $(2g-1) - (g-2) = g + 1$ and cokernel of dimension $(2g-1) - (g+1) = g-2$.

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6. Proof of Theorem 1.2

At this point, nearly everything is in place to prove Theorem 1.2. The work of previous sections will show that $\phi _g$ is a monomorphism over $\operatorname {Spec}\mathbb{Z}[1/2]$, but fails to be a monomorphism when restricted to any field $k$ with $\operatorname {char}(k) = 2$. In order to show $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ is an immersion over $\operatorname {Spec}\mathbb{Z}[1/2]$, we will verify the valuative criterion for immersions (or radimmersions) in Proposition 6.3. We then use Corollary A.5 to deduce $\phi _g$ is an immersion over $\operatorname {Spec}\mathbb{Z}[1/2]$. This valuative criterion loosely says that a monomorphism $f: X \rightarrow Y$ is an immersion when, given a map from a the spectrum of a discrete valuation ring to $Y$ with its two points factoring through $X$, the map from the spectrum of the discrete valuation ring factors through $X$.

We next set up some notation. Let $\overline{\mathscr{M}}_g$ denote the Deligne-Mumford compactification of $\mathscr{M}_{g}$ and let $\mathscr{M}_{g}^{c}$ denote the moduli stack of stable compact type curves of genus $g$. Recall that $\mathscr{M}_{g}^{c}$ can be constructed as the open substack of $\overline{\mathscr{M}}_g$ which, loosely speaking, parameterizes curves whose dual graph of components is a tree. The geometric points of $\overline{\mathscr{M}}_g$ lie in $\mathscr{M}_{g}^{c}$ precisely when the Jacobian of the corresponding curve is an abelian variety, as follows from BLR90, §9.2, Example 8. The following lemma is well-known:

Lemma 6.1.

The Torelli map $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ factors as the composition of an immersion $\mathscr{H}_{g} \rightarrow \mathscr{M}_{g}^{c}$ and a proper map $\tau _g^c: \mathscr{M}_{g}^{c} \rightarrow \mathscr{A}_{g}$.

Proof.

It is shown in Ale04, Corollary 5.4 that there is a compactification of $\mathscr{A}_{g}$, which we denote $\overline{\mathscr{A}}_g$, and a map $\overline{\tau }_g : \overline{\mathscr{M}}_g \rightarrow \overline{\mathscr{A}}_g$ extending $\tau _g$. (In Ale04, Corollary 5.4, it is only stated that this yields a map of coarse spaces, but the map is in fact constructed as a map of stacks.) Since $\overline{\mathscr{M}}_g$ and $\overline{\mathscr{A}}_g$ are proper, $\overline{\tau }_g$ is as well, and therefore the resulting map $\mathscr{M}_{g}^{c} = \overline{\mathscr{M}}_g \times _{\overline{\mathscr{A}}_g} \mathscr{A}_{g} \rightarrow \mathscr{A}_{g}$ is also proper. Finally, $\mathscr{H}_{g} \rightarrow \mathscr{M}_{g}^{c}$ is an immersion because it is a composition of the immersions $\mathscr{H}_{g} \rightarrow \mathscr{M}_{g} \rightarrow \mathscr{M}_{g}^{c}$.

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Remark 6.2.

We used the rather difficult result of Ale04, Corollary 5.4 in the proof of Lemma 6.1, but Lemma 6.1 can also be verified directly. One can extend the principal polarization on the universal Jacobian over $\mathscr{M}_{g}$ to that over $\mathscr{M}_{g}^{c}$, and then use the valuative criterion of properness and BLR90, §7.4, Proposition 3 to verify properness of $\mathscr{M}_{g}^{c} \rightarrow \mathscr{A}_{g}$. See Lan for further discussion of this. For the sake of brevity, we omit this more direct proof.

We now verify $\phi _g$ satisfies the valuative criterion for radimmersions.

Proposition 6.3.

The map $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ satisfies the valuative criterion for radimmersions for traits as in Definition A.3.

Proof.

Let $T$ be a trait (the spectrum of a discrete valuation ring) with $2$-commutative diagrams as in A.1 for $X = \mathscr{H}_{g}$ and $Y = \mathscr{A}_{g}$. By Lemma 6.1 and the valuative criterion for properness Ols16, Theorem 11.5.1 there is a dominant map of traits $T' \rightarrow T$ so that the resulting map $T' \rightarrow \mathscr{A}_{g}$ factors through $\mathscr{M}_{g}^{c}$. Let $\mathscr{J} \rightarrow T'$ denote the family of principally polarized abelian varieties of dimension $g$ corresponding to the map $T' \rightarrow \mathscr{A}_{g}$. The factorization $T' \rightarrow \mathscr{M}_{g}^{c}$ yields a family of stable curves $\mathscr{C} \rightarrow T'$, so that the principally polarized Jacobian of the generic fiber of $\mathscr{C}$ agrees with the generic fiber of $\mathscr{J} \rightarrow T'$.

With notation for $T, T', t, t', \eta , \eta '$ as in Definition A.3, we have diagrams

$$\begin{equation} \vcenter{\img[][232pt][85pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} \eta' \ar{r} \ar{dd} & \eta\ar{r} \ar{dd} & \mathscr{H}_{g} \ar{d} & t' \ar{r} \ar{dd} & t \ar{r} \ar{dd} & \mathscr{H}_{g} \ar{d} \\ && \mathscr{M}_{g}^{c} \ar{d}{\tau_g^c} & && \mathscr{M}_{g}^{c}\ar{d}{\tau_g^c} \\ T' \ar{urr} \ar{r} & T \ar{r} & \mathscr{A}_{g} & T' \ar{urr}\ar{r} & T \ar{r} & \mathscr{A}_{g} \end{tikzcd}}]{Images/img5256bb40706f2988aae9ce12d11d5b03.svg}}{\tag{6.1}} \end{equation}$$

The first diagram in 6.1 $2$-commutes by the valuative criterion for properness. We claim the second diagram also $2$-commutes. Granting this claim, observe that $\mathscr{H}_{g} \rightarrow \mathscr{M}_{g}^{c}$ is an immersion, being the composition of a closed immersion $\mathscr{H}_{g} \rightarrow \mathscr{M}_{g}$ and an open immersion $\mathscr{M}_{g} \rightarrow \mathscr{M}_{g}^{c}$. Therefore, the valuative criterion for immersions Corollary A.5 implies $T' \rightarrow \mathscr{M}_{g}^{c}$ lifts to a map $T' \rightarrow \mathscr{H}_{g}$ making the diagrams A.2 $2$-commute for $X = \mathscr{H}_{g}$ and $Y = \mathscr{A}_{g}$.

So, to conclude, we just need to check the second diagram in 6.1 $2$-commutes. This will necessarily follow if we verify the fiber product $[C] \times _{\phi _g, \mathscr{A}_{g}, \tau _g^c} \mathscr{M}_{g}^{c}$ contains a unique geometric point for any geometric point $[C] \in \mathscr{H}_{g}$. First, we show $[C] \times _{\phi _g, \mathscr{A}_{g}, \tau _g^c} \mathscr{M}_{g}^{c}$ does not contain any geometric points mapping to $\mathscr{M}_{g}^{c} - \mathscr{M}_{g}$. Indeed, the theta divisor associated to a singular compact type curve is always geometrically reducible, while that associated to a curve in $\mathscr{M}_{g}$ is geometrically irreducible. Hence, no geometric points of $\mathscr{M}_{g}^{c} - \mathscr{M}_{g}$ can map under $\tau _g^c$ to $\phi _g([C])$. Therefore, it suffices to show $[C] \times _{\phi _g, \mathscr{A}_{g}, \tau _g^c} \mathscr{M}_{g}^{c}$ contains a unique geometric point in $\mathscr{M}_{g}$, which follows from the classical Torelli theorem Cor86, Chapter VII, Theorem 12.1.

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Corollary 6.4.

The map

$$\begin{equation*} \phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g} \end{equation*}$$

is a radimmersion (as defined in Definition A.1).

Proof.

This will follow from Proposition 6.3 and Theorem A.4 once we verify $\phi _g$ is representable and induces a universal homeomorphism on isotropy groups at every point of $\mathscr{H}_{g}$. We first check that $\phi _g$ induces isomorphisms on isotropy groups at every point of $\mathscr{H}_{g}$. Observe that $\phi _g$ induces a bijection on geometric points of isotropy groups at every point of $\mathscr{H}_{g}$ by the Torelli theorem Cor86, Chapter VII, Theorem 12.1(b). Since both $\mathscr{H}_{g}$ and $\mathscr{A}_{g}$ are Deligne-Mumford stacks, their isotropy groups at any geometric point are constant group schemes. Therefore, $\phi _g$ induces an isomorphism on isotropy groups at geometric points of $\mathscr{H}_{g}$. As a consequence, $\phi _g$ is representable, as follows by applying Con07, Theorem 2.2.5 to the pullback of $\phi _g$ along a schematic cover of $\mathscr{A}_{g}$. Note that Con07, Theorem 2.2.5 assumes stacks have finite type separated diagonals, but these assumptions apply in this case as $\mathscr{M}_{g}$ and $\mathscr{A}_{g}$ even have finite diagonals.

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Combining Corollary 6.4 with Proposition 3.1, Lemma 4.1, and Lemma 5.3, we now prove Theorem 1.2.

Proof of Theorem 1.2.

We know $\phi _g$ is a radimmersion by Corollary 6.4. Next, we show $\phi _g$ is an immersion over $\operatorname {Spec}\mathbb{Z}[1/2]$ or when $g =2$. Using Theorem A.4 and Corollary A.5, we only need to check $\phi _g$ is a monomorphism over $\operatorname {Spec}\mathbb{Z}[1/2]$ or when $g = 2$. Equivalently, we just need to verify $\phi _g$ is injective on geometric points and tangent vectors Gro67, 17.2.6. It follows from the classical Torelli theorem Cor86, Chapter VII, Theorem 12.1(a) that the map $\phi _g: \mathscr{H}_{g} \rightarrow \mathscr{A}_{g}$ is injective on geometric points. Since smooth hyperelliptic curves $C$ in over a field $k$ of any characteristic can be deformed to smooth hyperelliptic curves of characteristic $0,$ (as can be seen by using explicit equations,) we obtain an identification of the kernel of $T_{[C]} \phi _g: T_{[C]}\mathscr{H}_{g} \rightarrow T_{[C]} \mathscr{A}_{g}$ with $\ker \left(T_{[C]} \phi _{g,k} \right)$. Note that $T_{[C]} \phi _{g,k}$ is given as the composition 3.2. The vanishing of $\ker T_{[C]} \phi _{g,k}$ over $\operatorname {Spec}\mathbb{Z}[1/2]$ follows from combining Proposition 3.1 with Lemma 4.1. Therefore, $\phi _g$ is injective on tangent vectors over $\operatorname {Spec}\mathbb{Z}[1/2]$. In the case $g = 2$ we find $\phi _g$ is a monomorphism by combining the above with Lemma 5.3.

To conclude the proof, we just need to check that the restriction of $\phi _g$ to a field of characteristic $2$ induces a map on tangent spaces with kernel of dimension $g -2$. Again using the identification $\ker T_{[C]} \phi _g = \ker T_{[C]} \phi _{g,k}$ mentioned above, this follows from combining Proposition 3.1 with Lemma 5.3.

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