2. Notation

Throughout, we work with Noetherian schemes over a fixed algebraically closed ground field k.

A curve over an extension K of k is a $\operatorname {\mathrm {Spec}}(K)$ -scheme X that is proper over $\operatorname {\mathrm {Spec}}(K)$ , geometrically connected, and of pure dimension $1$ . The curve X is a nodal curve if it is geometrically reduced and when passing to an algebraic closure $\overline {K}$ , its local ring at every singular point is isomorphic to $\overline {K}[[x,y]]/(xy)$ .

A coherent sheaf on a nodal curve X has rank $\boldsymbol {1}$ if its localisation at each generic point of X has length $1$ . It is torsion-free if it has no embedded components.

If F is a rank $1$ torsion-free sheaf on a nodal curve X, we denote by $ {\operatorname {N}}(F)$ the subset of X where F fails to be locally free. Note that $ {\operatorname {N}}(F)$ is contained in the singular locus of X. If F is a rank $1$ torsion-free sheaf on X, we say that F is simple if its automorphism group is $\mathbf {G}_m$ , or equivalently if $X\setminus {\operatorname {N}}(F)$ is connected.

A family of curves over a k-scheme S is a proper, flat morphism $\mathcal {X} \to S$ whose fibers are curves. A family of curves $\mathcal {X} \to S$ is a family of nodal curves if the fibers over all geometric points are nodal curves.

If X is a nodal curve over K, we denote by $\Gamma (X)$ its dual graph – that is, the labeled graph where each vertex v corresponds to an irreducible component $X_{\overline {K}}^v$ of the base change of X to (the spectrum of) an algebraic (equivalently, a separable) closure $\overline {K}$ , and edges corresponding to the nodes of $X_{\overline {K}}$ . Note that if $X^v_K$ is an irreducible component defined over K, then it is also defined over any extension L of K, and the corresponding vertices of the dual graphs $\Gamma (X_K)$ and $\Gamma (X_L)$ are canonically identified.

The dual graph is labeled by the geometric genus $p_g(X^v_{\overline {K}})$ . The definition of dual graph extends to the case where $(X,p_1,\dots ,p_n)$ is an n-pointed curve. In this case, the dual graph $\Gamma (X)$ also has n half-edges labeled from $1$ to n, corresponding to the marked points $p_1, \ldots , p_n$ . We refer to [Reference Arbarello, Cornalba and Griffiths5] and [Reference Melo, Molcho, Ulirsch and Viviani27] for a detailed definition and for the notion of graph morphisms.

Recall from [Reference Cavalieri, Chan, Ulirsch and Wise14, § 7.2] that if $\mathcal {X}/S$ is a family of nodal curves and $s,t$ are geometric points of S, then every étale specialization of t to s (written as $t \rightsquigarrow s$ ) induces a morphism of dual graphs $\Gamma (X_s)\rightarrow \Gamma (X_t)$ . (For the definition of étale specialization in this context, we refer to [Reference Cavalieri, Chan, Ulirsch and Wise14, Appendix A]).

For a graph G and H a subgraph of G, we denote by $G\setminus H$ and by $G/H$ the graph obtained from G by removing the edges of H and the graph obtained from G by contracting the edges of H, respectively.

We denote by $c(G)$ the complexity of the graph G (i.e., the number of spanning trees in G). (In particular, if G is disconnected, then $c(G)=0$ ).

If G is a graph and V is a subset of $\operatorname {\mathrm {Vert}}(G)$ , we denote by $\Gamma (V)$ the induced subgraph on the vertex set V. We denote by $\operatorname {\mathrm {Edges}}(V,V^c)$ the subset of $\operatorname {\mathrm {Edges}}(G)$ of the edges that connect some element of V to some element of $V^c= \operatorname {\mathrm {Vert}}(G) \setminus V$ (equivalently, $\operatorname {\mathrm {Edges}}(V,V^c)$ consists of the edges of G that are neither in $\Gamma (V)$ nor in $\Gamma (V^c)$ ).

We will denote by $\Delta $ the spectrum of a DVR with residue field K, and by $0$ (resp. by $\eta $ ) its closed (resp. its generic) point. A smoothing of a nodal curve $X/K$ over $\Delta $ is a flat family $\mathcal {X}/\Delta $ whose generic fiber $\mathcal {X}_{\eta }/\eta $ is smooth and with an isomorphism of K-schemes $\mathcal {X}_0 \cong X$ . The smoothing is regular if so is its total space $\mathcal {X}$ .

A family of rank $\boldsymbol {1}$ torsion-free sheaves over a family of curves $\mathcal {X} \to S$ is a coherent sheaf on $\mathcal {X}$ , flat over S, whose fibers over the geometric points have rank $1$ and are torsion-free.

If F is a rank $1$ torsion-free sheaf on a nodal curve X with irreducible components $X_i$ , we denote by $F_{\widetilde {X}_{i}}$ the maximal torsion-free quotient of the pullback of F to the normalization $\widetilde {X}_i$ of $X_i$ , and then define the multidegree of F by

$$\begin{align*}{\underline{\deg}}(F) := ({\deg}(F_{\widetilde{X}_{i}})) \in \mathbf{Z}^{\operatorname{Vert}(\Gamma(X))}.\end{align*}$$

We define the (total) degree of F to be $\deg _X(F):=\chi (F)-1+p_a(X)$ , where $p_a(X)= h^1(X, \mathcal {O}_X)$ is the arithmetic genus of X. The total degree and the multidegree of F are related by the formula $\deg _X(F) = \sum \deg _{X_i} F + \delta (F)$ , where $\delta (F)=\# {\operatorname {N}}(F)$ denotes the number of nodes of X where F fails to be locally free.Footnote ³

If $X' \subseteq X$ is a subcurve (by which we will always mean a union of irreducible components), then $\deg _{X'}(F)$ is defined as $\deg (F_{X'})$ , where $F_{X'}$ is the maximal torsion-free quotient of $F \otimes \mathcal {O}_{X'}$ . The total degree on X is related to the degree on a subcurve by the formula

(2.1) $$ \begin{align} \deg_X(F)= \deg_{X'}(F) + \deg_{\overline{X \setminus X'}}(F) + \# ( {\operatorname{N}}(F) \cap (X' \cap \overline{(X \setminus X')})), \end{align} $$

where the overline denotes the (Zariski) closure.

From now on, we fix an integer d once and for all.

2.1. Spaces of multidegrees

Here, we define the space of multidegrees on a graph $\Gamma $ at a connected spanning subgraph as the set of all possible ways of labeling its vertices with integral weights, suitably organized by total weight (or total degree). We then define the notion of an assignment on $\Gamma $ .

Let $\Gamma $ be a graph and let $G \subseteq \Gamma $ be a spanning subgraph of $\Gamma $ . We will denote by $n_\Gamma (G)$ or simply by $n(G)$ the number of elements in $\operatorname {Edges}(\Gamma ) \setminus \operatorname {Edges}(G)$ .

Define the space of multidegrees of total degree d of $\Gamma $ at G as the set

(2.2) $$ \begin{align} S^d_{\Gamma}(G) := \{\underline{\mathbf d} \in \mathbf{Z}^{\operatorname{\mathrm{Vert}}(\Gamma)} : \sum_{v \in \operatorname{\mathrm{Vert}}(\Gamma)} \underline{\mathbf d}(v) = d - n(G)\} \subset \mathbf{Z}^{\operatorname{\mathrm{Vert}}(\Gamma)}. \end{align} $$

The elements of $S^d_{\Gamma }(\Gamma )$ are also known as degree $d-n(G)$ divisors on $\Gamma $ .

According to our convention for the multidegree, if X is a nodal curve with dual graph $\Gamma $ and F is a rank $1$ torsion-free sheaf on X, then the subgraph $G(F)$ obtained from $\Gamma $ by removing the edges $ {\operatorname {N}}(F)$ is a spanning subgraph of $\Gamma $ , and we have

$$\begin{align*}\underline{\deg}(F) \in S^d_{\Gamma}(G(F)). \end{align*}$$

We are now ready for the definition of an assignment on a graph $\Gamma $ , which will play a role in the definition of a stability assignment (Definition 4.3).

Definition 2.1. A degree $\boldsymbol {d}$ assignment for the graph $\boldsymbol {\Gamma }$ is a subset

$$\begin{align*}\sigma = \{(G,\underline{\mathbf d}):\;\underline{\mathbf d}\in S^d_\Gamma(G)\}\subset \{\text{connected spanning subgraphs of }\Gamma\}\times \mathbf{Z}^{\operatorname{\mathrm{Vert}}(\Gamma)}.\end{align*}$$

If $\sigma $ is a degree d assignment for the graph $\Gamma $ and $G \subseteq \Gamma $ is a subgraph, we define

$$\begin{align*}\sigma(G) := \{\underline{\mathbf d}:\;(G,\underline{\mathbf d})\in\sigma\} \subset S^d_{\Gamma}(G)\end{align*}$$

(and the latter is empty unless G is connected and spanning).

3. Fine compactified Jacobians

In this section, we introduce the notion of a (smoothable) fine compactified Jacobian for a nodal curve (Definition 3.1) and for a flat family of nodal curves (Definition 3.7).

Let $\mathcal {X}/S$ be a flat family of nodal curves over a k-scheme S. Then there is an algebraic space $\operatorname {Pic}^d(\mathcal {X}/S)$ parameterizing line bundles on $\mathcal {X}/S$ of relative degree d (see [Reference Bosch, Lütkebohmert and Raynaud9, Chapter 8.3]). By [Reference Altman and Kleiman2] and [Reference Esteves16], the space $\operatorname {Pic}^d(\mathcal {X}/S)$ embeds in an algebraic space $ {\operatorname {Simp}}^{d}(\mathcal {X}/S)$ parameterizing flat families of degree d rank $1$ torsion-free simple sheaves on $\mathcal {X}/S$ . The latter is locally of finite type over S and satisfies the existence part of the valuative criterion of properness. However, it can fail to be of finite type and separated. In the special case of $S= {\operatorname {Spec}}(K)$ and $X=\mathcal {X}/S$ , we will simply write $ \operatorname {Pic}^d(X)$ (resp. $ {\operatorname {Simp}}^d(X)$ ) for $\operatorname {Pic}^d(\mathcal {X}/S)$ (resp. for $ {\operatorname {Simp}}^d(\mathcal {X}/S)$ ).

Let X be a nodal curve over some field extension K of k. The main point of this paper is to describe well-behaved subspaces of $ {\operatorname {Simp}}^{d}(X)$ , generalizing existing notions of compactified Jacobians in the literature. Specifically, we study the following subschemes.

Note that the fiber over the generic point $\eta $ of a nonempty, open and $\Delta $ -proper subscheme of $ {\operatorname {Simp}}^{d}(\mathcal {X}/\Delta )$ is necessarily the moduli space of degree d line bundles $\operatorname {Pic}^d(\mathcal {X}_{\eta }/{\eta })$ . As openness and properness are stable under base change, the fiber over $0 \in \Delta $ is open in $ {\operatorname {Simp}}^d(X)$ and K-proper. The axiom ‘geometrically connected’ is redundant in the smoothable case because the moduli space of degree d line bundles on the generic point is geometrically connected and dense in $ {\operatorname {Simp}}^{d}(\mathcal {X}/\Delta )$ .

In this paper, we will focus on smoothable fine compactified Jacobians, since they are better behaved and occur more often in applications.

Remark 3.6. The moduli space $ {\operatorname {Simp}}^d(X)$ of a nodal curve X admits a natural stratification (see, for example, [Reference Melo and Viviani29]) into locally closed subsets

(3.1) $$ \begin{align} {\operatorname{Simp}}^d(X) = \bigsqcup_{(\Gamma_{0},\underline{\mathbf d})}\mathcal J_{(\Gamma_{0},\underline{\mathbf d}),}\end{align} $$

where the union runs over all connected spanning subgraphs $\Gamma _{0}\subseteq \Gamma (X)$ and all multidegrees $\underline {\mathbf d}\in S^d_{\Gamma (X)}(G)$ . Each subspace $\mathcal J_{(\Gamma _{0},\underline {\mathbf d})}\subset {\operatorname {Simp}}^d(X)$ is defined as the locus whose points are sheaves F that fail to be locally free on $ {\operatorname {N}}(F)=\operatorname {\mathrm {Edges}}(\Gamma ) \setminus \operatorname {\mathrm {Edges}}(\Gamma _{0})$ , and whose multidegree $\underline {\deg }(F)$ equals $ \underline {\mathbf {d}}$ .

We now extend the notion of a fine compactified Jacobian to the case of a family of nodal curves $\mathcal {X}/S$ . Recall that the moduli space $ {\operatorname {Simp}}^d(\mathcal {X}/S)$ is also defined in [Reference Melo26] when $\mathcal {X}/S$ is the universal curve $\overline {\mathcal {C}}_{g,n}/\overline {\mathcal {M}}_{g,n}$ over the moduli stack of stable n-pointed curves of arithmetic genus g. In this case, $ {\operatorname {Simp}}^d(\overline {\mathcal {C}}_{g,n}/\overline {\mathcal {M}}_{g,n})$ is a Deligne–Mumford stack representable (by algebraic spaces) and flat over $\overline {\mathcal {M}}_{g,n}$ .

Note that the assumption that the generic fiber is smooth implies that all fibers over S of a degree d fine compactified Jacobian are smoothable.

4. Stability assignments for smoothable fine compactified Jacobians

Here, we define the combinatorial data identifying smoothable fine compactified Jacobians. We first do so for a single nodal curve – that is, for a fixed dual graph (Definition 4.3) – and then we generalize the definition to families (Definition 4.9).

If X is a nodal curve over an algebraically closed field, a sheaf $F\in {\operatorname {Simp}}^d(X)$ has two natural combinatorial invariants, given by its multidegree and by the subset $ {\operatorname {N}}(F)\subseteq \operatorname {Sing}(X)=\operatorname {\mathrm {Edges}}(\Gamma (X))$ of points of the curve where F fails to be locally free. Hence, it makes sense to study a fine compactified Jacobian $\overline {J}$ on X by looking at all pairs $\left (G(F)= \Gamma (X) \setminus {\operatorname {N}}(F),\underline {\deg }(F)\right )$ with $F\in \overline {J}$ . Recall that, with the notation introduced in Equation 2.2, we can regard $\underline {\deg }(F)$ as an element of the space of multidegrees $S_{\Gamma (X)}^d(G(F))$ .

For a single curve X, we identify, in Definition 4.3, the two properties characterizing the set of such pairs. One is related to properness, combined with the smoothability of the Jacobian, and it requires that the set of stable multidegrees should be a minimal complete set of representatives for the natural chip-firing action on the dual graph (see Definition 4.1). The other corresponds to openness. In combinatorial terms, this means that if we add an edge e to G, the set of stable multidegrees on $\Gamma \cup \{e\}$ should contain all multidegrees obtained by ‘adding a chip’ to either endpoints of e.

For a family of curves, we further require compatibility with all contractions of the dual graphs involved.

4.1. Stability assignments for a single curve

We start by introducing the twister group of a graph, which will play a role in characterizing smoothable compactified Jacobians.

Definition 4.1. Let G be a graph. For each $v \in \operatorname {\mathrm {Vert}}(G)$ , define the twister of G at v to be the element of $\mathbf {Z}^{\operatorname {\mathrm {Vert}}(G)}$ defined by

$$\begin{align*}\operatorname{\mathrm{Tw}}_{G, v} (w) = \begin{cases} \text{ # of edges of } G \text{ having } v \text{ and } w \text{ as endpoints }& \text{when } w \neq v, \\ - \text{ # of nonloop edges of } G \text{ having } v \text{ as an endpoint} & \text{when } w =v.\end{cases} \end{align*}$$

The twister group (or chip-firing group) $\operatorname {\mathrm {Tw}}(G)$ is the subgroup of $\mathbf {Z}^{\operatorname {\mathrm {Vert}}(G)}$ generated by the set $\{\operatorname {\mathrm {Tw}}_{G, v}\}_{ v \in \operatorname {\mathrm {Vert}}(G)}$ .

Recall from Equation (2.2) the definition of the space of multidegrees $S^d_{G}(G)$ of total degree equal to d. The twister group of G is contained in the sum zero submodule $S^0_{G}(G)$ of $\mathbf {Z}^{\operatorname {\mathrm {Vert}}(G)}$ . Hence, the group structure on $\mathbf {Z}^{\operatorname {\mathrm {Vert}}(G)}$ restricts to an action of $\operatorname {\mathrm {Tw}}(G)$ on $S^d_{G}(G)$ . The quotient group

(4.1) $$ \begin{align} J^d(G):=S^d_{G}(G)/\operatorname{\mathrm{Tw}}(G)\end{align} $$

is then a torsor over $J^0(G)$ , which is a finite abelian group. The latter is also known as the Jacobian of the graph G, and it has a number of element equal to the complexity $c(G)$ of the graph G. (It is also known by other names in the literature, such as the degree class group, or the sandpile group, or the critical group of the graph G).

Remark 4.2. Let $\mathcal {X}$ be a regular smoothing of X over some discrete valuation ring $\Delta $ with generic point $\eta $ . Let T be the image under the restriction map $\operatorname {\mathrm {Pic}}(\mathcal X)\rightarrow \operatorname {\mathrm {Pic}}(X)$ of the kernel of the surjection $\operatorname {\mathrm {Pic}}(\mathcal X)\rightarrow \operatorname {\mathrm {Pic}}(\mathcal X_\eta )$ (the restriction to the generic point). We claim that the restriction to T of the multidegree homomorphism $\operatorname {\mathrm {Pic}}(X) \to \mathbf {Z}^{\operatorname {\mathrm {Vert}}(\Gamma )}$ defines an isomorphism $T \to \operatorname {\mathrm {Tw}}(\Gamma )$ .

Since $\mathcal X$ is regular, the irreducible components $\{X_v\}_{v\in \operatorname {\mathrm {Vert}}(\Gamma )}$ of X are Cartier divisors on $\mathcal X$ , and it is easy to check that the elements of T are of the form

$$\begin{align*}\mathcal O_{\mathcal X}\left(\sum_{v\in\operatorname{\mathrm{Vert}}(\Gamma)}{d_vX_v}\right)\otimes \mathcal O_X \end{align*}$$

with $\underline {\mathbf d}\in \mathbf {Z}^{\operatorname {\mathrm {Vert}}(\Gamma )}$ . One can explicitly compute that the restriction to T of the multidegree map $\operatorname {\mathrm {Pic}}(X)\rightarrow \mathbf {Z}^{\operatorname {\mathrm {Vert}}(\Gamma )}$ is given by

$$\begin{align*}\begin{array}{rcl} \mathcal O_{\mathcal X}\left(\sum_{v\in\operatorname{\mathrm{Vert}}(\Gamma)}{d_vX_v}\right)\otimes \mathcal O_X&\longmapsto&\sum_{v\in\operatorname{\mathrm{Vert}}(\Gamma)}d_v\operatorname{\mathrm{Tw}}_{\Gamma,v}. \end{array}\end{align*}$$

This homomorphism is injective by [Reference Raynaud34, (5.2)], and its image is $\operatorname {\mathrm {Tw}}(\Gamma )$ (by definition). This proves that $T \to \operatorname {\mathrm {Tw}}(\Gamma )$ is an isomorphism. In particular, T as an abstract group is independent of the choice of the regular smoothing (though one could see that the embedding $T \subset \operatorname {Pic}(X)$ is not independent of the smoothing).

We are now ready to define our notion of a stability assignment.

Definition 4.3. A degree $\boldsymbol {d}$ smoothable fine compactified Jacobian stability assignment, or shortly, a degree $\boldsymbol {d}$ stability assignment for the graph $\Gamma $ , is a degree d assignment for $\Gamma $ (as in Definition 2.1) that satisfies the following conditions:

1. 1. For all edges e of $\operatorname {\mathrm {Edges}}(\Gamma )\setminus \operatorname {\mathrm {Edges}}(G)$ with endpoints $v_1$ and $v_2$ , we have

   $$\begin{align*}(G,\underline{\mathbf d})\in\sigma \Rightarrow (G\cup\{e\},\underline{\mathbf d}+\underline{\mathbf e}_{v_1}),(G\cup\{e\},\underline{\mathbf d}+\underline{\mathbf e}_{v_2})\in\sigma, \end{align*}$$
   where $\underline {\mathbf e}_{v_i}$ denotes the vector in the standard basis of $\mathbf {Z}^{\operatorname {\mathrm {Vert}}(\Gamma )}$ corresponding to $v_i$ .

2. 2. For every connected spanning subgraph G, the subset

   $$\begin{align*}\sigma(G) := \{\underline{\mathbf d}:\;(G,\underline{\mathbf d})\in\sigma\} \subset S^d_{\Gamma}(G)\end{align*}$$
   is a minimal complete set of representatives for the action of the twister group $\operatorname {\mathrm {Tw}}(G)$ on $S^d_{\Gamma }(G)$ .

If X is a nodal curve, a degree d stability assignment on X is a degree d stability assignment on its dual graph $\Gamma (X)$ .

Note that the genus of each of the components of X does not play any role in the above definition.

Remark 4.4. With the notation as above, the number of elements of $\sigma (G)$ equals the number of elements of the Jacobian $J^0(G)$ . By the Kirchoff–Trent theorem, this number equals the complexity $c(G)$ of the graph G. In particular, it is finite.

It is, in general, hard to classify all stability assignments on a given stable graph. However, the task is within reach when the number of vertices is small.

Example 4.5. If $\Gamma $ only has $1$ vertex (i.e., it is the dual graph of an irreducible curve), there is exactly $1$ stability assignment on $\Gamma $ . If there are t edges, the unique degree d stability assignment is

$$\begin{align*}\sigma=\bigcup_{0 \leq i \leq t} \bigcup_{E \subseteq \operatorname{\mathrm{Edges}}(\Gamma), |E|=i} \{(\Gamma \setminus E,d-i)\}.\end{align*}$$

Example 4.6. If instead $\Gamma $ consists of $2$ vertices $v_1, v_2$ connected by t edges, and no other edges (i.e., $\Gamma $ is the dual graph of a vine curve of type t; see Example 3.5), let $\Gamma _1, \ldots , \Gamma _t$ be the spanning trees. Then it follows from the definition that for every stability assignment $\sigma $ , there exists a unique integer $\lambda $ such that

1. 1. $\sigma (\Gamma _i)=\{(\lambda ,d+1-\lambda -t)\}$ for all $i=1, \ldots , t$ ;

2. 2. $\sigma (\Gamma )=\{(\lambda , d-\lambda ) ,(\lambda +1, d-\lambda -1), \ldots , (\lambda +t-1, d+1-\lambda -t)\}$ .

The following result will be used later as a tool to prove that certain collections are stability assignments. It was first proved by Barmak [Reference Barmak7]. A first publicly available proof of part of this result, based upon Barmak’s argument, appeared in Yuen’s Phd thesis, [Reference Yuen38, Theorem 3.5.1].

Proposition 4.7 [Reference Viviani37, Theorem 1.17 Part (1) and (2.a)]

Let $\sigma $ be a degree d assignment on the graph $\Gamma $ (see Definition 2.1), and assume that $\sigma $ satisfies Part (1) of Definition 4.3, and that $\sigma (T)$ is nonempty for every spanning tree $T \subseteq \Gamma $ .

Then for all spanning subgraphs $G \subseteq \Gamma $ , we have $|\sigma (G)| \geq c(G)$ (the complexity of G). Moreover, if $|\sigma (\Gamma )|=c(\Gamma )$ , then $|\sigma (G)|= c(G)$ for each spanning subgraph $G \subseteq \Gamma $ .

4.2. Stability assignments for families

So far, we have discussed the notion of a stability assignment for a curve in isolation. The flatness condition for families of sheaves imposes an additional compatibility constraint.

Definition 4.8. Let $f\colon \Gamma \to \Gamma '$ be a morphism of graphs and let $\sigma $ and $\sigma '$ be degree d stability assignments on $\Gamma $ and $\Gamma '$ , respectively. We say that $\sigma $ is f-compatible with $\sigma '$ if for every connected spanning subgraph G of $\Gamma $ , we have

$$\begin{align*}(G,\underline{\mathbf d})\in\sigma \implies (G',\underline{\mathbf d'}) \in \sigma', \end{align*}$$

where $G'$ is the image of G under f (and therefore, it is a connected spanning subgraph of $\Gamma '$ ), and $\underline {\mathbf d'}$ is defined by

(4.2) $$ \begin{align} \underline{\mathbf d'}(w) = \sum_{f(v)=w} \underline{\mathbf d}(v) + \# \left\{\textrm{edges of } \Gamma\setminus G \textrm{ that are contracted to } w \textrm{ by } f\right\}. \end{align} $$

Note that the notion of f-compatibility only depends upon the map that f induces on the set of vertices of the two graphs. We are now ready for the definition of the compatibility constraint.

Definition 4.9. Let $\mathcal {X}/S$ be a family of nodal curves. A family of degree $\boldsymbol {d}$ stability assignments (for fine compactified Jacobians) for $\mathcal X/S$ consists of associating a degree d stability assignment $\sigma _s$ on $\Gamma (X_s)$ (as in Definition 4.3) with every geometric point s in S, in a way that is compatible with all morphisms $\Gamma (X_s) \to \Gamma (X_t)$ arising from any étale specialization $t \rightsquigarrow s$ occurring on S.

We will say that a family of degree d stability assignments for the universal family $\overline {\mathcal {C}}_{g,n}$ over $\overline {\mathcal {M}}_{g,n}$ is a degree $\boldsymbol {d}$ universal stability assignment of type $\mathbf {(g,n)}$ (and often omit ‘of type $(g,n)$ ’ when clear from the context). These stability assignments will be studied in Section 9.

Universal stability assignments can be defined purely in terms of the category $G_{g,n}$ of stable graphs:

Remark 4.10. In the case of universal stability assignments, the étale specializations of points of $\overline {\mathcal {M}}_{g,n}$ induce all morphisms in the category $G_{g,n}$ of stable graphs of genus g with n marked half-edges.

In particular, if $\sigma $ is a degree d universal stability assignment of type $(g,n)$ and $\alpha \colon \Gamma (X_1) \to \Gamma (X_2)$ is an isomorphism of the dual graphs of two pointed curves $[X_1],[X_2]\in \overline {\mathcal {M}}_{g,n}$ , then $\alpha $ identifies $\sigma _{[X_1]}$ with $\sigma _{[X_2]}$ .

We conclude that a degree d universal stability assignment of type $(g,n)$ is a collection $\{\sigma _\Gamma \}_{\Gamma \in G_{g,n}}$ such that $\sigma _\Gamma $ is a degree d stability assignment on $\Gamma $ and $\sigma _{\Gamma }$ is f-compatible with $\sigma _{\Gamma '}$ for any morphism $f:\;\Gamma \rightarrow \Gamma '$ in $G_{g,n}$ .

5. Combinatorial preparation: Perturbations and lifts

This section contains combinatorial results that will be used in later proofs. First, we introduce the notion of a perturbation for the multidegree of a sheaf that fails to be locally free at some nodes. This corresponds to deforming a simple sheaf into a line bundle (Lemma 5.2). Then we investigate the relationship between the chip-firing action on a graph $\Gamma $ and the chip-firing action on the graph obtained from $\Gamma $ by subdividing each of its edges a certain number of times. In terms of dual graphs, this construction corresponds to blowing up a certain number of times the nodes of our curve. Our main techincal result here is Proposition 5.4.

Definition 5.1. Let $G \subseteq \Gamma $ be a connected spanning subgraph. A G-perturbation is an element in $S^{n(G)}_{G}(G)$ (for $n(G)=|\operatorname {Edges}(\Gamma ) \setminus \operatorname {Edges}(G)|$ ) of the form

$$\begin{align*}\sum_{e \in \operatorname{\mathrm{Edges}}(\Gamma) \setminus \operatorname{\mathrm{Edges}}(G)} {\underline{\mathbf e}}_{t(e),} \end{align*}$$

where t is some choice of an orientation on $\operatorname {\mathrm {Edges}}(\Gamma ) \setminus \operatorname {\mathrm {Edges}}(G)$ .

(If D is a G-perturbation for some G, then D is also known in the literature as a semibreak divisor; see [Reference Gross, Shokrieh and Tóthmérész19, Definition 3.1]).

By the description in Remark 3.6 of the stratification of $ {\operatorname {Simp}}^d(X)$ , and by Lemma 5.2, if G is a connected spanning subgraph of $\Gamma $ , a G-perturbation is the same as the multidegree of a line bundle on X that specializes to a sheaf F with $G(F)=G$ , $\underline {\deg }(F)=\underline {\mathbf 0}$ . We are now ready for the following technical lemma.

Lemma 5.2. Let $\mathcal X$ be a family of nodal curves over the spectrum $\Delta $ of a DVR with algebraically closed residue field $\overline {K}$ , and consider a section $\sigma :\;\Delta \rightarrow {\operatorname {Simp}}^d(\mathcal X/\Delta )$ . Let us fix a geometric point $\overline \eta $ lying over the generic point of $\Delta $ and denote by $\Gamma $ and $\Gamma '$ the dual graphs of the geometric fibers $X_0$ and $X_{\overline \eta }$ of $\mathcal X$ , respectively, and by $f:\;\Gamma \rightarrow \Gamma '$ the morphism of graphs induced by the specialization of $\overline \eta $ to $0$ . Write $F_0$ and $F_{\overline {\eta }}$ for $\sigma (0)$ and $\sigma (\overline \eta )$ , respectively, and set $G=G(F_0)\subseteq \Gamma $ and $G'=G(F_{\overline {\eta }})\subseteq \Gamma '$ for the connected subgraphs where the corresponding sheaf is locally free. Then there exists a $G\subseteq f^{-1}(G')$ -perturbation T such that the multidegrees $\underline {\mathbf d}=\underline {\deg }(F_0)$ and $\underline {\mathbf d}' = \underline {\deg }(F_{\overline \eta })$ satisfy the relation

$$\begin{align*}\underline{\mathbf d}'= f\left(\underline{\mathbf d}+T\right).\end{align*}$$

Note that if $G'\subseteq \Gamma '$ is connected and spanning, then so is $f^{-1}(G') \subseteq \Gamma $ .

We will now prove some combinatorial ingredients that relate the chip-firing on a graph with the chip-firing on its blow-up. We fix a graph $\Gamma $ and a function $m \colon \operatorname {\mathrm {Edges}}(\Gamma ) \to \mathbf {N}$ and let ${{\widetilde {\Gamma }}}_m$ be the graph obtained by subdividing each edge e of $\Gamma $ into $m(e)+1$ edges (called an exceptional chain) by adding $m(e)$ vertices (called exceptional vertices) in the middle.

We start by relating the complexities of $\Gamma $ and ${\widetilde {\Gamma }}$ .

Lemma 5.3 [Reference Busonero, Melo and Stoppino11, Theorem 3.4]

The following formula relates the number of spanning trees (i.e., the complexity) of ${{\widetilde {\Gamma }}}_m$ with the complexity of the spanning subgraphs of $\Gamma $ :

(5.1) $$ \begin{align} c({{\widetilde{\Gamma}}}_m)= \sum_{\substack{G \subseteq \Gamma \text{ connected}\\ \text{ spanning subgraph}}} \left( \prod_{e \in \operatorname{\mathrm{Edges}}(\Gamma) \setminus \operatorname{\mathrm{Edges}}(G)} m(e)\right) \cdot c(G). \end{align} $$

In order to state and then prove Proposition 5.4, we first need to define the lift of an assignment to $\widetilde {\Gamma }$ .

Let $\sigma $ be a degree d assignment for $\Gamma $ (see Definition 2.1). For each $m \colon \operatorname {\mathrm {Edges}}(\Gamma ) \to \mathbf {N}$ , let $\widetilde {\sigma }_m$ , and then define the lift $\widetilde {\sigma }_m \subset S^d_{\widetilde {\Gamma }}(\widetilde {\Gamma })$ of $\sigma $ as follows. First of all, for every element $(\Gamma ,\underline {\mathbf d}) \in \sigma $ , we can naturally identify $\underline {\mathbf d}$ with the element of $\mathbf {Z}^{\operatorname {\mathrm {Vert}}(\widetilde {\Gamma })}$ obtained by extending $\underline {\mathbf {d}}$ to zero on all exceptional vertices. Then a lift of $(G,\underline {\mathbf d})$ is any element $\widetilde {\underline {\mathbf d}}\in S^d_{\widetilde {\Gamma }}(\widetilde {\Gamma })$ of the form

(5.2) $$ \begin{align}\widetilde{\underline{\mathbf d}} = \underline{\mathbf d}+\sum_{e\in \operatorname{\mathrm{Edges}}(\Gamma)\setminus\operatorname{\mathrm{Edges}}(G)}\underline{\mathbf e}_{v(e)}, \end{align} $$

for any choice of a function $v\colon \operatorname {\mathrm {Edges}}(\Gamma )\setminus \operatorname {\mathrm {Edges}}(G) \to \operatorname {\mathrm {Vert}}(\widetilde {\Gamma })$ such that each vertex $v(e)$ is one of the $m(e)$ interior vertices of the exceptional chain in $\widetilde {\Gamma }$ that corresponds to the edge e.

Then we have the following:

Proof. We first prove (1) (i.e., we assume m is constant and equal to $1$ and prove that $\sigma $ also satisfies Part (2) of Definition 4.3). By [Reference Viviani37, Theorem 1.20, (c) $\kern2.00em \Rightarrow \kern2.00em$ (b)], it suffices to prove that $|\sigma (G)|=c(G)$ for all $G \subseteq \Gamma $ . By Proposition 4.7, it suffices to prove the inequality $|\sigma (\Gamma )|\leq c(\Gamma )$ . Arguing by contradiction, it is enough to show that if any two different elements $\mathbf {\underline {e}}$ and $\mathbf {\underline {e}}'$ of $\sigma (\Gamma )$ are in the same orbit for the action of $\operatorname {\mathrm {Tw}}(\Gamma )$ , then the corresponding lifts $\mathbf { \underline {\tilde {e}}}$ and $\mathbf {\underline {\tilde {e}}}'$ to $\widetilde {\sigma }$ via Equation 5.2 are different and in the same $\operatorname {\mathrm {Tw}}(\widetilde {\Gamma })$ -orbit.

Therefore, we assume that there exists $f \colon \operatorname {\mathrm {Vert}}(\Gamma) \to \mathbf {Z}$ such that

$$\begin{align*}\mathbf{\underline{e}} - \mathbf{\underline{e}}' = \sum_{v \in \operatorname{\mathrm{Vert}}(\Gamma)} f(v) \operatorname{\mathrm{Tw}}_{\Gamma,v}. \end{align*}$$

We now defineFootnote ⁴ a lift $g \colon \operatorname {\mathrm {Vert}}({{\widetilde {\Gamma }}}) \to \mathbf {Z}$ of f as follows:

$$\begin{align*}g(v)= \begin{cases} 2 f(v') & \textrm{ when } v \textrm{ is not exceptional and corresponds to }v' \in \operatorname{\mathrm{Vert}}(\Gamma), \\ f(w)+f(w') &\textrm{ when } v \textrm{ is exceptional and lies between } w,w' \in \operatorname{\mathrm{Vert}}(\Gamma).\end{cases} \end{align*}$$

With this definition, we have

$$\begin{align*}\mathbf{\tilde{\underline{e}}}- \mathbf{\tilde{\underline{e}}}' = \sum_{v \in \operatorname{\mathrm{Vert}}({{\widetilde{\Gamma}}})} g(v) \operatorname{\mathrm{Tw}}_{{{\widetilde{\Gamma}}},v}. \end{align*}$$

This proves that the lifts $\mathbf {\tilde {\underline {e}}}$ and $ \mathbf {\tilde {\underline {e}}}'$ , which are different elements of $\widetilde {\sigma }$ , are in the same $\operatorname {\mathrm {Tw}}({{\widetilde {\Gamma }}})$ -orbit. This completes our proof of (1).

To prove (2), we begin by observing that if $\sigma $ is a stability assignment (as in Definition 4.3), then the number of elements of $\widetilde {\sigma }_m$ equals the right-hand side of (5.1), which equals the complexity of ${{\widetilde {\Gamma }}}_m$ by Lemma (5.3). Thus, it suffices to prove that any two different elements $\mathbf {\underline {\tilde {e}}}$ and $\mathbf { \underline {\tilde {e}}}'$ of $\widetilde {\sigma }_m$ belong to different $\operatorname {\mathrm {Tw}}(\widetilde {\Gamma }_m)$ -orbits. By Lemma 5.5 below, it suffices to prove the inequality

(5.3) $$ \begin{align} \left| \sum_{v \in \widetilde{V}} \mathbf{\underline{\tilde{e}}}(v)-\mathbf{\underline{\tilde{e}}}'(v) \right|< \left|\operatorname{\mathrm{Edges}}(\widetilde{V},\widetilde{V}^c)\right| \end{align} $$

for all $\widetilde {V} \subset \operatorname {\mathrm {Vert}}(\widetilde {\Gamma }_m)$ such that the induced subgraph $\Gamma (\widetilde {V})$ is connected.

Let $j \colon \operatorname {\mathrm {Vert}}(\Gamma ) \to \operatorname {\mathrm {Vert}}(\widetilde {\Gamma })$ be the inclusion of the nonexceptional vertices. If $\widetilde {V}$ is not in the image of j, then $\widetilde {V}$ is a subset of an exceptional chain, so the right-hand side of (5.3) equals $2$ and by the definition of a lift, we have that the left-hand side of (5.3) equals $0$ or $1$ . From now on, we will assume that $\widetilde {V}$ contains some nonexceptional vertices.

We now prove (5.3) under the additional assumption that $\mathbf {\underline {\tilde {e}}}$ and $\mathbf {\underline {\tilde {e}}}'$ are zero on all exceptional vertices – namely, that they are lifts of elements $\mathbf { \underline {{e}}}$ and $\mathbf {\underline {{e}}}'$ that belong to $\sigma (\Gamma )$ . Because $\sigma $ is a degree d stability assignment for $\Gamma $ , by [Reference Viviani37, Theorem 1.20, (b) $\Rightarrow $ (a)], it is also a ‘V-stability condition’ in the sense of [Reference Viviani37]. By [Reference Viviani37, Remark 1.10]), this means that, for all $V \subseteq \operatorname {\mathrm {Vert}}(\Gamma )$ such that $\Gamma (V)$ is connected, there exist integers $n_{\mathbf {\vec {e}}}(V)$ and $n_{\mathbf {\vec {e}'}}(V)$ such that the inequalities

$$\begin{align*}n_{\mathbf{\vec{e}}}(V) \leq \sum_{v \in V} \mathbf{\underline{{e}}}(v) \leq n_{\mathbf{\vec{e}}}(V)+ \left|\operatorname{\mathrm{Edges}}(V,V^c)\right| -1 \end{align*}$$

and

$$\begin{align*}n_{\mathbf{\vec{e}'}}(V) \leq \sum_{v \in V} \mathbf{\underline{{e}}}'(v) \leq n_{\mathbf{\vec{e}'}}(V)+ \left|\operatorname{\mathrm{Edges}}(V,V^c)\right| -1 \end{align*}$$

hold. The combination of these two inequalities applied to $V:=j^{-1}(\widetilde {V})$ implies Inequality (5.3).

If $\mathbf {\underline {\tilde {e}}}$ and $\mathbf {\underline {\tilde {e}}}'$ are not zero on all of the exceptional vertices of $\widetilde {\Gamma }$ , we first replace them by elements of $\widetilde {\sigma }_m$ that satisfy this additional hypothesis. This can be achieved by replacing each $1$ on each exceptional vertex in $\tilde {V}$ (resp. in $\tilde {V}^c$ ) with a $0$ , and for each such replacement by adding $+1$ on (one of) the closest nonexceptional vertex in $\Gamma (\tilde {V})$ (resp. in $\Gamma (\tilde {V}^c)$ ). This replacement does not change the left-hand side of (5.3). That these replacements are also elements of $\widetilde {\sigma }_m$ follows from the fact that $\sigma $ satisfies Part (1) of Definition 4.3. This concludes our proof.

We conclude by proving the following, which was used in the proof of the above.

Lemma 5.5. Let $\mathbf {\underline {\tilde {e}}}$ and $\mathbf {\underline {\tilde {e}}}'$ be elements of $\tilde {\sigma }_m$ . If the inequality

(5.4) $$ \begin{align} \left| \sum_{v \in W} \mathbf{\underline{\tilde{e}}}(v)-\mathbf{\underline{\tilde{e}}}'(v) \right|< \left|\operatorname{\mathrm{Edges}}(W,W^c)\right| \end{align} $$

holds for all $W \subset \operatorname {\mathrm {Vert}}(\widetilde {\Gamma }_m)$ such that the induced subgraph $\Gamma (W)$ is connected, then $\mathbf {\underline {\tilde {e}}}$ and $\mathbf {\underline {\tilde {e}}}'$ belong to different $\operatorname {\mathrm {Tw}}(\widetilde {\Gamma }_m)$ -orbits.

Proof. Assume for a contradiction that there exists $f \colon \operatorname {\mathrm {Vert}}(\widetilde {\Gamma }_m) \to \mathbf {Z}$ such that

$$\begin{align*}\mathbf{\underline{\tilde{e}}'}= \mathbf{\underline{\tilde{e}}}+ \sum_{v \in \operatorname{\mathrm{Vert}}(\widetilde{\Gamma}_m)} f(v) \operatorname{\mathrm{Tw}}_{\widetilde{\Gamma}_m, v}. \end{align*}$$

Defining W to be a subset of $\{w: \ f(w)=\min _{v\in \operatorname {\mathrm {Vert}}(\Gamma )}(f)\}$ such that both induced subgraph $\Gamma (W)$ is connected (one such W subset always exists), we then deduce

$$\begin{align*}\sum_{v \in W}\mathbf{\underline{\tilde{e}}'}(v)\geq \sum_{v \in W} \mathbf{\underline{\tilde{e}}}(v)+ |E(W,W^c)|. \end{align*}$$

This contradicts (5.4), thus concluding our proof.

6. From stability assignments to smoothable fine compactified Jacobians

In this section, we show how to construct a fine compactified Jacobian from a given stability assignment. We define a sheaf to be stable if its multidegree is as prescribed by the stability assignment, both in the case of a single nodal curve and for families. Then we show that the moduli space of stable sheaves thus defined is a fine compactified Jacobian as introduced in Definitions 3.1 and 3.7. The case of a curve in isolation is Corollary 6.4, and the more general case of families with smooth generic element is Theorem 6.3. The most difficult part is the proof of properness (Lemma 6.5).

Throughout this section, let X be a nodal curve over some field extension K of k, and let $\Gamma $ be its dual graph.

Let now $\mathcal {X}/S$ be a family of nodal curves over a k-scheme S. For every geometric point s of S, denote by $\Gamma _s$ the dual graph of the fiber $\mathcal {X}_s$ .

The main result of this section is the following:

We immediately deduce the following:

The most delicate part of the proof of Theorem 6.3 is the following result, which builds on the combinatorial preparation of the previous section.

Proof. By Definitions 4.3 and 6.1, the scheme $\overline {\mathcal {J}}^d_{\mathfrak {S}}$ is the union of finitely many locally closed strata and hence of finite type and quasi-separated over S. To prove properness, we apply the refined valuative criterion [36, Lemma 70.22.3].

Given the spectrum $\Delta $ of a DVR and a commutative diagram (of solid arrows)

we need to prove that there exists exactly one dotted arrow that keeps the diagram commutative. (Note that $\operatorname {Pic}^d(\mathcal {X}_{\theta })$ is open and dense in $\overline {\mathcal {J}}^d_{\mathfrak {S}}$ ).

Let us recall that morphisms to moduli spaces of sheaves (line bundles, simple sheaves) correspond to sheaves on the corresponding family of curves, but this correspondence is a bijection only after identifying two such sheaves whenever they differ by the pullback of a line bundle from the base. In the remainder of this proof, we will slightly abuse the notation and assume this identification.

Let us denote by $\mathcal X_{\Delta }/\Delta $ the base change of $\mathcal X/S$ under $\Delta \rightarrow S$ and by $\overline {\mathcal {J}}^d_{\Delta }\subseteq {\operatorname {Simp}}^d(\mathcal X/\Delta )$ the base change of $\overline {\mathcal {J}}^d_{\mathfrak {S}}$ . Let s be the image of $0 \in \Delta $ under $\Delta \to S$ . Denoting by $X= \mathcal {X}_s=\mathcal {X}_{\Delta ,0}$ the fiber over $0 \in \Delta $ of the family $\mathcal {X}_{\Delta }/\Delta $ , by the hypothesis that $\mathcal {X}_{\theta }/\theta $ is smooth, we deduce that $\mathcal {X}_{\Delta }/\Delta $ is a smoothing of X. Moreover, we have that $\overline {\mathcal {J}}^d_{\Delta }$ is the disjoint union of $\operatorname {Pic}^d(\mathcal {X}_{\Delta , \eta })= {\operatorname {Simp}}^d(\mathcal {X}_{\Delta , \eta })$ and of $\overline {J}_\sigma \subseteq {\operatorname {Simp}}^d(X)$ , where $\sigma $ is the stability assignment for $\mathcal {X}_s$ .

The morphism $f\colon \Delta \setminus \{0\}\rightarrow \operatorname {Pic}^d(\mathcal {X}_\theta )$ corresponds to a line bundle $\mathcal [\mathcal {L}_{\eta }] \in \operatorname {Pic}^d(\mathcal {X}_{\Delta , \eta }/\eta )$ . We know from [Reference Esteves16] that $ {\operatorname {Simp}}^d(\mathcal X_\Delta /\Delta )$ satisfies the existence part of the valuative criterion of properness; hence, $\mathcal L_{\eta }$ extends (possibly nonuniquely) to an element $[\mathcal L'] \in {\operatorname {Simp}}^d(\mathcal {X}_\Delta /\Delta )$ . Our proof is concluded if we can show that $\mathcal L_{\eta }$ extends to a unique $[\mathcal L] \in \overline {\mathcal {J}}^d_{ \Delta }$ . When the total space $\mathcal {X}_{\Delta }$ of the smoothing $\mathcal {X}_\Delta /\Delta $ is regular, this follows immediately from Remark 4.2.

In general, each node e of the central fiber $X \subset \mathcal {X}_\Delta $ is a singularity of type $A_{m(e)}$ of the total space $\mathcal X_\Delta $ , for some $m(e) \in \mathbf {N}$ . Let $g \colon \widetilde {\mathcal {X}} \to \mathcal {X}_\Delta $ be the resolution of singularities obtained by blowing up $m(e)$ times each singularity e of $\mathcal X_\Delta $ . The central fiber $\widetilde {X} \subset \widetilde {\mathcal {X}}$ is therefore obtained by replacing each node e of the central fiber X of $\mathcal {X}_\Delta $ with a rational bridge of length $m(e)$ . In other words, the dual graph $\widetilde {\Gamma }$ of $\widetilde {X}$ is obtained by subdividing $m(e)$ times each edge of $\Gamma =\Gamma (X)$ by adding $m(e)$ additional genus $0$ vertices.

We have now achieved that $\widetilde {\mathcal {X}}$ is regular. Moreover, by [Reference Esteves and Pacini17, Proposition 5.5], for each $[F] \in {\operatorname {Simp}}^d(X)$ , there exists $[L] \in \operatorname {Pic}^d(\widetilde {X})$ such that $g_*(L)=F$ , and the multidegree $\mathbf {d}$ of F and the multidegree $\widetilde {\mathbf {d}}$ of any such L are related by Equation (5.2), where $G:=G(F)$ is the spanning subgraph consisting of all nodes where F is locally free.

In order to prove the existence and uniqueness of an extension of $\mathcal L_{\eta }$ to $\mathcal {X}_\Delta /\Delta $ with a sheaf in $\overline {\mathcal {J}}^d_{\Delta }$ , we prove that $g^*(\mathcal L_{\eta })$ extends uniquely on $\widetilde {\mathcal {X}}/\Delta $ to a line bundle whose restriction to $\widetilde {X}$ has multidegree in $\widetilde {\sigma }$ , where the elements of $\widetilde {\sigma }$ are defined by lifting elements of $\sigma $ to $\widetilde {\Gamma }$ as in Equation (5.2). Note that taking the direct image via g maps line bundles on $\widetilde {X}$ whose multidegree is in $\widetilde {\sigma }$ to rank $1$ , torsion free, simple sheaves on X whose multidegree is in $\sigma (G)$ for $G \subseteq \Gamma $ the spanning subgraph whose edges correspond to the exceptional rational bridges in $\widetilde {\Gamma }$ where the multidegree of the line bundle is equal to zero on all vertices.

By Part (2) of Proposition 5.4, we deduce that $\widetilde {\sigma }$ is a minimal complete set of representatives for the action of the twister group $\operatorname {\mathrm {Tw}}(\widetilde {\Gamma })$ on $S^d(\widetilde {\Gamma })$ .

We conclude, as in the regular case, that $g^*(\mathcal L_{\eta })$ extends to a unique line bundle $\widetilde {\mathcal L}$ on $\widetilde {\mathcal {X}}/\Delta $ whose multidegree on the central fiber $\widetilde {X}$ is an element of $\widetilde {\sigma }$ . Therefore, $g_*{\widetilde {\mathcal {L}}}$ is the unique extension of $\mathcal L_{\eta }$ whose multidegree on X is an element of $\sigma $ . This completes the proof.

We are now ready to complete the proof of Theorem 6.3.

7. From smoothable fine compactified Jacobians to stability assignments

In this section, we show that a smoothable fine compactified Jacobian defines a stability assignment in a natural way.

We start by considering the case of a curve in isolation and prove that the stability assignment associated with a (not necessarily smoothable) fine compactified Jacobian satisfies the first condition of Definition 4.3. This enables us to describe in Section 7.2 all fine compactified Jacobians for two classes of examples: vine curves and curves whose dual graph has topological genus $1$ . As an application, we prove in Lemma 7.8 that the stability assignment of a fine compactified Jacobian of an arbitrary nodal curve is nonempty for every connected spanning subgraph of the dual graph $\Gamma $ of the curve (a result needed in the proof of our main results, Corollaries 7.13 and 7.16).

Then we restrict ourselves to the case where the fine compactified Jacobian is smoothable and proceed to prove the second condition of Definition 4.3 for the associated stability assignment (Corollary 7.13). Finally, we consider the case of families (Proposition 7.14), and we conclude by proving our main result, Corollary 7.16, which establishes that the operations described in this section and in the previous one are inverse to each other.

Throughout this section, we will consider a fixed nodal curve X over a field extension K of k, with dual graph $\Gamma $ , and denote by $\overline {J} \subset \operatorname {Simp}^{d}(X)$ a degree d fine compactified Jacobian of X.

7.1. Definition and first properties

Definition 7.1. We define the associated assignment $\sigma _{\overline {J}}$ of ${\overline {J}}$ as

$$\begin{align*}\sigma_{\overline{J}}=\{(G(F),\underline{\deg}(F)):\;[F]\in\overline{J}\}, \end{align*}$$

where $G(F)=\Gamma \setminus {\operatorname {N}}(F)$ is the connected spanning subgraph of $\Gamma $ obtained by removing the edges corresponding to the nodes where F fails to be locally free.

A key point is that if a sheaf is an element of a given fine compactified Jacobian, then so are all other sheaves that fail to be locally free on the same set of nodes and that have the same multidegree:

Lemma 7.2. Let $\overline {J} \subset {\operatorname {Simp}}^d(X)$ be a degree d fine compactified Jacobian, and assume that $(G,\underline {\mathbf d}) \in \sigma _{\overline {J}}$ . Then $\overline {J}$ contains all sheaves $[F]\in {\operatorname {Simp}}^d(X)$ with $ {\operatorname {N}}(F) =\Gamma \setminus G$ and $\underline {\deg }(F)=\underline {\mathbf d}$ .

Proof. By possibly passing to the partial normalization of X at the nodes in $\Gamma \setminus G$ , we may assume that $G=\Gamma $ . Thus, we aim to prove that if $\mathcal {J}^{\underline {\mathbf d}}(X)\cap \overline {J} \neq \emptyset $ , then $\mathcal {J}^{\underline {\mathbf d}}(X) \subset \overline {J}$ .

The moduli space of line bundles $\mathcal {J}^{\underline {\mathbf d}}(X)$ of multidegree $\underline {\mathbf d}$ is irreducible and $\overline {J}$ is open, so $\mathcal {J}^{\underline {\mathbf d}}(X) \cap \overline {J} \neq \emptyset $ implies that $\mathcal {J}^{\underline {\mathbf d}}(X)\cap \overline {J}$ is dense in $\mathcal {J}^{\underline {\mathbf d}}(X)$ . For $[F]$ in $\mathcal {J}^{\underline {\mathbf d}}(X)$ , we can then find a line bundle L on $X \times \Delta $ , for some $\Delta $ , spectrum of a DVR, whose generic element $L_{\eta }$ is in $\mathcal {J}^{\underline {\mathbf d}}(X) \cap \overline {J}$ and whose special fiber $L_0$ equals F.

By properness of $\overline {J}$ , there exists a sheaf $L'$ on $X\times \Delta $ with the property that $L^{\prime }_{X \times \eta }=L_{X \times \eta }$ (here, $\eta $ is the generic point of $\Delta $ ) and whose special fiber $L^{\prime }_0$ is in $\overline {J}$ . Then $L' \otimes L^{-1}$ defines a morphism $\Delta \to {\operatorname {Simp}}^0(X)$ that maps the generic point $\eta $ to a closed point; hence, it must be the constant morphism. We conclude, in particular, that $[F]=[L_0']$ and so $[F] \in \overline {J}$ .

As a consequence of Lemma 5.2, the associated assignment to a fine compactified Jacobian satisfies the first condition of Definition 4.3:

Corollary 7.3. The associated assignment $\sigma _{\overline {J}}$ satisfies

(7.1) $$ \begin{align} (G,\underline{\mathbf d})\in\sigma_{\overline{J}} \Rightarrow (G\cup\{e\},\underline{\mathbf d}+\underline{\mathbf e}_{v_1}),(G\cup\{e\},\underline{\mathbf d}+\underline{\mathbf e}_{v_2})\in\sigma_{\overline{J}}, \end{align} $$

for all spanning subgraphs $G\subseteq \Gamma $ and all edges e of $\operatorname {\mathrm {Edges}}(\Gamma )\setminus \operatorname {\mathrm {Edges}}(G)$ with endpoints $v_1$ and $v_2$ .

Note that the above statement remains valid (with the same proof) for an arbitrary open subscheme of $ {\operatorname {Simp}}^d(X)$ ; properness does not play any role here.

For every connected spanning subgraph G of $\Gamma $ , the pairs $(G,\underline {\mathbf d})\in \sigma $ with $G\subset G$ define a fine compactified Jacobian on a partial normalization of the curves X.

Lemma 7.4. Let $f \colon X' \to X$ be a partial normalization of X at some nodes $e_1, \ldots , e_k \in \operatorname {\mathrm {Edges}}(\Gamma )$ and let $j_f \colon {\operatorname {Simp}}^{d-k}(X') \to {\operatorname {Simp}}^d(X)$ be the morphism induced by taking the pushforward along f. If $\overline {J}$ is a fine compactified Jacobian of X, then any geometrically connected component of $j_f^{-1} (\overline {J})$ is a fine compactified Jacobian of $X'$ . Moreover, for every $G \subseteq \Gamma (X')$ , we have

(7.2) $$ \begin{align} \bigcup_{J'\subset j_f^{-1}(\overline{J})}\sigma_{J'}(G)=\sigma_{\overline{J}}(G),\end{align} $$

where the union is over all geometrically connected components $J'$ of $j_f^{-1}(\overline {J})$ .

Note that Lemma 7.4 does not exclude the possibility that $j_f^{-1} (\overline {J})= \emptyset $ . This possibility will be excluded later, in Lemma 7.8. Moreover, we observe that the pullback $j_f^{-1} (\overline {J})$ is not necessarily connected: an example is given by nonsmoothable fine compactified Jacobians of curves of arithmetic genus $1$ ; see the $\rho \geq 2$ case of [Reference Pagani and Tommasi32, Lemma 3.7].

Proof. The fact that $j_f^{-1} (\overline {J})$ is open in $ {\operatorname {Simp}}^{d-k}(X')$ follows from the fact that openness is stable under base change.

Moreover, $j_f^{-1} (\overline {J})$ is separated and of finite type because $\overline {J}$ is. To complete the first part of the statement, it remains to show that $j_f^{-1} (\overline {J})$ satisfies the existence part of the valuative criterion of properness.

Let $\Delta \setminus \{0 \} \to j_f^{-1} (\overline {J})$ be a morphism out of a DVR without its closed point, which corresponds to an element $\mathcal {F'}$ in $ {\operatorname {Simp}}^{d-k}( X'\times \eta /\eta )$ . Because $\overline {J}$ satisfies the existence part of the valuative criterion of properness, there exists a family of sheaves $\tilde {\mathcal {F}}$ in $ {\operatorname {Simp}}^{d-k}(X \times \Delta /\Delta )$ whose restriction to $\eta $ equals $f_* (\mathcal {F})$ . Then the family $f^*(\tilde {\mathcal {F}})/\text {torsion}$ is an extension of the original family $\mathcal {F}'$ , and with central fiber belonging to $j_f^{-1} (\overline {J})$ (because $f^*(f_*(\tilde {\mathcal {F}}))/\text {torsion}=\tilde {\mathcal {F}}$ – see [Reference Alexeev1, Lemma 1.5]). This concludes the first part.

The second part, or Equation (7.2), follows immediately from the definition of an associated assignment.

7.3. Stability assignments of smoothable fine compactified Jacobians

We now go back to our main line of reasoning and show that the assignment associated to a smoothable fine compactified Jacobian also satisfies the second condition of Definition 4.3.

Proposition 7.10. If $\overline {J} \subseteq {\operatorname {Simp}}^d(X)$ is also smoothable, then for all spanning subgraphs $G \subseteq \Gamma $ , the subset $\sigma _{\overline {J}}(G)\subset S^d_{\Gamma }(G)$ is a minimal complete set of representatives for the action of the twister group $\operatorname {\mathrm {Tw}}(G)$ on $S^d_{\Gamma }(G)$ .

Proof. We first fix some notation for our proof. Let $\mathcal {X}/\Delta $ be a regular smoothing of X such that there exists an open $\overline {\mathcal J}_{\mathcal {X}/\Delta } \subseteq {\operatorname {Simp}}^d(\mathcal {X}/\Delta )$ and $\Delta $ -proper subscheme, whose special fiber equals $\overline {J}$ . Then consider the degree $2$ base change $\Delta \to \Delta $ of $\mathcal {X}/\Delta $ , giving a nonregular smoothing $\mathcal {X}'/ \Delta $ with $A_1$ singularities at all nodes of the special fiber X. By functoriality of the Picard functor, and by stability under base change of openness and properness, we have that the base change $\overline {\mathcal J}^{\prime }_{\mathcal {X}'/\Delta }$ is also an open and $\Delta $ -proper subscheme of $ {\operatorname {Simp}}^d(\mathcal {X}'/\Delta )$ . Finally, let $f \colon \widetilde {\mathcal {X}} \to \mathcal {X}' $ be the blow-up at all singularities. The family $\widetilde {\mathcal {X}}$ is then a regular smoothing of the special fiber $\widetilde {X}$ , the curve obtained from X by replacing each node with an irreducible rational bridge.

We are now ready for the proof. First, observe that, by Corollary 7.3 and by Lemma 7.8 combined with Proposition 4.7, the set $\sigma _{\overline {J}}$ satisfies the hypothesis of Proposition 5.4 Part (1). By applying loc. cit., it is enough to prove that the collection $\widetilde {\sigma }_{\overline {J}}$ obtained by lifting every element of $\sigma _{\overline {J}}$ via Equation 5.2 is a minimal complete set of representatives for the action of $\operatorname {\mathrm {Tw}}({\Gamma (\widetilde {X}}))$ .

Let $\mathbf {\vec {d}}$ be the lift via (5.2) of some ${\mathbf d} \in S^d_\Gamma (G)$ for some $G \subseteq \Gamma $ . There exists $[L] \in \operatorname {\mathrm {Pic}}^d(\widetilde {X})$ such that $\underline {\deg }(f_*(L)) \in S^{\mathbf{d}}_{\Gamma }(G)$ . By Hensel’s lemma, L extends to a family $[\mathcal {L}] \in \operatorname {Pic}^d(\widetilde {\mathcal {X}}/\Delta )$ . Since $\overline {J}_{\mathcal {X}'/\Delta }$ is universally closed over $\Delta $ , there exists $\mathcal {F}'\in {\operatorname {Simp}}^d(\mathcal {X}'/\Delta )$ such that $\mathcal {F}'|_{\mathcal {X}^{\prime }_\eta }= f_* \mathcal {L}_{\mathcal {X}^{\prime }_\eta }$ and $[\mathcal {F}'|_X] \in \overline {J}_{X'}$ . We take $[\mathcal {L}'] \in \operatorname {Pic}^d(\widetilde {\mathcal {X}}/\Delta )$ such that $f_*(\mathcal {L}')= \mathcal {F}'$ . Thus, we have that $\left (\mathcal {L}'\otimes \mathcal {L}^{-1}\right )_{\widetilde {\mathcal {X}}_{\eta }}=\mathcal {O}_{\mathcal {X}_{\eta }}$ . We let then $\mathbf {\vec {t}}= \underline {\deg }(\mathcal {L}'\otimes \mathcal {L}^{-1}|_{\widetilde {X}})$ , and so $\mathbf {\vec {d}}+\mathbf {\vec {t}}= \underline {\deg } (\mathcal {L}'|_{\widetilde {X}})$ is in $\widetilde {\sigma }_{\overline {J}}$ . This proves that $\widetilde {\sigma }_{\overline {J}}$ is a complete set of representatives.

To prove minimality, assume that there exist lifts $\mathbf {\vec {d}_1}, \mathbf {\vec {d}_2} \in \widetilde {\sigma }_{\overline {J}}$ such that $\mathbf {\vec {d}_2}=\mathbf {\vec {d}_1}+ \mathbf {\vec {t}}$ for some $\mathbf {\vec {t}} \in \operatorname {\mathrm {Tw}}(\Gamma (\widetilde {X}))$ . By Hensel’s lemma and by Lemma 7.2, there exist $[\mathcal {L}_1], [\mathcal {L}_2] \in \operatorname {\mathrm {Pic}}^d(\widetilde {\mathcal {X}}/\Delta )$ , coinciding on the generic fiber, and whose multidegrees on $\widetilde {X}$ equal $\mathbf {\vec {d}_1}, \mathbf {\vec {d}_2}$ . The pushforwards $f_*(\mathcal {L}_1)$ and $f_*(\mathcal {L}_2)$ coincide on $\mathcal {X}^{\prime }_{\eta }$ , and because $\overline {J}_{\mathcal {X}'/\Delta }$ is separated over $\Delta $ , their central fibers must coincide, which implies that $\mathbf {\vec {t}}=0$ . This proves minimality.

Remark 7.11. In [Reference Caporaso13], Caporaso introduced the notion of ‘being of Néron type’ for a degree d compactified Jacobian of a nodal curve X. In our notation, this can be restated as the property that $\sigma _{\overline {J}}(\Gamma (X))$ is a minimal complete set of representatives for the action of $\operatorname {\mathrm {Tw}}(\Gamma (X))$ on $S^d_{\Gamma (X)}(\Gamma (X))$ .

Proposition 7.10 proves, in particular, that all smoothable fine compactified Jacobians are of Néron type. Similar results were obtained in [Reference Kass23] and [Reference Melo and Viviani29] for fine compactified Jacobians obtained from some numerical polarizations (see Section 8 for the notion of a numerical polarization).

We refer the reader to [Reference Kass23] for the definition of a Néron model and its relations with compactified Jacobians.

Remark 7.12. The smoothability assumption in the statement of Proposition 7.10 is crucial. In [Reference Pagani and Tommasi32], the authors give an example, for X a nodal curve of genus $1$ , of degree $0$ fine compactified Jacobians $\overline {J} \subset {\operatorname {Simp}}^0(X)$ whose collection of line bundle multidegrees contain an arbitrary number $r\geq 2$ of elements for each orbit of the action of $\operatorname {\mathrm {Tw}}(\Gamma )$ on $S^0_{\Gamma }(\Gamma )$ . Such compactified Jacobians can always be extended to a universally closed family over a regular smoothing, but such extensions are separated if and only if r equals $1$ .

We conclude with the main result:

Corollary 7.13. The associated assignment (Definition 7.1) to a smoothable degree d fine compactified Jacobian of X is a degree d stability assignment for the dual graph $\Gamma (X)$ (as in Definition 4.3).

Proof. This is obtained as a combination of Corollary 7.3 and Proposition 7.10.

We conclude by observing that the same result holds for families.

Proposition 7.14. Let $\mathcal {X}/S$ be a family of nodal curves over a base scheme (or Deligne–Mumford stack) S, and let $\overline {\mathcal J}=\overline {\mathcal J}_{\mathcal {X}/S}$ be a degree d fine compactified Jacobian for the family. Assume that for each geometric point s of S, the fine compactified Jacobian $\overline {J}_s$ of the fiber $X_s$ over $s \in S$ is smoothable.

Then the collection $\{\sigma _{\overline {J}_s}\}$ of associated assignments of $\overline {J}_s$ for all geometric points s of S is a family of degree d stability assignments (as in Definition 4.9).

Proof. The case where S is a single geometric point is Corollary 7.13.

To complete our proof, we will show that the assignment of $\sigma _{\overline {J}_s}$ is compatible with all morphisms $f \colon \Gamma (\mathcal {X}_{s}) \to \Gamma (\mathcal {X}_t)$ arising from étale specializations $t \rightsquigarrow s$ .

Assume that $F_{t}$ is a simple sheaf on the nodal curve $\mathcal {X}_{t}$ that specializes to $F_s$ on $\mathcal {X}_s$ . Suppose that the subcurve $\mathcal {X}_{t, 0} \subseteq \mathcal {X}_{t}$ generalizes the subcurve $\mathcal {X}_{s, 0} \subseteq \mathcal {X}_s$ . By flatness and by continuity of the Euler characteristic, and because we are passing to the torsion-free quotients, the degrees are related by

(7.3) $$ \begin{align} \deg_{\mathcal{X}_{t, 0}}(F_{t})= \deg_{\mathcal{X}_{s, 0}}(F_{s}) + n(F_s,f), \end{align} $$

where $n(F_s,f)$ is the number of nodes of $\mathcal {X}_{s,0}$ that are smoothened in t and where $F_s$ fails to be locally free. (See Equation (2.1)). Formula (7.3) is precisely the condition of Equation (4.2).

Remark 7.15. Assume that the dual graph of the fibers of $\mathcal {X}/S$ is constant in S, and let S be irreducible with $\eta \in S$ its generic point. Then the family $\mathcal X/S$ induces a group homomorphism from the group of étale specializations of $\eta $ to itself, which equals the Galois group $\operatorname {Gal}(k(\eta )^{\operatorname {sep}}, k(\eta )))$ , to the automorphism group $\operatorname {Aut}(\Gamma (X_{k(\eta )^{\operatorname {sep}}}))$ of the dual graph of the generic fiber. Let G be the image of this group homomorphism. In this case, Proposition 7.14 implies that the associated assignment $\sigma _{\overline {J}_{k(\eta )}}$ , which is defined as a collection of discrete data on the dual graph $\Gamma (X_{k(\eta )^{\operatorname {sep}}})=\Gamma (X_{\overline {k(\eta )}})$ , is invariant under the action of G.

In the particular case where S is a stratum of $\overline {\mathcal {M}}_{g,n}$ (i.e., S is the Deligne–Mumford moduli stack of curves whose dual graph is isomorphic to a fixed graph $\Gamma \in G_{g,n}$ ), we have $G=\operatorname {Aut}(\Gamma )$ .

By combining Theorem 6.3/Corollary 6.4 with Proposition 7.14 and Lemma 7.2, we immediately obtain that the two operations of (1) taking the associated assignment to a fine compactified Jacobian, and (2) constructing the moduli space of stable sheaves associated with a given stability assignment, are inverses of each other.

Corollary 7.16. Let $\mathcal {X}/S$ be a family of nodal curves over an irreducible scheme (or Deligne–Mumford stack) S, and assume that $\mathcal {X}_{\theta }/\theta $ is smooth for $\theta $ the generic point of S.

If $\overline {\mathcal J} \subseteq {\operatorname {Simp}}^d(\mathcal {X}/S)$ is a degree d fine compactified Jacobian and $\sigma _{\overline {\mathcal J}}$ is its associated assignment (Definition 7.1), then the moduli space $\overline {\mathcal {J}}_{\sigma _{\overline {\mathcal J}}}$ of $\sigma _{\overline {\mathcal J}}$ -stable sheaves (Definitions 6.1 and 6.2) equals $\overline {\mathcal J}$ .

Conversely, if $\tau $ is a family of degree d stability assignments, and $\overline {\mathcal J}_\tau $ is the moduli space of $\tau $ -stable sheaves, then the associated assignment $\sigma _{\overline {\mathcal J}_\tau }$ equals $\tau $ .

8. Numerical polarizations

An example of a (smoothable fine compactified Jacobian) stability assignment on a nodal curve $X/K$ (as in Definition 4.3) comes from numerical polarizations, introduced by Oda–Seshadri in [Reference Oda and Seshadri31]. (In fact, the Oda–Seshadri formalism permits to also construct compactified Jacobians that are not necessarily fine in the sense of Definition 3.1).

In this section, we review the notion of numerical stability for a single curve (Definition 8.1) and extend it to families (Definition 8.10) following [Reference Kass and Pagani25].

We let $\Gamma $ be the dual graph of X.

Definition 8.1. Let $V^d(\Gamma ) \subset \mathbf {R}^{\operatorname {\mathrm {Vert}}(\Gamma )}$ be the sum-d affine subspace. Let ${\phi } \in V^d(\Gamma )$ , and let G be a (not necessarily connected) spanning subgraph of $\Gamma $ , with $E_0:= \operatorname {\mathrm {Edges}}(G)$ and $E_0^c:= \operatorname {\mathrm {Edges}}(\Gamma ) \setminus E_0$ . We say that $\underline {\mathbf d} \in S^d_{\Gamma }(G)$ is ${\boldsymbol {\phi }}$ -semistable (resp. ${\boldsymbol {\phi }}$ -stable) on G when the inequality

(8.1) $$ \begin{align} \left|\sum_{v \in V}( \underline{\mathbf{d}}(v) -\phi(v)) + \left|E_0^c \cap E(\Gamma(V)) \right|+\frac{\left|E_0^c \cap \operatorname{E}(V,V^c)\right|}{2}\right|\leq \frac{\left|E_0 \cap \operatorname{E}(V,V^c) \right|}{2} \end{align} $$

(resp. $<$ ) is satisfied for all $\emptyset \neq V \subsetneq \operatorname {\mathrm {Vert}}(\Gamma )$ . (To keep the notation compact, in the inequality, we have denoted the edge sets by $\operatorname {E}$ instead of $\operatorname {\mathrm {Edges}}$ ).

For $\phi \in V^d(\Gamma )$ , we define the (numerical, Oda–Seshadri) assignment associated with $\phi $ , denoted $\operatorname {\mathrm {\sigma }}_{\Gamma , \phi }$ , by setting

(8.2) $$ \begin{align} \operatorname{\mathrm{\sigma}}_{\Gamma, {\phi}}(G):= \{ \underline{\mathbf d} \in S^d_{\Gamma}(G): \underline{\mathbf d} \text{ is } {\phi} \text{-semistable on } G \} \subset S^d_{\Gamma}(G) \end{align} $$

for all spanning subgraphs $G \subseteq \Gamma $ .

We define ${\phi } \in V^d(\Gamma )$ to be nondegenerate when for every spanning subgraph $G\subseteq \Gamma $ , all elements of $\operatorname {\mathrm {\sigma }}_{\Gamma , {\phi }}(G)$ are ${\phi }$ -stable.

Elements of $V^d(\Gamma )$ are called numerical polarizations.

Every nondegenerate numerical polarization gives a stability assignment:

This leads to the following natural question:

Below, we give three classes of examples where we can answer Question 8.4 in the positive.Footnote ⁶

Example 8.6 (Vine curves of type t)

Let $\Gamma $ consist of $2$ vertices $v_1, v_2$ connected by t edges (and no loops). The complexity of $\Gamma $ equals t. With the notation as in Example 4.6, it is straightforward to check that a stability assignment $\sigma _{\Gamma }$ determined by some $\lambda \in \mathbf {Z}$ as described in loc. cit. equals $\operatorname {\mathrm {\sigma }}_{\Gamma , \phi _\Gamma }$ for

$$\begin{align*}\phi_\Gamma(v_1, v_2):= \left(\lambda- \frac{t-1}{2}, d -\lambda + \frac{t-1}{2} \right),\end{align*}$$

and that $\phi _\Gamma \in V^d(\Gamma )$ is nondegenerate.

Example 8.8 (Integral Break Divisors)

Let $\Gamma $ be a (not necessarily stable) graph and assume $d=g(\Gamma )$ . Define the stability assignment $\sigma _{\operatorname {IBD}}$ by

$$\begin{align*}\sigma_{\operatorname{IBD}}(G)=\{ \text{integral break divisors for } G \},\end{align*}$$

for all connected spanning subgraphs $G \subseteq \Gamma $ . Recall that $\underline {\mathbf {d}} \in S^{g(\Gamma )}_{\Gamma }(G)$ is an integral break divisor for G if and only if it is of the form

$$\begin{align*}\underline{\mathbf{d}} (v) =g(v) + \sum_{e \in \operatorname{\mathrm{Edges}}(G) \setminus \operatorname{\mathrm{Edges}}(T)} \underline{\mathbf{e}}_{t(e)}(v) \end{align*}$$

for some choice of a spanning tree $T \subseteq G$ and of an orientation $t \colon \operatorname {\mathrm {Edges}}(G) \setminus \operatorname {\mathrm {Edges}}(T) \to \operatorname {\mathrm {Vert}}(G)$ (for $w \in \operatorname {\mathrm {Vert}}(G)$ , we denote by $\underline {\mathbf {e}}_w$ the function that is $1$ on the vertex w and $0$ elsewhere).

We claim that $\sigma _{\operatorname {IBD}}$ is indeed a degree $d=g(\Gamma )$ stability assignment. Condition (1) of Definition 4.3 follows directly from the definition of an integral break divisor. The second axiom follows immediately from [Reference An, Baker, Kuperberg and Shokrieh4, Theorem 1.3] (earlier proved in [Reference Mikhalkin and Zharkov30]).

When $\Gamma $ is stable, it follows from [Reference Christ, Payne and Shen15, Lemma 5.1.5] that $\sigma _{\operatorname {IBD}}= \sigma _{\Gamma , \phi _{\operatorname {can}}^{\Gamma }}$ , for

(8.3) $$ \begin{align} \phi_{\operatorname{can}}^{\Gamma}(v):= \frac{g(\Gamma) }{2 g(\Gamma) -2} \cdot \left(2g(v)-2+\operatorname{val}_{\Gamma}(v)\right), \end{align} $$

where $\operatorname {val}_{\Gamma }(v)$ is the valence of the vertex v in $\Gamma $ . (This follows from loc. cit. as $\phi ^{\Gamma }_{\operatorname {can}}$ is induced from the canonical divisor, which is ample when $\Gamma $ is stable).

When $\Gamma $ is not necessarily stable, we define

(8.4) $$ \begin{align} \phi_{\operatorname{IBD}}(v):= \frac{g(\Gamma) + |\operatorname{\mathrm{Vert}}(\Gamma)|}{2 \left( g(\Gamma) + |\operatorname{\mathrm{Vert}}(\Gamma)|\right) -2} \cdot (2g(v)+\operatorname{val}_{\Gamma}(v)) -1. \end{align} $$

We claim that $\sigma _{\operatorname {IBD}}= \sigma _{\Gamma , \phi _{\operatorname {IBD}}}$ .

Let $\Gamma '$ be the graph obtained from $\Gamma $ by increasing the genus of each vertex by $1$ . Then the integral break divisors of $\Gamma '$ are the integral break divisors on $\Gamma $ increased by $1$ on each vertex and $\Gamma '$ is stable. Therefore, the integral break divisors on $\Gamma $ are stable for the stability assignment $\phi _{\operatorname {can}}^{\Gamma '}-1$ , which equals $\phi _{\operatorname {IBD}}$ . This proves our claim.

We are now ready to define families of numerical polarizations, similarly to what was done in Definition 4.9 for families of stability assignments. We shall see that there are two ways of doing so, and that they are not equivalent.

Definition 8.9. Let $f \colon \Gamma \to \Gamma '$ be a morphism of stable graphs. We say that $\phi \in V^d(\Gamma )$ is f-compatible with $\phi ' \in V^d(\Gamma ')$ if

$$\begin{align*}\phi'(w)= \sum_{f(v)=w} \phi(v). \end{align*}$$

For a strongly compatible collection, one can define the notion of a $\Phi $ -stable sheaf for all fibers as in Definition 8.1. We shall not do that, as this would be a repetition of our Definition 6.2. Instead, we observe the following:

In [Reference Kass and Pagani25], the authors studied the theory of universal Oda–Seshadri stability assignments for compactified Jacobians. In loc. cit., they defined, for fixed $(g,n)$ such that $2g-2+n>0$ , the space $V_{g,n}^d$ of universal numerical polarizations – that is, elements $\Phi =(\phi _s)_{s \in \overline {\mathcal {M}}_{g,n}}$ that are strongly compatible for morphisms $f \colon \Gamma _1 \to \Gamma _2$ between any two elements of ${G}_{g,n}$ (a skeleton of the category of stable n-pointed graphs of genus g).

In loc. cit., the authors also constructed, for each such nondegenerate $\Phi \in V_{g,n}^d$ , a compactified Jacobian $\overline {\mathcal {J}}_{g,n}(\Phi )$ . In the language of this paper, this can be rephrased as follows.

A natural question that we will address next is whether every fine compactified universal Jacobian arises as in Corollary 8.12, or if one can construct fine compactified universal Jacobian from nondegenerate weakly compatible numerical stability conditions that are not strongly compatible.

Our main result in the next section, Theorem 9.7, settles in the affirmative the analogous question for the case of the universal family $\overline {\mathcal {C}}_{g}/ \overline {\mathcal {M}}_{g}$ for all $g \geq 2$ . More explicitly, when $n=0$ , every fine compactified universal Jacobian $\overline {\mathcal {J}}_{g}$ arises as in Corollary 8.12 (i.e., it is of the form $\overline {\mathcal {J}}_{g}(\Phi )$ for some $\Phi \in V_g^d=V_{g,0}^d$ ).

9. Classification of universal stability assignments

The main result in this section is a classification of universal stability assignments – that is, the case where the family $\mathcal {X}/S$ is the universal family over $S= \overline {\mathcal {M}}_g$ , the moduli stack of stable curves of genus g (and no marked points). As an intermediate step, we will also produce results that are valid for the case of the universal family over $S= \overline {\mathcal {M}}_{g,n}$ for arbitrary n.

One way to generate universal stability assignments is by means of strongly compatible (universal) numerical polarizations; see Definition 8.10 and Corollary 8.12. Our main result here is Theorem 9.7, where we classify all universal stability assignments with $n=0$ and for every genus, by proving that they all arise from strongly compatible numerical polarizations (i.e., they all are of the form $\operatorname {\mathrm {\sigma }}_{\Phi }$ for some $\Phi \in V_g^d$ (see Corollary 8.11 and Corollary 8.12)). Combining with the fact, shown in Proposition 7.14, that universal stability assignments classify fine compactified universal Jacobians, we deduce in Corollary 9.8 that, in the absence of marked points, there are no fine compactified universal Jacobians other than the classical ones constructed in the nineties by Caporaso, Pandharipande and Simpson.

As discussed in Remark 8.14, an analogous result does not hold, in general, when $n>0$ .

Recall from Example 3.5 that for $t \in \mathbf {N}$ , a vine curve of type $\boldsymbol {t}$ is a curve with $2$ nonsingular irreducible components joined by t nodes. These curves and their dual graphs will play an important role in this section.

We will break down our argument in $3$ parts.

9.1. First part: Universal stability assignments are uniquely determined by their restrictions to vine curves

The main result of this subsection, summarized in the title, is Corollary 9.3 below. The result is obtained as a combination of Lemma 9.1 and Lemma 9.2. The first establishes that a stability assignment for a given curve is uniquely determined by its restriction to the spanning trees of the dual graph of that curve. The second one shows that an assignment of integers on all spanning trees of all elements of $G_{g,n}$ that is compatible with graph morphisms is uniquely determined by its values over the dual graphs of all vine curves.

Let us fix a degree d universal stability assignment $\sigma $ of type $(g,n)$ . Recall from Definition 4.9 (and Remark 4.10) that this is the datum of a collection

$$\begin{align*}\operatorname{\mathrm{\sigma}}=\{\operatorname{\mathrm{\sigma}}_{\Gamma}\}_{\Gamma \in {G}_{g,n}}\end{align*}$$

that is compatible with all graph morphisms in $G_{g,n}$ .

By Condition (1) of Definition 4.3, if $T \subseteq \Gamma $ is a spanning tree, then $\operatorname {\mathrm {\sigma }}_{\Gamma }(T)= \{\underline {\mathbf d}_{T}\}$ contains a unique multidegree $\underline {\mathbf d}_{T} \in S^d_\Gamma (T)$ . The following lemma shows that a stability assignment is ‘overdetermined’ by the collection of data obtained by extracting this unique value from all spanning trees.

Lemma 9.1. Let $\Gamma $ be a stable graph and let us fix, for all spanning trees T of $\Gamma $ , a multidegree $\underline {\mathbf d}_{T} \in S^d_\Gamma (T)$ . Then there exists at most one degree d stability assignment $\operatorname {\mathrm {\sigma }}_{\Gamma }$ whose value $\operatorname {\mathrm {\sigma }}_{\Gamma }(T)$ equals $\{\underline {\mathbf d}_{T}\}$ for all spanning trees T.

Proof. Let $\operatorname {\mathrm {\sigma }}_{\Gamma }$ be a degree d stability assignment that satisfies the hypothesis. Inductively define another collection $M_{\Gamma }$ by defining $M_{\Gamma }(G) \subset S_{\Gamma }^d(G)$ for all spanning subgraphs G of $\Gamma $ . Starting from the assignments $M_{\Gamma }(T):=\{d_{T}\}$ for all spanning trees $T \subseteq G$ , let $M_{\Gamma }$ be the minimal assignment that satisfies Condition (1) of Definition 4.3.

Because $\operatorname {\mathrm {\sigma }}_{\Gamma }$ satisfies Condition (1) of Definition 4.3, for all spanning subgraphs $G \subseteq \Gamma $ , we have the inclusion $M_{\Gamma }(G) \subseteq \operatorname {\mathrm {\sigma }}_{\Gamma }(G)$ . The inequality $|M_{\Gamma }(G)|\geq c(G)$ follows from Proposition 4.7. By Condition (2) of Definition 4.3, we also have the equality $|\operatorname {\mathrm {\sigma }}_{\Gamma }(G)| = c(G)$ . From this, we conclude that $M_{\Gamma }(G)$ and $\operatorname {\mathrm {\sigma }}_{\Gamma }(G)$ must coincide.

By applying the same idea as in [Reference Kass and Pagani24, Lemma 3.8] and [Reference Kass and Pagani25, Lemma 3.9], we now see how compatibility with graph morphisms propagates an assignment of multidegrees on all strata of vine curves (with the exception of the vine curves whose graph admits a symmetry that swaps the two vertices) to an assignment on all vertices of all spanning trees of all graphs $\Gamma \in G_{g,n}$ . The reason for excluding the symmetric graphs is that compatibility under automorphisms forces all universal degree d stability assignments to have the same value on those graphs.

Let $T_{g,n} \subseteq G_{g,n}$ be the collection of loopless graphs with $2$ vertices, and let $T^{\prime }_{g,n} \subseteq T_{g,n}$ be the subset of graphs such that each automorphism fixes the two vertices. For each element $G \in T_{g,n}$ , fix an ordering of its two vertices (i.e., $\operatorname {\mathrm {Vert}}(G)=\{v_1^G, v_2^G\}$ ). Let $C_{g,n} \subseteq T_{g,n}$ and be the subset of graphs with exactly $1$ edge (equivalently, the dual graphs of vine curves with no separating nodes, equivalently, the dual graphs of vine curves whose strata have codimension $1$ in $\overline {\mathcal {M}}_{g,n}$ ), and let $G_{g,n}^{\operatorname {NS}} \subseteq G_{g,n}$ be the subset of graphs without separating edges.

Lemma 9.2. For each function $\alpha \colon T^{\prime }_{g,n} \to \mathbf {Z}$ , there exists a unique collection of assignments $\underline {\mathbf d}_{\Gamma , T} \in S^d_{\Gamma }(T)$ for each spanning tree $T \subseteq \Gamma $ of each element $\Gamma \in G_{g,n}$ such that we have

(9.1) $$ \begin{align} \sum_{v \in \operatorname{\mathrm{Vert}}(\Gamma) : f(v) = v_1^G} \underline{\mathbf d}_{\Gamma, T}(v) = \alpha(G) \end{align} $$

for all morphisms $f \colon \Gamma \to G$ where G is in $T_{g,n}$ .

Similarly, for each function $\beta \colon T^{\prime }_{g,n}\setminus C_{g,n} \to \mathbf {Z}$ , there exists a unique collection of assignments $\underline {\mathbf d}_{\Gamma , T} \in S^d_{\Gamma }(T)$ for each spanning tree $T \subseteq \Gamma $ of each element $\Gamma \in G_{g,n}^{\operatorname {NS}}$ such that (9.1) is satisfied for all morphisms $f \colon \Gamma \to G$ where G is in $T_{g,n}$ .

Note that Equation (9.1) is the same as compatibility for graph morphisms defined in Definition 4.8, Equation (4.2).

Proof. We only prove the first statement and leave the second to the reader.

The statement is an extension of the proof given in [Reference Kass and Pagani24, Lemma 3.8]. The main point of the proof is that for a fixed spanning tree $T \subseteq \Gamma $ , contracting all but one edge of T induces a bijection from $\operatorname {\mathrm {Edges}}(T)$ to the set of morphisms from $\Gamma $ to some graph G isomorphic to an element of $T_{g,n}$ . The isomorphism is unique when $G \in T^{\prime }_{g,n}$ , and there are two isomorphisms when $G \in T_{g,n} \setminus T^{\prime }_{g,n}$ .

Each $e \in \operatorname {\mathrm {Edges}}(T)$ induces a morphism $\Gamma \to \Gamma _e$ obtained by contracting all edges $\operatorname {\mathrm {Edges}}(T) \setminus \{ e\}$ . There are two cases, depending on how each automorphism of $\Gamma _e$ acts on $\operatorname {\mathrm {Vert}}(\Gamma _e)$ : (1) If each acts trivially, then there exists a unique isomorphism from $\Gamma _e$ to an element of $T^{\prime }_{g,n}$ . (2) If some act nontrivially, there are two isomorphisms from $\Gamma _e$ to an element of $T_{g,n} \setminus T^{\prime }_{g,n}$ .

Now we view $\underline {\mathbf {d}}_{\Gamma , T}$ as a vector with $|\operatorname {\mathrm {Vert}}{T}|=|\operatorname {\mathrm {Vert}}{\Gamma }|$ unknown entries. In Case (1), we obtain an affine linear constraint among these unknowns by Equation (9.1). In Case (2), a linear constraint is given by compatibility for the extra automorphism, which imposes that the sum of the values of $\underline {\mathbf {d}}_{\Gamma , T}$ on the vertices on one side of e equals the sum of the values on the vertices on the other side of e. Altogether, this gives $\left |\operatorname {\mathrm {Edges}}(T)\right |=\left |\operatorname {\mathrm {Vert}}(\Gamma )\right |-1$ affine linear constraints on the $\left |\operatorname {\mathrm {Vert}}(\Gamma )\right |$ different entries of $\underline {\mathbf d}_{\Gamma , T}$ . One more affine linear constraint is given by the fact that $\underline {\mathbf {d}}_{\Gamma , T} \in S^d_{\Gamma }(T) \subset \mathbf {Z}^{\operatorname {\mathrm {Vert}}(\Gamma )}$ . All in all, we obtain an affine linear system of $|\operatorname {Vert}(T)|$ equations in $|\operatorname {Vert}(T)|$ unknowns, which one can check has invertible determinant over $\mathbf {Z}$ – hence a unique solution for $\underline {\mathbf d}_{\Gamma , T}$ .

Lemmas 9.1 and 9.2 allow us to regard every degree d universal stability assignment of type $(g,n)$ as an element of $\mathbf {Z}^{T^{\prime }_{g,n}}$ . However, not all elements of $\mathbf {Z}^{T^{\prime }_{g,n}}$ give rise to a universal stability assignment. For $\Gamma \in G_{g,n}$ , it can happen that the collection on all spanning trees $\{\underline {\mathbf {d}}_{\Gamma ,T}\}_{T}$ obtained from some $\alpha \in \mathbf {Z}^{T^{\prime }_{g,n}}$ as in Lemma 9.2 is not the restriction of any stability assignment on the graph $\Gamma $ (see Lemma 9.1).

Corollary 9.3. Let $\sigma , \tau $ be degree d universal stability assignments of type $(g,n)$ . If $\sigma _G=\tau _G$ for all $G \in T^{\prime }_{g,n}$ , then $\sigma =\tau $ . If $\sigma _G = \tau _G$ for all $G \in T^{\prime }_{g,n} \setminus C_{g,n}$ , then $\sigma $ and $\tau $ coincide on all curves that have no separating nodes.

Proof. Follows immediately from Lemmas 9.1 and 9.2.

9.2. Second part: When $n=0$ , over each vine curve with at least $2$ nodes there is at most $1$ stability assignment that extends to a universal stability

In this subsection, we fix $g \geq 2$ and $n=0$ . The main result here is Corollary 9.5.

We first need some combinatorial preparation. Let $\operatorname {GSym}_g$ be the trivalent graph with $2g-2$ vertices $v_1, \ldots , v_{2g-2}$ of genus $0$ , where each vertex $v_i$ is connected to $v_{i-1}$ , $v_{i+1}$ and $v_{i+g-1}$ (indices should be considered modulo $2g-2$ ). For $i\in \mathbf {Z}/(2g-2)\mathbf {Z}$ and $j=1,\dots ,g-1$ , we shall denote by $e_i$ the edge joining $v_i$ and $v_{i+1}$ and by $e_j'$ the edge joining $v_j$ and $v_{j+g-1}$ . (See Picture 1).

Figure 1 A picture of the graph $\operatorname {GSym}_g$ .

Let $\Gamma _g \subset \operatorname {GSym}_g$ be the maximal $1$ -cycle consisting of all edges of the form $e_i$ .

Lemma 9.4. For every $d \in \mathbf {Z}$ , there is [12pc]at most one assignment

$$\begin{align*}\operatorname{\mathrm{\sigma}}_{\operatorname{GSym}_g}(\Gamma_g)\subset S^d_{\operatorname{GSym}_g}(\Gamma_g)\end{align*}$$

that satisfies the two conditions of Definition 4.3 and that extends to a degree d universal stability assignment of type $(g,0)$ . When such an assignment exists, the integers $d-g+1$ and $2g-2$ are necessarily coprime.

Proof. Assume that $\operatorname {\mathrm {\sigma }}_{\operatorname {GSym}_g}$ is the restriction to $\operatorname {GSym}_g$ of some universal stability $\operatorname {\mathrm {\sigma }}$ . Since $\Gamma _g$ is a graph of genus $1$ , we can apply the results from [Reference Pagani and Tommasi32, Section 3] to describe the assignment $\operatorname {\mathrm {\sigma }}_{\operatorname {GSym}_g}(\Gamma _g)$ . Namely, in genus $1$ , it is known that all such assignments are induced by a polarization $\phi $ whose value $\phi (v_i)$ at each vertex $v_i$ is given by the average of the $2g-2$ admissible multidegrees in $\operatorname {\mathrm {\sigma }}_{\operatorname {GSym}_g}(\Gamma _g)$ . In particular, the polarization $\phi $ should be invariant under the automorphisms of $\operatorname {GSym}_g$ preserving $\Gamma _g$ , such as the cyclic automorphism mapping $e_i$ to $e_{i+1}$ for all i. From this, we deduce that all $\phi (v_i)$ are equal, and since we have obtained $\Gamma _g$ from $\operatorname {GSym}_g$ by removing the $g-1$ edges $e_i'$ , we have $\sum _{i\in \mathbf {Z}/(2g-2)\mathbf {Z}}\phi _i=d-g+1$ . It follows that there exists at most $1$ assignment, and that this assignment should be the one induced by $\phi (v_i)=\frac {d-g+1}{2g-2}$ for all $i=1, \ldots , 2g-2$ . Then the claim follows from the fact that this polarization is nondegenerate if and only if $d-g+1$ and $2g-2$ are coprime.

We can now prove the following.

Corollary 9.5. If $G\in T_{g,0} \setminus C_{g,0} \subseteq G_{g,0}$ is a loopless graph with $2$ vertices and at least $2$ edges, there exists at most one degree d stability assignment for the graph G that extends to a degree d universal stability assignment.

In the proof, we will denote by $\Gamma (t,i,j)$ the object of $G_{g}$ that consists of two vertices of genus i and j connected by t edges. (Note that $g=i+j+t-1$ ).

Proof. The claim is obtained by proving that the value of an assignment on vine curves of type $t \geq 2$ can be obtained, using compatibility with graph morphisms (Definition 4.8), from the assignment calculated on $\Gamma _g \subset \operatorname {GSym}_g$ (described in Lemma 9.4). Recall that a degree d stability assignment over a graph with $2$ vertices has the same value on all spanning trees and is uniquely determined by this value (Example 4.6).

First, we prove the claim when $G = \Gamma (t, i,0)$ has one vertex of genus zero. Apply Lemma 9.4 to deduce the uniqueness of an assignment $\operatorname {\mathrm {\sigma }}_{\operatorname {GSym}_g}(\Gamma _g)$ that satisfies the two conditions of Definition 4.3 and automorphism-invariance. Then apply to $\Gamma _g$ the first composition of contractions defined in [Reference Kass and Pagani25, Lemma 3.9].

The statement for an arbitrary graph $G= \Gamma (t, i, j)$ is obtained by applying the second set of contractions used in the proof of [Reference Kass and Pagani25, Lemma 3.9] to deduce the value of $\operatorname {\mathrm {\sigma }}$ on $\Gamma (t, i, j)$ from the value of $\operatorname {\mathrm {\sigma }}$ on $G=\Gamma ( i-j+2, t+2j-2, 0)$ . More precisely, consider the graph $G'$ with $4$ vertices $w_1,w_2,w_3,w_4$ defined in the proof of [Reference Kass and Pagani25, Lemma 3.9] and its spanning subgraph ${G}_0':=w_1-w_2-w_4-w_3-w_1$ . Let $\alpha \in \mathbf {Z}$ be defined by the condition

$$\begin{align*}\operatorname{\mathrm{\sigma}}_{\Gamma( i-j+2, t+2j-2, 0)}({G}_0)= \{(d-i+j -1-\alpha, \alpha) \}\end{align*}$$

for ${G}_0$ a spanning tree of $\Gamma ( i-j+2, t+2j-2, 0)$ . By combining the axioms of a degree d stability with compatibility with graph morphisms (Definition 4.8), we find out

$$\begin{align*}\operatorname{\mathrm{\sigma}}_{G'}({G}_0')= \{(\beta+1, \beta, \alpha, \alpha), (\beta, \beta+1, \alpha, \alpha), (\beta, \beta, \alpha+1, \alpha), (\beta,\beta, \alpha, \alpha+1) \} \end{align*}$$

for the unique $\beta \in \mathbf {Z}$ such that $2\alpha + 2 \beta = d+1 - t - i + j$ (observe that $d+1-t - i + j$ must be even since so is $d-g+1$ , because by Lemma 9.4, it is coprime with $2g-2$ , and by using $g=t + i + j -1$ ). From Condition (1) of Definition 4.3, we deduce that the unique assignment on the spanning tree ${G}_0" \subset {G}_0'$ defined by $w_2- w_1 -w_3 - w_4$ equals

$$\begin{align*}\operatorname{\mathrm{\sigma}}_{G'}({G}_0") = \{(\beta, \beta, \alpha, \alpha) \}, \end{align*}$$

and applying the second set of contractions used in the proof of [Reference Kass and Pagani25, Lemma 3.9], we deduce the value of $\operatorname {\mathrm {\sigma }}$ on the graph $G= \Gamma (t, i, j)$ .