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# On the height of Gross–Schoen cycles in genus three

- Robin de Jong
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**Received:**16 November 2017**Accepted:**1 September 2018**Published:**17 September 2018

## Abstract

We show that there exists a sequence of genus three curves defined over the rationals in which the height of a canonical Gross–Schoen cycle tends to infinity.

## Keywords

- Beilinson–Bloch height
- Faltings height
- Gross–Schoen cycle
- Height jump
- Horikawa index

## Mathematics Subject Classification

- 11G50
- 14G40
- 14H25

## 1 Introduction

Let *X* be a smooth, projective and geometrically connected curve of genus \(g \ge 2\) over a field *k* and let \(\alpha \) be a divisor of degree one on *X*. The Gross–Schoen cycle \(\Delta _\alpha \) associated to \(\alpha \) is a modified diagonal cycle in codimension two on the triple product \(X^3\), studied in detail in [18] and [49]. The cycle \(\Delta _\alpha \) is homologous to zero, and its class in \({\text {CH}}^2(X^3)\) depends only on the class of \(\alpha \) in \({\text {Pic}}^1 X\).

Assume that *k* is a number field or a function field of a curve. Gross and Schoen show in [18] the existence of a Beilinson–Bloch height \(\langle \Delta _\alpha ,\Delta _\alpha \rangle \in \mathbb {R}\) of the cycle \(\Delta _\alpha \), under the assumption that *X* has a “good” regular model over *k*. A good regular model exists after a suitable finite extension of the base field *k*, and one can unambiguously define a height \(\langle \Delta _\alpha ,\Delta _\alpha \rangle \) of the Gross–Schoen cycle for all *X* over *k* and all \(\alpha \in {\text {Div}}^1 X\) by passing to a finite extension of *k* where *X* has a good regular model, computing the Beilinson–Bloch height over that extension, and dividing by the degree of the extension.

Standard arithmetic conjectures of Hodge Index type [16] predict that one should always have the inequality \(\langle \Delta _{\alpha },\Delta _{\alpha }\rangle \ge 0\), and that equality should hold if and only if the class of the cycle \(\Delta _{\alpha }\) vanishes in \({\text {CH}}^2(X^3)_\mathbb {Q}\). Zhang [49] has proved formulae that connect the height \(\langle \Delta _\alpha ,\Delta _\alpha \rangle \) of a Gross–Schoen cycle with more traditional invariants of *X*, namely the stable self-intersection of the relative dualizing sheaf, and the stable Faltings height. Zhang’s formulae feature some new interesting local invariants of *X*, called the \(\varphi \)-invariant and the \(\lambda \)-invariant.

For \(\alpha \in {\text {Div}}^1 X\) let \(x_\alpha \) be the class of the divisor \(\alpha - K_X/(2g-2)\) in \({\text {Pic}}^0(X)_\mathbb {Q}\), where \(K_X\) is a canonical divisor on *X*. Then a *canonical* Gross–Schoen cycle on \(X^3\) is a Gross–Schoen cycle \(\Delta _\alpha \) for which the class \(x_\alpha \) vanishes in \({\text {Pic}}^0(X)_\mathbb {Q}\). A corollary of Zhang’s formulae in [49] is that for given *X*, the height \(\langle \Delta _{\alpha },\Delta _{\alpha }\rangle \) is minimized for \(\Delta _\alpha \) a canonical Gross–Schoen cycle. The question as to the non-negativity of \(\langle \Delta _{\alpha },\Delta _{\alpha }\rangle \) is therefore reduced to the cases where \(\Delta _\alpha \) is canonical.

As an example, for *X* a hyperelliptic curve and \(\alpha \) a Weierstrass point on *X* one has by [18], Proposition 4.8] that \(\Delta _\alpha \) is zero in \({\text {CH}}^2(X^3)_\mathbb {Q}\). It follows that the height \(\langle \Delta _{\alpha },\Delta _{\alpha }\rangle \) vanishes, and by Zhang’s formulae the height of any Gross–Schoen cycle on a hyperelliptic curve is non-negative.

When *k* is a function field in characteristic zero the inequality \(\langle \Delta _{\alpha },\Delta _{\alpha }\rangle \ge 0\) is known to hold by an application of the Hodge Index Theorem [50]. It seems that only very little is known though beyond the hyperelliptic case when *k* is a function field in positive characteristic, or a number field. Yamaki shows in [46] that \(\langle \Delta _{\alpha },\Delta _{\alpha }\rangle \ge 0\) if *X* is a non-hyperelliptic curve of genus three with semistable reduction over a function field, under the assumption that certain topological graph types do not occur as dual graph of a special fiber of the semistable regular model.

The purpose of this paper is to prove the following theorem.

### Theorem A

There exists a sequence of genus three curves over \(\mathbb {Q}\) in which the height of a canonical Gross–Schoen cycle tends to infinity.

To the best of the author’s knowledge, Theorem A is the first result to prove unconditionally the existence of a curve *X* over a number field such that a canonical Gross–Schoen cycle on \(X^3\) has strictly positive height.

Our proof of Theorem A is, like Yamaki’s work, based on Zhang’s formulae. More precisely we use the formula that relates the height of a canonical Gross–Schoen cycle on \(X^3\) with the stable Faltings height of *X*. We then express the Faltings height of a non-hyperelliptic curve of genus three in terms of the well-known modular form \(\chi _{18}\) of level one and weight 18, defined over \(\mathbb {Z}\). Combining both results we arrive at an expression for the height of a canonical Gross–Schoen cycle on a non-hyperelliptic genus three curve *X* with semistable reduction as a sum of local contributions ranging over all places of *k*, cf. Theorem 8.2.

The local non-archimedean contributions can be bounded from below by some combinatorial data in terms of the dual graphs associated to the stable model of *X* over *k*. This part of the argument is heavily inspired by Yamaki’s work [47] dealing with the function field case. In fact, the differences with [47] at this point are only rather small: the part of [47] that works only in a global setting, by an application of the Hirzebruch-Riemann-Roch theorem, is replaced here by a more local approach, where the application of Hirzebruch-Riemann-Roch is replaced by an application of Mumford’s functorial Riemann-Roch [41].

The modular form \(\chi _{18}\) is not mentioned explicitly in [47] but clearly plays a role in the background. As an intermediate result, we obtain an expression for the local order of vanishing of \(\chi _{18}\) in terms of the Horikawa index [35, 44] and the discriminant, cf. Proposition 9.3. This result might be of independent interest.

We will then pass to a specific family of non-hyperelliptic genus three curves \(C_n\) defined over \(\mathbb {Q}\), for \(n \in \mathbb {Z}_{>0}\) and \(n \rightarrow \infty \), considered by Guàrdia in [19]. In the paper [19], the stable reduction types of the curves \(C_n\) are determined explicitly. By going through the various cases, we will see that the local non-archimedean contributions to the height of a canonical Gross–Schoen cycle on \(C_n\), as identified by Theorem 8.2, are all non-negative.

We show that the archimedean contribution to the height of a canonical Gross–Schoen cycle of \(C_n\) is bounded from below by a quantity that tends to infinity like \(\log n\). In order to do this we recall the work [21] by Hain and Reed on the Ceresa cycle, which allows us to study the archimedean contribution as a function of \(\kappa \in \mathbb {P}^1 \setminus \{0,1,\infty \}\). For \(n \rightarrow \infty \) we have \(\kappa \rightarrow 0\). The stable reduction of the family \(D_\kappa \) near \(\kappa =0\) is known, see for instance [24], Proposition 8] and the asymptotic behavior of the archimedean contribution near \(\kappa =0\) can then be determined by invoking an asymptotic result due to Brosnan and Pearlstein [6].

The paper is organized as follows. In Sects. 2 and 3 we recall the non-archimedean and archimedean \(\varphi \)- and \(\lambda \)-invariants from Zhang’s paper [49]. The main formulae from [49] relating the height of the Gross–Schoen cycle to the self-intersection of the relative dualizing sheaf and the Faltings height are then stated in Sect. 4. In Sect. 5 we display Zhang’s \(\lambda \)-invariant for a couple of polarized metrized graphs that we will encounter in our proof of Theorem A.

In Sect. 6 we recall a few general results on analytic and algebraic modular forms, and in Sect. 7 we recall the work of Hain and Reed, and Brosnan and Pearlstein that we shall need on the asymptotics of the archimedean contribution to the height. In Sect. 8 we discuss the modular form \(\chi _{18}\). The first new results are contained in Sect. 9, where we recall the Horikawa index for stable curves in genus three and show how it can be expressed in terms of the order of vanishing of \(\chi _{18}\) and the discriminant. This leads to a useful lower bound for the order of vanishing of \(\chi _{18}\). Sections 10–12 contain the proof of Theorem A.

## 2 Non-archimedean invariants

*metrized graph*is a connected compact metric space \(\Gamma \) such that \(\Gamma \) is either a point or for each \(p \in \Gamma \) there exist a positive integer

*n*and \(\epsilon \in \mathbb {R}_{>0}\) such that

*p*possesses an open neighborhood

*U*together with an isometry \(U \xrightarrow {\sim }S(n,\epsilon )\), where \(S(n,\epsilon )\) is the star-shaped set

*n*is uniquely determined, and is called the

*valence*of

*p*, notation

*v*(

*p*). We set the valence of the unique point of the point-graph to be zero. Let \(V_0 \subset \Gamma \) be the set of points \(p \in \Gamma \) with \(v(p) \ne 2\). Then \(V_0\) is a finite subset of \(\Gamma \), and we call any finite non-empty set \(V \subset \Gamma \) containing \(V_0\) a

*vertex set*of \(\Gamma \).

Let \(\Gamma \) be a metrized graph and let *V* be a vertex set of \(\Gamma \). Then \(\Gamma \setminus V\) has a finite number of connected components, each isometric with an open interval. The closure in \(\Gamma \) of a connected component of \(\Gamma \setminus V\) is called an *edge* associated to *V*. We denote by *E* the set of edges of \(\Gamma \) resulting from the choice of *V*. When \(e \in E\) is obtained by taking the closure in \(\Gamma \) of the connected component \(e^\circ \) of \(\Gamma \setminus V\) we call \(e^\circ \) the *interior* of *e*. The assignment \(e \mapsto e^\circ \) is unambiguous, given the choice of *V*, as we have \(e^\circ = e \setminus V\). We call \(e \setminus e^\circ \) the set of *endpoints* of *e*. For example, assume \(\Gamma \) is a circle, and say *V* consists of \(n>0\) points on \(\Gamma \). Then \(\Gamma \setminus V\) has *n* connected components, and \(\Gamma \) has *n* edges. In general, an edge is homeomorphic to either a circle or a closed interval, and thus has either one endpoint or two endpoints.

Let \(e \in E\), and assume that \(e^\circ \) is isometric with the open interval \((0,\ell (e))\). Then the positive real number \(\ell (e)\) is well-defined and called the *weight* of *e*. The total weight \(\delta (\Gamma )=\sum _{e \in E} \ell (e)\) is called the *volume* of \(\Gamma \). We note that the volume \(\delta (\Gamma )\) of a metrized graph \(\Gamma \) is independent of the choice of a vertex set *V*.

*degree*in \(\mathbb {Z}\). Assume we have fixed a map \(\mathbf {q} :V \rightarrow \mathbb {Z}\). The associated

*canonical divisor*\(K=K_\mathbf {q}\) is by definition the element \(K \in \mathbb {Z}^V\) such that for all \(p \in V\) the equality \(K(p)=v(p)-2+2 \,\mathbf {q}(p)\) holds. We call the pair \(\overline{\Gamma }=(\Gamma ,\mathbf {q})\) a

*polarized metrized graph*, abbreviated

*pm-graph*, if \(\mathbf {q}\) is non-negative, and the canonical divisor \(K_\mathbf {q}\) is effective. Let \(\overline{\Gamma }=(\Gamma , \mathbf {q})\) be a pm-graph with vertex set

*V*. We call the integer

*genus*of \(\overline{\Gamma }\). Here \(b_1(\Gamma ) \in \mathbb {Z}_{\ge 0}\) is the first Betti number of \(\Gamma \). We see that \(g(\overline{\Gamma }) \in \mathbb {Z}_{\ge 1}\). We occasionally call \(\mathbf {q}(p)\) the

*genus*of the vertex \(p \in V\).

An edge \(e \in E\) is called *of type 0* if removal of its interior results into a connected graph. Let \(h \in [1,g/2]\) be an integer. An edge \(e \in E\) is called *of type h* if removal of its interior yields the disjoint union of a pm-graph of genus *h* and a pm-graph of genus \(g-h\). The total weight of edges of type 0 is denoted by \(\delta _0(\overline{\Gamma })\), and the total weight of edges of type *h* is denoted \(\delta _h(\overline{\Gamma })\). We have \(\delta (\Gamma ) = \sum _{h=0}^{[g/2]} \delta _h(\overline{\Gamma })\).

*E*. We define \(\Gamma _{ \{e_1 \} }\) to be the topological space obtained from \(\Gamma \) by contracting the subspace \(e_1\) to a point. Then \(\Gamma _{ \{e_1 \} }\) has a natural structure of metrized graph, and the natural projection \(\Gamma \rightarrow \Gamma _{ \{e_1\} }\) endows \(\Gamma _{ \{e_1\} }\) with a designated vertex set, and maps each edge \(e_i\) for \(i=2,\ldots ,n\) onto an edge of \(\Gamma _{ \{e_1\} }\). Continuing by induction we obtain after

*n*steps a metrized graph \(\Gamma _S\) with natural projection \(\pi :\Gamma \rightarrow \Gamma _S\) and designated vertex set \(V_S\). The result is independent of the ordering of the edges in

*S*and is called the metrized graph obtained by

*contracting*the edges in

*S*.

Consider the pushforward divisor \(\pi _* K_\mathbf {q}\) on \(\Gamma _S\). It is then clear that \(\pi _* K_\mathbf {q}\) is effective and has the same degree as \(K_{\mathbf {q}}\). The associated map \(\mathbf {q}_S :V_S \rightarrow \mathbb {Z}\) is non-negative, and thus we obtain a pm-graph \(\overline{\Gamma }_S=(\Gamma _S,\mathbf {q}_S)\) canonically determined by *S*. Clearly we have \(g(\overline{\Gamma }_S)=g(\overline{\Gamma })\). The pm-graph obtained by contracting all edges in \(E \setminus S\) is denoted by \(\overline{\Gamma }^S=(\Gamma ^S,\mathbf {q}^S)\).

Assume \(\Gamma \) is not a point. When \(\Gamma _1, \Gamma _2\) are subgraphs of \(\Gamma \) such that \(\Gamma =\Gamma _1 \cup \Gamma _2\) and \(\Gamma _1 \cap \Gamma _2\) consists of one point, we say that \(\Gamma \) is the *wedge sum* of \(\Gamma _1, \Gamma _2\), notation \(\Gamma = \Gamma _1 \vee \Gamma _2\). By induction one has a well-defined notion of wedge sum \(\Gamma _1 \vee \ldots \vee \Gamma _n\) of subgraphs \(\Gamma _1,\ldots ,\Gamma _n\) of \(\Gamma \). We say that \(\Gamma \) is *irreducible* if the following holds: write \(\Gamma = \Gamma _1 \vee \Gamma _2\) as a wedge sum. Then one of \(\Gamma _1, \Gamma _2\) is a one-point graph. The graph \(\Gamma \) has a unique decomposition \(\Gamma =\Gamma _1 \vee \ldots \vee \Gamma _n\) as a wedge sum of irreducible subgraphs. We call the \(\Gamma _i\) the *irreducible components* of \(\Gamma \). Each \(\Gamma _i\) can be canonically seen as the contraction of some edges of \(\Gamma \), and hence has a natural induced structure of pm-graph \(\overline{\Gamma }_i\) of genus *g*, where \(g=g(\overline{\Gamma })\) is the genus of \(\overline{\Gamma }\).

We call an invariant \(\kappa =\kappa (\overline{\Gamma })\) of pm-graphs of genus *g* *additive* if the invariant \(\kappa \) is compatible with decomposition into irreducible components. More precisely, let \(\overline{\Gamma }\) be a pm-graph of genus *g* and let \(\overline{\Gamma } = \overline{\Gamma }_1 \vee \ldots \vee \overline{\Gamma }_n\) be its decomposition into irreducible components, where each \(\overline{\Gamma }_i\) has its canonical induced structure of pm-graph of genus *g*. Then we should have \(\kappa (\overline{\Gamma })= \kappa (\overline{\Gamma }_1) + \cdots + \kappa (\overline{\Gamma }_n)\). It is readily seen that each of the invariants \(\delta _h(\overline{\Gamma })\) where \(h=0,\ldots ,[g/2]\) is additive on pm-graphs of genus *g*. By [49], Theorem 4.3.2] the \(\varphi \)-invariant, the \(\epsilon \)-invariant and the \(\lambda \)-invariant are all additive on pm-graphs of genus *g*.

Let \(G=(V,E)\) be a connected graph (multiple edges and loops are allowed) and let \(\ell :E \rightarrow \mathbb {R}_{>0}\) be a function on the edge set *E* of *G*. We then call the pair \((G,\ell )\) a *weighted graph*. Let \((G,\ell )\) be a weighted graph. Then to \((G,\ell )\) one has naturally associated a metrized graph \(\Gamma \) by glueing together finitely many closed intervals \(I(e)=[0,\ell (e)]\), where *e* runs through *E*, according to the vertex assignment map of *G*. Note that the resulting metrized graph \(\Gamma \) comes equipped with a distinguished vertex set \(V \subset \Gamma \).

Let *R* be a discrete valuation ring and write \(S={\text {Spec}}R\). Let \(f :\mathcal {X}\rightarrow S\) be a generically smooth stable curve of genus \(g \ge 2\) over *S*. We can canonically attach a weighted graph \((G,\ell )\) to *f* in the following manner. Let *C* denote the geometric special fiber of *f*. Then the graph *G* is to be the dual graph of *C*. Thus the vertex set *V* of *G* is the set of irreducible components of *C*, and the edge set *E* is the set of nodes of *C*. The incidence relation of *G* is determined by sending a node *e* of *C* to the set of irreducible components of *C* that *e* lies on. Each \(e \in E\) determines a closed point on \(\mathcal {X}\). We let \(\ell (e) \in \mathbb {Z}_{>0}\) be its so-called *thickness* on \(\mathcal {X}\).

Let \(\Gamma \) denote the metrized graph associated to \((G,\ell )\) with designated vertex set *V*. We have a canonical map \(\mathbf {q} :V \rightarrow \mathbb {Z}\) given by associating to \(v \in V\) the geometric genus of the irreducible component *v*. The map \(\mathbf {q}\) is non-negative, and the associated canonical divisor \(K_\mathbf {q}\) is effective. We therefore obtain a canonical pm-graph \(\overline{\Gamma }=(\Gamma ,\mathbf {q})\) from *f*. The genus \(g(\overline{\Gamma })\) is equal to the genus of the generic fiber of *f*.

*g*determined by

*f*. For \(h=0,\ldots ,[g/2]\) we have canonical boundary divisors \(\Delta _h\) on \(\overline{\mathcal {M}}_g\) whose generic points correspond to irreducible stable curves of genus

*g*with one node (in the case \(h=0\)), or to reducible stable curves consisting of two irreducible components of genus

*h*and \(g-h\), joined at one point (in the case \(h>0\)). Let

*v*denote the closed point of

*S*. Then for each \(h=0,\ldots , [g/2]\) we have the equality

## 3 Archimedean invariants

*C*be a compact and connected Riemann surface of genus \(g \ge 2\). Let \(\mathrm {H}^0(C,\omega _C)\) denote the space of holomorphic differentials on

*C*, equipped with the hermitian inner product

*C*. Let \(\Delta _\mathrm {Ar}\) be the Laplacian operator on \(L^2(C,\mu _C)\), i.e. the endomorphism of \(L^2(C,\mu _C)\) determined by setting

*C*is then defined to be the real number

*C*as defined by Faltings in [14], p. 401], and put \(\delta (C)=\delta _F(C)-4g \log (2\pi )\). Then the \(\lambda \)-invariant \(\lambda (C)\) of

*C*is defined to be the real number

## 4 Zhang’s formulae for the height of the Gross–Schoen cycle

The non-archimedean and archimedean \(\varphi \)- and \(\lambda \)-invariants as introduced in the previous two sections occur in [49] in formulae relating the height of a Gross–Schoen cycle on a curve over a global field with more traditional invariants, namely the self-intersection of the relative dualizing sheaf, and the Faltings height, respectively. The purpose of this section is to recall these formulae. In view of our applications, we will be solely concerned here with the number field case.

Let *k* be a number field and let *X* be a smooth projective geometrically connected curve of genus \(g \ge 2\) defined over *k*. Let \(\alpha \in {\text {Div}}^1 X\) be a divisor of degree one on *X*. Following [49], Sect. 1.1] we have an associated Gross–Schoen cycle \(\Delta _{\alpha }\) in the rational Chow group \({\text {CH}}^2(X^3)_\mathbb {Q}\). The cycle \(\Delta _\alpha \) is homologous to zero, and has by [18] a well-defined Beilinson–Bloch height \(\langle \Delta _\alpha ,\Delta _\alpha \rangle \in \mathbb {R}\). The height \(\langle \Delta _\alpha ,\Delta _\alpha \rangle \) vanishes if \(\Delta _\alpha \) is rationally equivalent to zero.

Assume now that *X* has semistable reduction over *k*. Let \(\hat{\omega }\) denote the admissible relative dualizing sheaf of *X* from [48], viewed as an adelic line bundle on *X*. Let \( \langle \hat{\omega },\hat{\omega } \rangle \in \mathbb {R}\) be its self-intersection as in [48]. Let \(O_k\) be the ring of integers of *k*. Denote by \(M(k)_0\) the set of finite places of *k*, and by \(M(k)_\infty \) the set of complex embeddings of *k*. We set \(M(k) = M(k)_0 \sqcup M(k)_\infty \). For \(v \in M(k)_0\) we set *Nv* to be the norm of the residue field of \(O_k\) at *v*, and for \(v \in M(k)_\infty \) we set \(Nv=1\).

Let \(S={\text {Spec}}O_k\) and let \(f :\mathcal {X}\rightarrow S\) denote the stable model of *X* over *S*. For \(v \in M(k)_0\) we denote by \(\varphi (X_v)\) the \(\varphi \)-invariant of the pm-graph of genus *g* canonically associated to the base change of \(f :\mathcal {X}\rightarrow S\) along the inclusion \(O_k \rightarrow O_{k,v}\). For \(v \in M(k)_\infty \) we denote by \(\varphi (X_v)\) the \(\varphi \)-invariant of the compact and connected Riemann surface \(X_v = X \otimes _v \mathbb {C}\) of genus *g*.

Let \(x_\alpha \) be the class of the divisor \(\alpha - K_X/(2g-2)\) in \({\text {Pic}}^0(X)_\mathbb {Q}\), where \(K_X\) is a canonical divisor on *X*. Let \(\hat{\mathrm {h}}\) denote the canonical Néron-Tate height on \({\text {Pic}}^0(X)_\mathbb {Q}\). With these notations Zhang has proved the following identity [49], Theorem 1.3.1].

### Theorem 4.1

*X*be a smooth projective geometrically connected curve of genus \(g \ge 2\) defined over the number field

*k*. Let \(\alpha \in {\text {Div}}^1 X\) be a divisor of degree one on

*X*, and assume that

*X*has semistable reduction over

*k*. Then the equality

*X*, the height \(\langle \Delta _\alpha ,\Delta _\alpha \rangle \) attains its minimum precisely when \(x_\alpha \) is zero in \({\text {Pic}}^0(X)_\mathbb {Q}\). We refer to \(\Delta _\alpha \) where \(x_\alpha \) is zero as a

*canonical*Gross–Schoen cycle. Also, by Theorem 4.1, the non-negativity of the height of a canonical Gross–Schoen cycle (as predicted by standard arithmetic conjectures of Hodge Index type [16]) is equivalent to the lower bound

We recall that the strict inequality \(\langle \hat{\omega },\hat{\omega } \rangle > 0\) is equivalent to the Bogomolov conjecture for *X*, canonically embedded in its jacobian. A conjecture by Zhang [49], Conjecture 4.1.1], proved by Cinkir [11], Theorem 2.9], implies that for \(v \in M(k)_0\) one has \(\varphi (X_v) \ge 0\). As \(\varphi (X_v)>0\) for \(v \in M(k)_\infty \) we find that the right hand side of (4.1) is strictly positive. Hence, the non-negativity of the height of a canonical Gross–Schoen cycle implies the Bogomolov conjecture for *X*.

*X*.

*S*. Its arithmetic degree is given by choosing a non-zero rational section

*s*of \(\mathcal {L}\) and by setting

*s*, by the product formula. As before let \(f :\mathcal {X}\rightarrow S\) denote the stable model of

*X*over

*S*. Let \(\omega _{\mathcal {X}/S}\) denote the relative dualizing sheaf on \(\mathcal {X}\). We endow the line bundle \(\det f_*\omega _{\mathcal {X}/S}\) on

*S*with the metrics \(\Vert \cdot \Vert _{\mathrm {Hdg}, v}\) at the infinite places determined by the inner product in (3.1). The resulting metrized line bundle is denoted \(\det f_*\bar{\omega }_{\mathcal {X}/S}\). Its arithmetic degree \(\deg \det f_*\bar{\omega }_{\mathcal {X}/S}\) is the (non-normalized) stable Faltings height of

*X*.

*g*.

*X*are related by the identity

### Corollary 4.2

*X*be a smooth projective geometrically connected curve of genus \(g \ge 2\) defined over the number field

*k*. Assume that

*X*has semistable reduction over

*k*. Let \(\Delta \in {\text {CH}}^2(X^3)_\mathbb {Q}\) be a canonical Gross–Schoen cycle on \(X^3\). Then the equality

## 5 The \(\lambda \)-invariants for some pm-graphs

The purpose of this section is to display the \(\lambda \)-invariants of a few pm-graphs that we will encounter in the sequel. We refer to the papers [9–11] by Cinkir for an extensive study of the \(\varphi \)- and \(\lambda \)-invariants of pm-graphs. The reference [9] focuses in particular on pm-graphs of genus three.

*r*(

*p*,

*q*) denote the effective resistance between points \(p, q \in \Gamma \). Fix a point \(p \in \Gamma \). We then put

By [8], Lemma 2.16] the number \(\tau (\Gamma )\) is independent of the choice of \(p \in \Gamma \). It is readily verified that for a circle \(\Gamma \) of length \(\delta (\Gamma )\) we have \(\tau (\Gamma )=\frac{1}{12}\delta (\Gamma )\), and for a line segment of length \(\delta (\Gamma )\) we have \(\tau (\Gamma )=\frac{1}{4}\delta (\Gamma )\). The \(\tau \)-invariant is an additive invariant.

*g*, with vertex set

*V*, and canonical divisor

*K*. We set

### Proposition 5.1

*g*. Then the equality

### Proof

See [11], Corollary 4.4]. \(\square \)

We will need the following particular cases.

### Example 5.2

Let \(\overline{\Gamma }\) be a pm-graph of genus *g* consisting of one vertex of genus \(g-1\) and with one loop attached of length \(\delta (\Gamma )\). Then \(\tau (\Gamma )=\frac{1}{12}\delta (\Gamma )\), \(\theta (\overline{\Gamma })=0\) and hence \((8g+4)\lambda (\overline{\Gamma })=g\,\delta (\Gamma )\).

### Example 5.3

Let \(\overline{\Gamma }\) be a pm-graph of genus *g* consisting of two vertices of genera *h* and \(g-h\) joined by one edge of length \(\delta (\Gamma )\). Then \(\tau (\Gamma )=\frac{1}{4}\delta (\Gamma )\), \(\theta (\overline{\Gamma })=2(2h-1)(2g-2h-1)\delta (\Gamma )\) and hence \((8g+4)\lambda (\overline{\Gamma })=4h(g-h)\delta (\Gamma )\).

### Example 5.4

*g*. Then we have

### Example 5.5

*g*consisting of two vertices of genera

*h*and \(g-h-1\) and joined by two edges of weights \(m_1, m_2\). We have

## 6 Algebraic and analytic modular forms

*g*, and denote by \(p :\mathcal {U}_g \rightarrow \mathcal {A}_g\) the universal abelian variety. Let \(\Omega _{\mathcal {U}_g/\mathcal {A}_g}\) denote the sheaf of relative 1-forms of

*p*. Then we have the Hodge bundle \(\mathcal {E}=p_* \Omega _{\mathcal {U}_g/\mathcal {A}_g}\) and its determinant \(\mathcal {L}= \det p_* \Omega _{\mathcal {U}_g/\mathcal {A}_g}\) on \(\mathcal {A}_g\). Kodaira-Spencer deformation theory gives a canonical isomorphism

*R*and all \(h \in \mathbb {Z}_{\ge 0}\) we let

*R*-module of algebraic Siegel modular forms of degree

*g*and weight

*h*.

*g*. We have a natural uniformization map \(u :\mathbb {H}_g \rightarrow \mathcal {A}_g(\mathbb {C})\) and hence a universal abelian variety \(\tilde{p} :\mathbb {U}_g \rightarrow \mathbb {H}_g\) over \(\mathbb {H}_g\). The Hodge bundle \(\tilde{\mathcal {E}}=\tilde{p}_*\Omega _{\mathbb {U}_g/\mathbb {H}_g}\) over \(\mathbb {H}_g\) has a standard trivialization by the frame \((\mathrm {d}\zeta _1/\zeta _1,\ldots , \mathrm {d}\zeta _g /\zeta _g) = (2\pi i \,\mathrm {d}z_1,\ldots , 2\pi i \,\mathrm {d}z_g)\), where \(\zeta _i = \exp (2\pi i z_i)\). In particular, the determinant of the Hodge bundle \(\tilde{\mathcal {L}} = \det \tilde{\mathcal {E}}\) is trivialized by the frame \(\omega = \frac{\mathrm {d}\zeta _1}{\zeta _1} \wedge \ldots \wedge \frac{\mathrm {d}\zeta _g}{\zeta _g} = (2\pi i)^g (\mathrm {d}z_1 \wedge \ldots \wedge \mathrm {d}z_g)\). Let \(\mathcal {R}_{g,h}\) denote the usual \(\mathbb {C}\)-vector space of analytic Siegel modular forms of degree

*g*and weight

*h*. Then the map

*g*, then we have the identity

*g*. The Torelli map \(t :\mathcal {M}_g \rightarrow \mathcal {A}_g\) gives rise to the bundles \(t^*\mathcal {E}\) and \(t^*\mathcal {L}\) on \(\mathcal {M}_g\). Let \(\pi :\mathcal {C}_g \rightarrow \mathcal {M}_g\) denote the universal curve of genus

*g*, and denote by \(\Omega _{\mathcal {C}_g/\mathcal {M}_g}\) its sheaf of relative 1-forms. Then we have locally free sheaves \(\mathcal {E}_\pi = \pi _* \Omega _{\mathcal {C}_g/\mathcal {M}_g}\) and \(\mathcal {L}_\pi = \det \mathcal {E}_\pi \) on \(\mathcal {M}_g\), and natural identifications \(\mathcal {E}_\pi \xrightarrow {\sim }t^*\mathcal {E}\) and \(\mathcal {L}_\pi \xrightarrow {\sim }t^*\mathcal {L}\). Kodaira-Spencer deformation theory gives a canonical isomorphism

Over \(\mathbb {C}\), the pullback of the Hodge metric \(\Vert \cdot \Vert _{\mathrm {Hdg}}\) to \(\mathcal {L}_\pi \) coincides with the metric derived from the inner product (3.1) introduced before.

Let \(\overline{\mathcal {M}}_g \supset \mathcal {M}_g\) denote the moduli stack of stable curves of genus *g*, and consider the universal stable curve \(\bar{\pi } :\overline{\mathcal {C}}_g \rightarrow \overline{\mathcal {M}}_g\). Let \(\omega _{\overline{\mathcal {C}}_g/\overline{\mathcal {M}}_g}\) be the relative dualizing sheaf of \(\bar{\pi }\), and put \(\mathcal {E}_{\bar{\pi }}= \bar{\pi }_*\omega _{\overline{\mathcal {C}}_g/\overline{\mathcal {M}}_g}\) and \(\mathcal {L}_{\bar{\pi }}= \det \mathcal {E}_{\bar{\pi }}\). Then \(\mathcal {E}_{\bar{\pi }}\) resp. \(\mathcal {L}_{\bar{\pi }}\) are natural extensions of \(\mathcal {E}_\pi \) resp. \(\mathcal {L}_\pi \) over \(\overline{\mathcal {M}}_g\). When *S* is a scheme or analytic space and \(f :\mathcal {X}\rightarrow S\) is a stable curve of genus *g*, we usually denote by \(\mathcal {E}_f = f_* \omega _{\mathcal {X}/S}\) and \(\mathcal {L}_f = \det f_* \omega _{\mathcal {X}/S}\) the sheaves on *S* induced from \(\mathcal {E}_{\bar{\pi }}\) and \(\mathcal {L}_{\bar{\pi }}\) by the classifying map \(J :S \rightarrow \overline{\mathcal {M}}_g\) associated to *f*.

### Lemma 6.1

*f*is smooth over \(\mathbb {D}^*\). Let

*s*be a Siegel modular form over \(\mathbb {D}^*\) of degree

*g*and weight

*h*. Let \(\Gamma \subset {\text {Sp}}(2g,\mathbb {Z})\) denote the image of the monodromy representation \(\rho :\pi _1(\mathbb {D}^*) \rightarrow {\text {Sp}}(2g,\mathbb {Z})\) induced by

*f*, and let \(\Omega :\mathbb {D}^* \rightarrow \mathbb {H}_g/\Gamma \) denote the induced period map. Then the frame \( \Omega ^*(\mathrm {d}z_1 \wedge \ldots \wedge \mathrm {d}z_g) \) of \(\mathcal {L}_f|_{\mathbb {D}^*}\) extends as a frame of \(\mathcal {L}_f\) over \(\mathbb {D}\). Furthermore, we have the asymptotics

*O*(1) if the special fiber \(X_0\) is a stable curve of compact type.

### Proof

By (6.2) we have \( \log \Vert \Omega ^*( \mathrm {d}z_1 \wedge \ldots \wedge \mathrm {d}z_g) \Vert _\mathrm {Hdg}(t) =\frac{1}{2} \log \det \mathrm {Im}\,\Omega (t) \) for all \(t \in \mathbb {D}^*\). By the Nilpotent Orbit Theorem there exists an element \(c \in \mathbb {Z}_{\ge 0}\) such that \( \det \mathrm {Im}\,\Omega (t) \sim - c \log |t| \) as \(t \rightarrow 0\). We conclude that \(\Omega ^*( \mathrm {d}z_1 \wedge \ldots \wedge \mathrm {d}z_g) \) extends as a frame of Mumford’s canonical extension [42] of \(\mathcal {L}_f|_{\mathbb {D}^*}\) over \(\mathbb {D}\). By [13], p. 225] this canonical extension is equal to \(\mathcal {L}_f\). We thus obtain the first assertion. Also we obtain the equality \({\text {ord}}_0(s,\mathcal {L}_f)={\text {ord}}_0(\tilde{s})\), which then leads to the asymptotic \(-\log |\tilde{s}| \sim -{\text {ord}}_0(s,\mathcal {L}_f) \log |t|\) as \(t \rightarrow 0\). Combining with (6.3) we find the stated asymptotics for \(-\log \Vert s\Vert _\mathrm {Hdg}\). The element \(c \in \mathbb {Z}_{\ge 0}\) vanishes if the special fiber \(X_0\) is a stable curve of compact type. This proves the last assertion. \(\square \)

## 7 Asymptotics of the biextension metric

In this section we continue the spirit of the asymptotic analysis from Lemma 6.1 by replacing the Hodge metric \(\Vert \cdot \Vert _{\mathrm {Hdg}}\) with the biextension metric \(\Vert \cdot \Vert _\mathcal {B}\). We recall the necessary ingredients, and finish with a specific asymptotic result due to Brosnan and Pearlstein [6]. General references for this section are [20, 21] and [22]. We continue to work in the analytic category.

Let \(g \ge 2\) be an integer. Let *H* denote the standard local system of rank 2*g* over \(\mathcal {M}_g\). Following Hain and Reed in [21] we have a canonical normal function section \(\nu :\mathcal {M}_g \rightarrow J\) of the intermediate jacobian \(J=J(\bigwedge ^3 H/H)\) over \(\mathcal {M}_g\), given by the Abel-Jacobi image of a Ceresa cycle on the usual jacobian *J*(*H*).

Let \(\mathcal {B}\) denote the natural biextension line bundle on *J*, equipped with its natural biextension metric [22]. By pulling back along the section \(\nu :\mathcal {M}_g \rightarrow J\) we obtain a natural line bundle \( \mathcal {N}=\nu ^* \mathcal {B}\) over \(\mathcal {M}_g\), equipped with the pullback metric from \(\mathcal {B}\). By functoriality we obtain a canonical smooth hermitian line bundle \(\mathcal {N}\) on the base of any family \(\rho :\mathcal {C}\rightarrow B\) of smooth complex curves of genus *g*.

As it turns out, the underlying line bundle of \(\mathcal {N}\) on \(\mathcal {M}_g\) is isomorphic with \(\mathcal {L}_\pi ^{\otimes 8g+4}\), where \(\mathcal {L}_\pi = \det \mathcal {E}_\pi \) is the determinant of the Hodge bundle as before. An isomorphism \(\mathcal {N}\xrightarrow {\sim }\mathcal {L}_\pi ^{\otimes 8g+4}\) is determined up to a constant depending on *g*, and by transport of structure we obtain a smooth hermitian metric \(\Vert \cdot \Vert _\mathcal {B}\) on \(\mathcal {L}_\pi \), well-defined up to a constant, that we will ignore from now on.

*s*be a non-zero rational section of \( \mathcal {L}_\pi ^{\otimes (8g+4)a}\) over \(\mathcal {M}_g\). Consider then the quantity

Here we discuss a set-up to study this question. Consider a base complex manifold *B* and a stable curve \(\rho :\mathcal {C}\rightarrow B\) of genus \(g \ge 2\) smooth over an open subset \(U\subset B\). We then have the canonical Hain–Reed line bundle \(\mathcal {N}\) on *U*, equipped with its natural metric. Assume that the boundary \(D=B \setminus U\) of *U* in *B* is a normal crossings divisor. D. Lear’s extension result [37], see also [20], Corollary 6.4] and [43], Theorem 5.19], implies that there exists a \(\mathbb {Q}\)-line bundle \([\mathcal {N},B]\) over *B* extending the line bundle \(\mathcal {N}\) on *U* in such a way that the metric on \(\mathcal {N}\) extends continuously over \([\mathcal {N},B]\) away from the singular locus of *D*. This property uniquely determines the \(\mathbb {Q}\)-line bundle \([\mathcal {N},B]\).

*D*be the origin of \(\mathbb {D}\) and let \(f :X \rightarrow \mathbb {D}\) be a stable curve smooth over \(\mathbb {D}^*=B\setminus D\). Let

*t*be the standard coordinate on \(\mathbb {D}\). The existence of the Lear extension \([\mathcal {N},\mathbb {D}]\) implies that there exists a rational number

*b*such that the asymptotics

*c*such that

*c*.

*X*is smooth one has that

*r*nodes the graph \(\Gamma \) has

*r*designated edges with weights equal to the thicknesses \((m_1,\ldots ,m_r)\) of the nodes on the total space

*X*. Let \(\lambda (\overline{\Gamma })\) be the \(\lambda \)-invariant of \(\overline{\Gamma }\). In general one expects that the asymptotic

We can characterize though when this asymptotics holds in terms of the classifying map \(I :\mathbb {D}\rightarrow B\) to the universal deformation space of \(X_0\), see Proposition 7.4 below. We hope that the criterion in Proposition 7.4 will be useful to prove the asymptotic in (7.4) in general. In the present paper, we are able to verify the criterion in a special case.

The proof of the following lemma is left to the reader.

### Lemma 7.1

Denote by \([\mathcal {N},\mathbb {D}]\) the Lear extension of the Hain-Reed line bundle \(\mathcal {N}\) over \(\mathbb {D}\). Suppose \(e \in \mathbb {Z}_{>0}\) is such that \([\mathcal {N},\mathbb {D}]^{\otimes e}\) is a line bundle on \(\mathbb {D}\). Denote this line bundle by *N*. Let *s* be a generating section of \(\mathcal {N}\) over \(\mathbb {D}^*\) and let \(k \in \mathbb {Z}\). The following assertions are equivalent: (a) the asymptotic \(-e \log \Vert s\Vert _\mathcal {B}\sim -k \log |t|\) holds as \(t \rightarrow 0\). (b) the section \(t^{-k} \cdot s^{\otimes e} \) extends as a generating section of the line bundle *N* over \(\mathbb {D}\). (c) the divisor of \(s^{\otimes e}\), when viewed as a rational section of *N*, is equal to \(k \cdot [0]\).

*U*. Let

*s*be a rational section of the line bundle \(\mathcal {L}_\rho ^{\otimes (8g+4)a}\) such that \(\phi ^*s\) has no zeroes or poles on \(\mathbb {D}^*\). Let \(q(m_1,\ldots ,m_r) \in \mathbb {Q}\) for all \(m=(m_1,\ldots , m_r) \in \mathbb {Z}_{> 0}^r\) be determined by the asymptotic

*q*is a rational homogeneous weight one function of \(m_1,\ldots , m_r\) which extends continuously over \(\mathbb {R}_{\ge 0}^r\). Write \(q_i = q(e_i)\) where \(e_i\) is the

*i*-th coordinate vector in \(\mathbb {R}^r\). Let \(D_i\) for \(i=1,\ldots ,r\) denote the divisor on

*B*given by the equation \(z_i=0\). Then for a holomorphic arc \(\bar{\psi } :\mathbb {D}\rightarrow B\) intersecting \(D_i\) transversally and intersecting none of the \(D_j\) where \(j \ne i\) we have the asymptotic

*B*. Applying part (c) of Lemma 7.1 we find the following.

### Lemma 7.2

The \(\mathbb {Q}\)-divisor of *s*, when seen as a rational section of \([\mathcal {N},B]^{\otimes a}\), is given by the \(\mathbb {Q}\)-divisor \(a\sum _{i=1}^r q_i D_i\).

*height jump*for the map \(\bar{\phi }\) is defined to be the rational homogeneous weight one function

*s*. It was conjectured by Hain [20], Conjecture 14.6] and proved by Brosnan and Pearlstein in [6] (combine [6], Corollary 11] and [6], Theorem 20]) and independently by J. I. Burgos Gil, D. Holmes and the author in [7], Theorem 4.1] that \(j\ge 0\). Note that if for some \(\bar{\phi } :\mathbb {D}\rightarrow B\) as above the height jump is strictly positive, no positive tensor power of the Hain–Reed line bundle \(\mathcal {N}\) on

*U*extends as a continuously metrized line bundle over

*B*.

Now assume that \(\rho :\mathcal {C}\rightarrow B\) is the universal deformation of the special fiber \(X_0\) of our stable curve \(f :X \rightarrow \mathbb {D}\). Recall [1], Sect. XI.6] that the base space *B* is a complex manifold, carrying an action of \({\text {Aut}}(X_0)\), and endowed with a canonical point \(b_0\) with fiber \(X_0\). Locally around the point \(b_0 \in B\) the divisor of singular curves in *B* is a normal crossings divisor. Hence, locally around \(b_0\) the family \(\mathcal {C}\rightarrow B\) can be identified with a stable curve over \(\mathbb {D}^d\) smooth over \((\mathbb {D}^*)^r \times \mathbb {D}^{d-r}\), for some integers *d*, *r*. Then for the classifying map \(I :\mathbb {D}\rightarrow B\) one has for \(i=1,\ldots ,r\) that \({\text {mult}}_0 I^*z_i = m_i\), where \(m_1,\ldots ,m_r\) are the thicknesses of the nodes of \(X_0\) on *X*. We find in particular a height jump \(j(m_1,\ldots ,m_r) \in \mathbb {Q}\) associated to *I*.

*slope*to be the invariant

### Lemma 7.3

*f*and let

*j*be the height jump (7.7) for the classifying map \(I :\mathbb {D}\rightarrow B\) to the universal deformation space

*B*of the special fiber \(X_0\). Let \(a \in \mathbb {Z}_{>0}\) and let

*s*be a rational section of \(\mathcal {L}_f^{\otimes (8g+4)a}\) such that

*s*has no zeroes or poles on \(\mathbb {D}^*\). Then the asymptotics

### Proof

*s*, and hence we may assume without loss of generality that

*s*is the pullback along

*I*of a rational section of \(\mathcal {L}_\rho ^{\otimes (8g+4)a}\), where \(\rho :\mathcal {C}\rightarrow B\) is the universal deformation of \(X_0\). Let \(m_1,\ldots ,m_r\) be the multiplicities at \(0 \in \mathbb {D}\) of the analytic branches through \(b_0 \in B\) determined by the locus of singular curves in

*B*. Then one has the asymptotics

*s*when seen as a rational section of \([\mathcal {N},B]^{\otimes a}\) is equal to \(a\sum _{i=1}^r q_iD_i\) where \(D_i\) for \(i=1,\ldots ,r\) denotes the divisor on

*B*given by \(z_i=0\). Since \({\text {mult}}_0 I^*z_i = m_i\) for \(i=1,\ldots ,r\) it follows that

We deduce the following criterion to verify whether (7.4) holds.

### Proposition 7.4

*X*are equal.

### Proof

The equivalence of (a) and (b) follows from Lemma 6.1. The equivalence of (b) and (c) follows from Lemma 7.3. \(\square \)

Now we have the following two results, that allow us to verify condition (c) in a special case.

### Theorem 7.5

*h*, one of genus \(g-h-1\), joined at two points. Then the height jump

*j*for the classifying map \(I :\mathbb {D}\rightarrow B\) to the universal deformation space of \(X_0\) is equal to

*B*.

### Proof

This follows from the calculation done in the proof of [6], Theorem 241]. We note that [6], Theorem 241] is about a stable curve *C* in \(\overline{\mathcal {M}}_g\) consisting of two smooth irreducible components joined in two points, and with trivial automorphism group, so that the statement of [6], Theorem 241] does not apply immediately to our setting if \(X_0\) has non-trivial automorphisms. However the calculation in the proof of [6], Theorem 241] is carried out effectively on the universal deformation space of *C*. Under the assumption that \({\text {Aut}}(C)\) is trivial, this deformation space maps locally isomorphically to \(\overline{\mathcal {M}}_g\). Now the calculation in the proof of [6], Theorem 241] on the universal deformation space of *C* puts no particular restrictions on \({\text {Aut}}(C)\), and we conclude that the expression for the height jump in [6], Theorem 241] is valid in our setting. \(\square \)

### Proposition 7.6

*g*consisting of two vertices of genera

*h*and \(g-h-1\) and joined by two edges of weights \(m_1, m_2\). Then the slope of \(\overline{\Gamma }\) is equal to

We observe that the height jump in Theorem 7.5 and the slope in Proposition 7.6 are equal. With Proposition 7.4 we thus obtain the following result.

### Corollary 7.7

*h*, one of genus \(g-h-1\), joined at two points. Then one has the asymptotics

We will use (7.11) with \(g=3\) and \(h=1\) for the proof of our main result.

## 8 The modular form \(\chi _{18}\)

*q*-expansion principle, cf. [13], p. 140], the modular form \(\chi _{18}\) is defined over \(\mathbb {Z}\), that is, we have a unique element in \(\mathcal {S}_{3,18}(\mathbb {Z})\) whose base change to \(\mathbb {C}\) is equal to \(\chi _{18}\). By a slight abuse of notation we also denote this element by \(\chi _{18}\). By [25], Proposition 3.4] the modular form \(\chi '_{18}=2^{-28} \chi _{18}\) is primitive, i.e. not zero modulo

*p*for all primes

*p*.

We recall that one has a natural structure of reduced effective Cartier divisor on the locus *H* of hyperelliptic curves in \(\mathcal {M}_3\). The following result seems to be well known.

### Proposition 8.1

The divisor of \(\chi '_{18}\) on \(\mathcal {M}_3\) equals 2*H*.

### Proof

Over \(\mathbb {C}\) this follows from (the proof of) [45], Theorem 1]. Recall that \(\mathcal {M}_3\) is smooth over \({\text {Spec}}(\mathbb {Z})\) with geometrically connected fibers. The primitivity of \(\chi '_{18}\) then gives the statement over \(\mathbb {Z}\). \(\square \)

*S*be a scheme. When \(f :\mathcal {X}\rightarrow S\) is a stable curve of genus three we can view \(\chi '_{18}\) as a rational section of the line bundle \(\mathcal {L}_f^{\otimes 18}\) on

*S*. In particular, let

*k*be a number field with ring of integers \(O_k\), and let

*X*be a non-hyperelliptic genus three curve with semistable reduction over

*k*. Let \(f :\mathcal {X}\rightarrow S={\text {Spec}}O_k\) denote the stable model of

*X*over

*k*. From Proposition 8.1 we obtain that \(\chi _{18}'\) is generically non-vanishing on

*S*, and from (4.2) we obtain the formula

*X*.

Combining with Corollary 4.2 we deduce the following result.

### Theorem 8.2

*X*be a non-hyperelliptic genus three curve with semistable reduction over the number field

*k*. Let \(f :\mathcal {X}\rightarrow {\text {Spec}}O_k\) denote the stable model of

*X*over

*k*and view \(\chi ^{\prime }_{18}\) as a rational section of the line bundle \(\mathcal {L}_f^{\otimes 18}\). Then the height of a canonical Gross–Schoen cycle \(\Delta \) on \(X^3\) satisfies

We will take Theorem 8.2 as a starting point in our proof of Theorem A.

## 9 The Horikawa index

Let \(S = {\text {Spec}}R\) be the spectrum of a discrete valuation ring *R*. Let \(f :\mathcal {X}\rightarrow S\) be a stable curve with generic fiber smooth and non-hyperelliptic of genus three. Denote by *v* the closed point of *S*. As above we view \(\chi '_{18}\) as a rational section of the line bundle \(\mathcal {L}_f^{\otimes 18}\) on *S*. Then \(\chi '_{18}\) is generically non-vanishing by Proposition 8.1. The aim of this section is to give a lower bound on the multiplicity \({\text {ord}}_v (\chi '_{18})\) in terms of the reduction graph of the special fiber. The result is displayed in Corollary 9.5. We start by writing down an expression for the divisor \({\text {div}}(\chi '_{18})\) of \(\chi '_{18}\) on the moduli stack \(\overline{\mathcal {M}}_3\).

*S*be a scheme and let \(f :\mathcal {X}\rightarrow S\) be a stable curve of genus three. Then on

*S*we have the locally free sheaves \( \mathcal {E}_f = f_* \omega _{\mathcal {X}/S} \) and \( \mathcal {G}_f = f_* \omega ^{\otimes 2}_{\mathcal {X}/S} \) as well as a natural map

*f*. The map \(\nu _f\) is surjective if

*f*is smooth and nowhere hyperelliptic. Both \({\text {Sym}}^2 \mathcal {E}_f\) and \(\mathcal {G}_f\) have rank six and we thus we have a natural map of invertible sheaves

*f*. We may and do view \(\det \nu _f\) as a global section \(s_f\) of the invertible sheaf \( (\det {\text {Sym}}^2 \mathcal {E}_{f})^{\otimes -1} \otimes \det \mathcal {G}_f \) on

*S*. It has support on the locus of hyperelliptic fibers on

*S*. Standard multilinear algebra yields a canonical isomorphism

*S*, where \(\mathcal {L}_f = \det \mathcal {E}_f\) as before, and this shows that we may as well view \(s_f\) as a global section of the invertible sheaf \( \mathcal {L}_{f}^{\otimes - 4} \otimes \det \mathcal {G}_{f} \) on

*S*.

### Proposition 9.1

Let \(\pi :\mathcal {C}_3 \rightarrow \mathcal {M}_3\) be the universal smooth curve of genus three. The section \(s_\pi \) is not identically equal to zero, and the divisor of \(s_\pi \) on \(\mathcal {M}_3\) is equal to the reduced hyperelliptic divisor *H*.

### Proof

*t*is finite. Let

*R*denote the ramification divisor of

*t*. By (6.1) and (6.4) we have canonical isomorphisms

*t*is ramified precisely along the hyperelliptic locus. As

*t*has generic degree two we find \(R=H\). Combining we obtain \(\Sigma = H\). \(\square \)

Let \(\bar{\pi } :\overline{\mathcal {C}}_3 \rightarrow \overline{\mathcal {M}}_3\) be the universal stable curve of genus three. Let *K* denote the divisor of *s* on \(\overline{\mathcal {M}}_3\) and let \(\Delta \) denote the divisor of singular curves on \(\overline{\mathcal {M}}_3\).

### Proposition 9.2

### Proof

*H*. We deduce that \(\chi '_{18} \otimes s^{\otimes -2}\) is a trivializing section of the invertible sheaf \(\mathcal {L}_\pi ^{\otimes 26} \otimes (\det \mathcal {G}_\pi )^{\otimes -2}\) over \(\mathcal {M}_3\). Mumford’s functorial Riemann-Roch, see [40], Théorème 2.1 and equation (2.1.2)], restricted to \(\mathcal {M}_3\) gives a canonical isomorphism

*R*. Let \(f :\mathcal {X}\rightarrow S\) be a stable curve with generic fiber smooth and non-hyperelliptic of genus three. The morphism \(\nu :{\text {Sym}}^2 \mathcal {E}_f \rightarrow \mathcal {G}_f \) is surjective at the generic point, hence is globally injective. Let \(\mathcal {Q}_f\) denote the cokernel of \(\nu \). Then \(\mathcal {Q}_f\) is a finite length \(\mathcal {O}_S\)-module, and we have an exact sequence of coherent sheaves on

*S*with canonical maps,

*v*denote the closed point of

*S*. Following Reid [44], Konno [35] and Yamaki [46, 47] we call the integer \({\text {length}}_{\mathcal {O}_{S}} \mathcal {Q}_f\) the

*Horikawa index*of

*f*at

*v*, notation \(\mathrm {Ind}_v(f)\). Let \(\Gamma _v\) denote the metrized graph associated to the stable curve

*f*.

### Proposition 9.3

*S*. Then the equality

### Proof

*S*a functorial invertible sheaf \(\det \mathcal {F}\) on

*S*, by using locally free resolutions. From the locally free resolution (9.1) of \(\mathcal {Q}_f\) we obtain a canonical isomorphism \((\det {\text {Sym}}^2 \mathcal {E}_f)^{\otimes -1} \otimes \det \mathcal {G}_f \xrightarrow {\sim }\det \mathcal {Q}_f\) of invertible sheaves, and we find that \(s=\det \nu \) can be viewed as a canonical non-zero global section of \(\det \mathcal {Q}_f\). By Proposition 9.2 its divisor

*K*satisfies the relation \({\text {div}}(\chi '_{18}) = 2\,K + 2\,\Delta \) in \({\text {Div}}(S)\). This gives the identity \( {\text {ord}}_v (\chi '_{18}) = 2 {\text {ord}}_v(s) + 2\,\delta (\Gamma _v)\). We are thus left to prove that \( \mathrm {Ind}_v(f)={\text {ord}}_v(s)\). By the structure theorem for finitely generated

*R*-modules we can find effective Cartier divisors \(K_i\) on

*S*uniquely determined by \(\mathcal {Q}_f\) together with a decomposition \(\mathcal {Q}_f = \bigoplus _i \mathcal {O}/\mathcal {O}(-K_i)\) of \(\mathcal {Q}_f\) as a direct sum of cyclic modules. The exact sequences

*S*. This implies

Let \(\overline{\Gamma }=(\Gamma ,\mathbf {q})\) be a pm-graph of genus three. From [46, 47] we recall the notion of a pair of edges of *h*-type on \(\overline{\Gamma }\), and the definition of an invariant \(h(\overline{\Gamma })\). We call a vertex \(v \in V(\overline{\Gamma })\) *eliminable* if *v* has valence two and satisfies \(\mathbf {q}(v)=0\). We assume that \(\overline{\Gamma }\) has no eliminable vertices. Then for \(e, e'\) two distinct edges of \(\overline{\Gamma }\) we denote by \(\overline{\Gamma }^{ \{e,e'\} }\) the contraction of all edges except \(e, e'\) on \(\overline{\Gamma }\). The pair \(\{e,e'\}\) is called a *pair of h-type* on \(\overline{\Gamma }\) if \(\overline{\Gamma }^{ \{e,e'\} }\) is an irreducible graph with precisely two vertices, and the induced polarization \(\mathbf {q}^{ \{e,e'\} }\) takes value 1 on both vertices. It can be shown [46], Lemma 2.1] that \(\overline{\Gamma }\) has at most one pair of edges of *h*-type.

Now, if \(\overline{\Gamma }\) has a pair \(\{e, e' \}\) of edges of *h*-type, then we set \(h(\overline{\Gamma }) = \min \{ m_1, m_2 \}\), where \(m_1, m_2\) are the weights of the edges \(e, e'\). If \(\overline{\Gamma }\) has no pair of edges of *h*-type, then we set \(h(\overline{\Gamma }) =0\).

We continue to work with the spectrum \(S={\text {Spec}}R\) of a discrete valuation ring *R* and a stable curve \(f :\mathcal {X}\rightarrow S\) of genus three, whose generic fiber is smooth and non-hyperelliptic. Let \(\overline{\Gamma }_v\) denote the pm-graph associated to *f*. Note that \(\overline{\Gamma }_v\) has no eliminable vertices. Let \(h(\overline{\Gamma }_v)\) be its *h*-invariant as above. Let \(e, e'\) be nodes of \(\mathcal {X}_v\). It is easy to see that the corresponding pair \(\{ e,e' \}\) of edges in \(\overline{\Gamma }_v\) is a pair of edges of *h*-type if and only if both \(e, e'\) are of type 0, and the partial normalization of \(\mathcal {X}_v\) at \(\{e ,e'\}\) has exactly two connected components, both of genus one.

### Proposition 9.4

### Proof

This is [47], Proposition 3.7]. \(\square \)

Combining Propositions 9.3 and 9.4 we find

### Corollary 9.5

We saw in Sect. 8 that one has a natural structure of reduced effective Cartier divisor on the hyperelliptic locus *H* in \(\mathcal {M}_3\). Let \(\overline{H}\) be the closure of *H* in \(\overline{\mathcal {M}}_3\). Then as \(\overline{\mathcal {M}}_3\) is smooth over \({\text {Spec}}\mathbb {Z}\) (see [33], Theorem 2.7]), one has a natural structure of reduced effective Cartier divisor on \(\overline{H}\). Not surprisingly, the Horikawa index at *v* can be directly expressed in terms of the multiplicity of \(\overline{H}\) at *v*. We do not need the next result, but we would like to mention it for completeness.

### Proposition 9.6

### Proof

*H*. To obtain the multiplicities of \(\chi '_{18}\) along \(\Delta _0\), \(\Delta _1\) it suffices to compute the multiplicities of \(\tilde{\chi }_{18}\) along these two divisors, as \( \mathrm {d}z_1 \wedge \mathrm {d}z_2 \wedge \mathrm {d}z_3 \) is trivializing in \( \mathcal {L}_{\bar{\pi }}\) by Lemma 6.1. The multiplicities of \(\tilde{\chi }_{18}\) along \(\Delta _0\), \(\Delta _1\) can be computed by writing down explicitly the Fourier expansion of \(\tilde{\chi }_{18}\), see for example [26], p. 852] for the multiplicity along \(\Delta _1\). \(\square \)

### Remark 9.7

## 10 Proof of Theorem A

We can now combine all previous results in order to prove Theorem A.

### Theorem 10.1

Consider the sequence of curves \(C_n\) with \(n \in \mathbb {Z}_{>0}\), \(n \equiv 2\, ( \bmod \, 3)\) and \(n \not \equiv 0, 1 \, (\bmod \, 2^5)\). Then the height of a canonical Gross–Schoen cycle on \(C_n^3\) tends to infinity as \(n \rightarrow \infty \).

*n*as in Theorem 10.1 the curve \(C_n\) acquires semistable reduction over \(k_n\).

### Theorem 10.2

For *n* as in Theorem 10.1 and for \(v \in M(k_n)_0\) we have that the local non-archimedean contribution \((\frac{1}{18}{\text {ord}}_v (\chi '_{18}) -\lambda (C_{n,v}))\log Nv\) is non-negative.

### Corollary 10.3

*n*as in Theorem 10.1 the inequality

In Sect. 12 we will show

### Theorem 10.4

*c*such that for \(n \rightarrow \infty \) the asymptotics

Combining Corollary 10.3 and Theorem 10.4 one finds Theorem 10.1.

## 11 Proof of Theorem 10.2

Assume that *n* is as in Theorem 10.1. The reduction types of \(C_n\) at all \(v \in M(k_n)_0\) are given in [19]. At a prime of \(O_n\) not dividing \(n(n-1)\) the curve \(C_n\) has good reduction, by [19], Proposition 3.1]. At a prime of \(O_n\) dividing 2 we have by [19], Theorem 7.4] that the special fiber consists of a smooth genus zero component, with three disjoint elliptic curves attached to it. The dual graph of the special fiber is thus a polarized tree in this case. Finally, at an odd prime of \(O_n\) dividing \(n(n-1)\) the special fiber is the union of two elliptic curves meeting in two distinct points, by [19], Theorem 5.3]. Hence in this case the polarized dual graph of the special fiber consists of two vertices of genus one, joined by two edges.

We now analyze each of these various cases. Let \(S = {\text {Spec}}R\) be the spectrum of a discrete valuation ring *R*. Let \(f :\mathcal {X}\rightarrow S\) be a stable curve with generic fiber smooth and non-hyperelliptic of genus three. Let \(\overline{\Gamma }\) denote the polarized dual graph associated to *f*.

Theorem 10.2 is proved by the following three lemmas.

### Lemma 11.1

Assume that \(f :\mathcal {X}\rightarrow S\) is smooth. Then \(\frac{1}{18} {\text {ord}}_v (\chi '_{18}) -\lambda (\overline{\Gamma }) \ge 0\).

### Proof

We have \({\text {ord}}_v (\chi '_{18})= 2 {\text {mult}}_v H \ge 0\) by Proposition 8.1, and \(\lambda (\overline{\Gamma })=0\). \(\square \)

### Lemma 11.2

Assume that \(\Gamma \) is a tree. Then \(\frac{1}{18} {\text {ord}}_v (\chi '_{18}) -\lambda (\overline{\Gamma }) \ge \frac{1}{21} \delta (\Gamma ) \).

### Proof

### Lemma 11.3

*f*is the union of two elliptic curves meeting in two distinct points. Let \(m_1, m_2 \in \mathbb {Z}_{>0}\) be the thicknesses on \(\mathcal {X}\) of the two singular points of the special fiber. Then we have

### Proof

### Remark 11.4

## 12 Proof of Theorem 10.4

Specializing (7.11) from Corollary 7.7 to the case \(g=3\), \(h=1\) and working with a suitable power of \(\chi '_{18}\) we obtain the following.

### Theorem 12.1

*f*. View \(\chi '_{18}\) as a rational section of the line bundle \(\mathcal {L}_f^{\otimes 18}\) on \(\mathbb {D}\). Then one has the asymptotics

*n*.

*d*a positive integer such that the family \(D_\kappa \cong C_n\) has semistable reduction near \(\kappa =0\) after a ramified base change of degree

*d*(we can take \(d=2\) but this is not important). Putting \(t = \root d \of {\kappa }=\root d \of {1/n}\) we deduce from Corollary 12.1 that

## Declarations

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## Authors’ Affiliations

## References

- Arbarello, E., Cornalba, M., Griffiths, P.: Geometry of algebraic curves. Volume II. Grundlehren der Mathematischen Wissenschaften, vol. 268. Springer, Heidelberg (2011)MATHGoogle Scholar
- Arakelov, S.Y.: An intersection theory for divisors on an arithmetic surface. Izv. Akad. USSR
**86**, 1164–1180 (1974)MathSciNetGoogle Scholar - Arbarello, E., Cornalba, M.: The Picard groups of the moduli spaces of curves. Topology
**26**(2), 153–171 (1987)MathSciNetView ArticleGoogle Scholar - Beilinson, A.: Height pairing between algebraic cycles. In: Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985). Contemp. Math. 67, 1–24 (1987)Google Scholar
- Bloch, S.: Height pairing for algebraic cycles. J. Pure Appl. Algebra
**34**, 119–145 (1984)MathSciNetView ArticleGoogle Scholar - Brosnan, P., Pearlstein, G.: Jumps in the archimedean height. Preprint, arxiv:1701.05527
- Burgos Gil, J.I., Holmes, D., de Jong, R.: Positivity of the height jump divisor. Int. Math. Res. Notices. https://doi.org/10.1093/imrn/rnx169 (2017)
- Chinburg, T., Rumely, R.: The capacity pairing. J. Reine Angew. Math.
**434**, 1–44 (1993)MathSciNetMATHGoogle Scholar - Cinkir, Z.: Admissible invariants of genus 3 curves. Manuscr. Math.
**148**(3–4), 317–339 (2015)MathSciNetView ArticleGoogle Scholar - Cinkir, Z.: Computation of polarized metrized graph invariants by using discrete Laplacian matrix. Math. Comput.
**84**(296), 2953–2967 (2015)MathSciNetView ArticleGoogle Scholar - Cinkir, Z.: Zhang’s conjecture and the effective Bogomolov conjecture over function fields. Invent. Math.
**183**, 517–562 (2011)MathSciNetView ArticleGoogle Scholar - Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Etudes Sci. Publ. Math.
**36**, 75–109 (1969)MathSciNetView ArticleGoogle Scholar - Faltings, G., Chai, C.-L.: Degeneration of Abelian Varieties. With an Appendix by David Mumford. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 3, p. 22. Springer, Berlin (1990)Google Scholar
- Faltings, G.: Calculus on arithmetic surfaces. Ann. Math.
**119**, 387–424 (1984)MathSciNetView ArticleGoogle Scholar - van der Geer, G.: Siegel Modular Forms and Their Applications. In: The 1-2-3 of Modular Forms, pp. 181–245. Springer, Berlin (2008)MATHGoogle Scholar
- Gillet, H., Soulé, C.: Arithmetic analogs of the standard conjectures. In: Motives (Seattle, WA, 1991), Proceedings of Symposia in Pure Mathematics, 55, Part 1. American Mathematical Society, Providence, RI, pp. 129–140 (1994)Google Scholar
- Forni, G.: On the Lyapunov exponents of the Kontsevich–Zorich cocycle. In: Katok, A. (ed.) Handbook of Dynamical Systems, vol. 1B, pp. 549–580. Elsevier, Amsterdam (2006)MATHGoogle Scholar
- Gross, B., Schoen, C.: The modified diagonal cycle on the triple product of a pointed curve. Ann. Inst. Fourier
**45**, 649–679 (1995)MathSciNetView ArticleGoogle Scholar - Guàrdia, J.: A family of arithmetic surfaces of genus 3. Pac. J. Math.
**212**, 71–91 (2003)MathSciNetView ArticleGoogle Scholar - Hain, R.: Normal functions and the geometry of moduli spaces of curves. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli. Advanced Lectures in Mathematics (ALM), 24, vol. I, pp. 527–578. International Press, Somerville, MA (2013)MATHGoogle Scholar
- Hain, R., Reed, D.: On the Arakelov geometry of the moduli space of curves. J. Differ. Geom.
**67**, 195–228 (2004)MathSciNetView ArticleGoogle Scholar - Hain, R.: Biextensions and heights associated to curves of odd genus. Duke Math. J.
**61**, 859–898 (1990)MathSciNetView ArticleGoogle Scholar - Harris, J., Mumford, D.: On the Kodaira dimension of the moduli space of curves. With an appendix by William Fulton. Invent. Math.
**67**(1), 23–88 (1982)MathSciNetView ArticleGoogle Scholar - Herrlich, F., Schmithüsen, G.: An extraordinary origami curve. Math. Nachr.
**281**, 219–237 (2008)MathSciNetView ArticleGoogle Scholar - Ichikawa, T.: Theta constants and Teichmüller modular forms. J. Number Theory
**61**, 409–419 (1996)MathSciNetView ArticleGoogle Scholar - Igusa, J.-I.: Modular forms and projective invariants. Am. J. Math.
**89**, 817–855 (1967)MathSciNetView ArticleGoogle Scholar - de Jong, R.: Néron-Tate heights of cycles on jacobians. J. Alg. Geom.
**27**, 339–381 (2018)View ArticleGoogle Scholar - de Jong, R.: Torus bundles and 2-forms on the universal family of Riemann surfaces. In: European Mathematical Society (ed.) Handbook of Teichmüller Theory. EMS IRMA Lectures in Mathematics and Theoretical Physics, 27, vol. VI, pp. 195–227. European Mathematical Society, Zürich (2016)Google Scholar
- de Jong, R.: Asymptotic behavior of the Kawazumi–Zhang invariant for degenerating Riemann surfaces. Asian J. Math.
**18**, 507–523 (2014)MathSciNetView ArticleGoogle Scholar - de Jong, R.: Normal functions and the height of Gross–Schoen cycles. Nagoya Math. J.
**213**, 53–77 (2014)MathSciNetView ArticleGoogle Scholar - de Jong, R.: Second variation of Zhang’s \(\lambda \)-invariant on the moduli space of curves. Am. J. Math.
**135**, 275–290 (2013)MathSciNetView ArticleGoogle Scholar - Kawazumi, N.: Johnson’s homomorphisms and the Arakelov-Green function. Preprint, arxiv:0801.4218
- Knudsen, F.: The projectivity of the moduli space of stable curves. II. The stacks \(M_{g, n}\). Math. Scand.
**52**(2), 161–199 (1983)MathSciNetView ArticleGoogle Scholar - Knudsen, F., Mumford, D.: The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div”. Math. Scand.
**39**(1), 19–55 (1976)MathSciNetView ArticleGoogle Scholar - Konno, K.: Clifford index and the slope of fibered surfaces. J. Algebraic Geom.
**8**(2), 207–220 (1999)MathSciNetMATHGoogle Scholar - Lachaud, G., Ritzenthaler, C., Zykin, A.: Jacobians among abelian threefolds: a formula of Klein and a question of Serre. Math. Res. Lett.
**17**, 323–333 (2010)MathSciNetView ArticleGoogle Scholar - Lear, D.: Extensions of normal functions and asymptotics of the height pairing. PhD thesis, University of Washington, (1990)Google Scholar
- Möller, M.: Shimura and Teichmüller curves. J. Mod. Dyn.
**5**(1), 1–32 (2011)MathSciNetView ArticleGoogle Scholar - Moonen, B., Oort, F.: The Torelli locus and special subvarieties. In: Farkas, G. (ed.) Handbook of Moduli. Advanced Lectures in Mathematics (ALM), 25, vol. II, pp. 549–594. International Press, Somerville, MA (2013)Google Scholar
- Moret-Bailly, L.: La formule de Noether pour les surfaces arithmétiques. Invent. Math.
**98**, 491–498 (1989)MathSciNetView ArticleGoogle Scholar - Mumford, D.: Stability of projective varieties. l’Ens. Math. (2)
**23**(1–2), 39–110 (1977)MathSciNetMATHGoogle Scholar - Mumford, D.: Hirzebruch’s proportionality theorem in the noncompact case. Invent. Math.
**42**, 239–272 (1977)MathSciNetView ArticleGoogle Scholar - Pearlstein, G.: \(\text{ SL }_2\)-orbits and degenerations of mixed Hodge structure. J. Differ. Geom.
**74**, 1–67 (2006)MathSciNetView ArticleGoogle Scholar - Reid, M.: Problems on pencils of small genus. Preprint (1990)Google Scholar
- Tsuyumine, S.: Thetanullwerte on a moduli space of curves and hyperelliptic loci. Math. Z.
**207**(4), 539–568 (1991)MathSciNetView ArticleGoogle Scholar - Yamaki, K.: Graph invariants and the positivity of the height of the Gross–Schoen cycle for some curves. Manuscr. Math.
**131**, 149–177 (2010)MathSciNetView ArticleGoogle Scholar - Yamaki, K.: Geometric Bogomolov’s conjecture for curves of genus 3 over function fields. J. Math. Kyoto Univ.
**42**, 57–81 (2002)MathSciNetView ArticleGoogle Scholar - Zhang, S.: Admissible pairing on a curve. Invent. Math.
**112**, 171–193 (1993)MathSciNetView ArticleGoogle Scholar - Zhang, S.: Gross–Schoen cycles and dualising sheaves. Invent. Math.
**179**, 1–73 (2010)MathSciNetView ArticleGoogle Scholar - Zhang, S.: Positivity of heights of codimension 2 cycles over function fields of characteristic 0. Preprint, arXiv:1001.4788