1. Introduction

Our concern in this paper is the extension of the universal Picard group to the boundary of the Deligne–Mumford moduli space of stable curves. Over the interior, the Picard group of a smooth, proper, connected curve is well known to be an extension of the integers by a smooth, proper, connected, commutative group scheme, the Jacobian. These properties do not persist over the boundary, and natural variants sacrifice one or another of them to obtain others.

The Deligne–Mumford compactification of the moduli space of curves admits curves with nodal singularities. As long as the dual graph of the curve is a tree, the Picard group remains an extension of a discrete, free abelian group – the group of multidegrees – by an abelian variety, but it becomes nonseparated in families because the multidegrees do. One can focus here on the component of multidegree $0$, which is an abelian variety and is well behaved in families.

Should a curve degenerate so that its dual graph contains nontrivial loops, the multidegree-$0$ component of the Picard group remains separated, but fails to be universally closed. The construction of compactifications of this group is the subject of a vast literature [Reference IshidaIsh78, Reference D'SouzaD'S79, Reference Oda and SeshadriOS79, Reference Altman and KleimanAK80, Reference Altman and KleimanAK79, Reference KajiwaraKaj93, Reference CaporasoCap94, Reference PandharipandePan96, Reference JarvisJar00, Reference EstevesEst01, Reference CaporasoCap08a, Reference CaporasoCap08b, Reference MeloMel11, Reference ChiodoChi15], of which the above references are only a sample. We must direct the reader to the references for a history of the subject.

While we do not attempt to summarize all of the different approaches to compactifying the Picard group, we emphasize that all operate in the category of schemes, and none produces a proper group scheme. Indeed, it is not possible to produce a proper group scheme, for the multidegree-$0$ component of the Picard group of a maximally degenerate curve is a torus, and there is no way of completing a torus to a proper group scheme.

On the other hand, K. Kato observed that the multiplicative group does have compactifications – with group structure – in the category of logarithmic schemes [Reference KatoKat, § 2.1]. This gives reason to hope that the Picard group might also find a natural compactification in the category of logarithmic schemes, as Kato himself anticipated. Kato proposed a definition for, and then calculated, the logarithmic Picard group of the Tate curve [Reference KatoKat, § 2.2.4]. Illusie advanced the natural generalization of Kato's calculation as a definition for the Picard group of an arbitrary logarithmic scheme [Reference Illusie, Cristante and MessingIll94, § 3.3]. In the analytic category, Kajiwara, Kato, and Nakayama constructed the logarithmic Picard group using Hodge-theoretic methods [Reference Kajiwara, Kato and NakayamaKKN08a]. Significantly, they discovered the need to restrict attention to a subfunctor of the one defined by Illusie in order to get the logarithmic Picard group, and logarithmic abelian varieties in general, to vary well geometrically over logarithmic base schemes. In the present work, we work entirely in the algebraic category – but the condition of Kajiwara, Kato, and Nakayama, which appears here under the heading of bounded monodromy, first introduced in § 3.5, will play an essential role throughout.

The provenance of logarithmic geometry. This section is intended to motivate the presence of logarithmic geometry in the compactification of the Picard group. Consider a family of logarithmic curves $X$ over a one-parameter base $S$ with generic point $\eta$ and a line bundle $L_\eta$ on the general fiber of $X$. Let $i : s \to S$ denote the inclusion of the closed point and also write $i : X_s \to X$ for the inclusion of the closed fiber; write $j : \eta \to S$ and $j : X_\eta \to X$ for the inclusion of the generic point and the generic fiber.

Let $\mathscr S$ denote the ringed space $(s, i^{-1} j_\ast \mathcal {O}_\eta )$ and let $\mathscr X$ denote the ringed space $(X_s, i^{-1} j_\ast \mathcal {O}_{X_\eta })$. Then $\mathscr L = i^{-1} j_\ast L_\eta$ is a line bundle on $\mathscr X$.

We can describe $\mathscr L$ by giving local trivializations and transition functions in $\mathbf {G}_m$. However, these cannot necessarily be restricted to $X_s$ because a unit of $i^{-1} j_\ast \mathcal {O}_{X_\eta }^\ast$ may have zeros or poles along components of the special fiber.

If the dual graph of $X_s$ is a tree then it is possible to modify the local trivializations to ensure that the transition functions have no zeros or poles, but in general such a modification may not exist.

The degeneration of transition functions suggests we might compactify the Picard group by allowing ‘line bundles’ whose transition functions are sometimes allowed to vanish or have poles. If transition functions are thus permitted not to take values in a group then the objects assembled from them will no longer have a group structure. However, this leads naturally to the consideration of rank-$1$, torsion-free sheaves.

Logarithmic geometry takes a different approach to the same idea. Instead of keeping track of only the zeros and poles of the transition functions, we instead keep track of their orders of vanishing and leading coefficients. Together, order of vanishing and leading coefficient have the structure of a group and therefore the objects glued with transition functions in this group can be organized into a group as well.

The way this is actually done is to take the image of a transition function $f \in i^{-1} j_\ast \mathcal {O}_{X_\eta }^\ast$, not in $\mathcal {O}_{X_s} \cup \{ \infty \}$, but instead in

\[ M_{X_s}^{\rm gp} = i^{-1} j_\ast \mathcal{O}_{X_\eta}^\ast / \ker (i^{-1} \mathcal{O}_X^\ast \to \mathcal{O}_{X_s}^\ast) \]

That is, we obtain a natural limit $M_{X_s}^{\rm gp}$-torsor $P$ for $L_\eta$, whose isomorphism class lies in $H^1(X_s, M_{X_s}^{\rm gp})$.

Taking transition functions in $M_{X_s}$ has an added benefit, even when the dual graph of the special fiber is a tree. Indeed, if $L_\eta$ extends to $L$ with limit $L_s$, one can always produce another limit $L(D)_s$ by twisting $L_s$ by a component $D$ of the special fiber. But the effect of twisting by $D$ on $\mathscr L$ is to modify the local trivializations of $\mathscr L$ by units of $\mathcal {O}_{X_\eta }$. This changes the local trivializations of $P$ by elements of $M_{X_s}^{\rm gp}$, but that only affects a cocycle representative by a coboundary. In other words, the class of $P$ in $H^1(X_s, M_{X_s}^{\rm gp})$ is independent of twisting by components of $X_s$.

Logarithmic line bundles. It is sensible to take $M_X^{\rm gp}$-torsors as a candidate for a compactification of $\mathcal {O}_X^*$ in general, even when the base $S$ of the family is an arbitrary logarithmic scheme. In this paper, we use this observation to define logarithmic line bundles on a family of logarithmic curves $X \rightarrow S$ as torsors under the logarithmic multiplicative group, as Kato and Illusie proposed, that satisfy the additional bounded monodromy condition. This definition produces a stack $\operatorname {\mathbf {Log\,Pic}}(X/S)$ – the logarithmic Picard stack – with respect to the strict étale topology on $S$, and an associated sheaf $\operatorname {Log\,Pic}(X/S)$ – the logarithmic Picard group – via rigidification. In § 4.18 we explain why the bounded monodromy condition is necessary if infinitesimal deformation of logarithmic line bundles is to have the expected relationship to formal families of logarithmic line bundles. Thus, the bounded monodromy condition can be considered to be the first subtlety of the theory. The second subtlety is that even with this condition, the logarithmic Picard stack and the logarithmic Picard group are not representable by an algebraic stack or algebraic space with a logarithmic structure, respectively. The reason is essentially that the logarithmic multiplicative group is itself not representable (see § 2.2.7). Nevertheless, $\operatorname {\mathbf {Log\,Pic}}(X/S)$ and $\operatorname {Log\,Pic}(X/S)$ do have all the formal properties of an algebraic stack and an algebraic space, albeit only in the logarithmic category. Specifically, $\operatorname {Log\,Pic}(X/S)$ has a logarithmically étale cover by a logarithmic scheme, and we prove that it is a smooth algebraic group object in the category of logarithmic schemes, with proper components.

Our results for the logarithmic Picard stack, which remembers automorphisms, are similar, but a bit more technical.

The difference between Theorems A and C and Olsson's result [Reference OlssonOls04, Theorem 4.4] is that Olsson works with a fixed logarithmic structure on the base while we allow the logarithmic structure to vary. This is necessary for the logarithmic Picard group to be proper. Our method of proof also differs from Olsson's: we do not rely on the Artin–Schlessinger representability criteria (for which there is not yet an analogue in logarithmic geometry) and instead construct logarithmically smooth covers directly.

Connection with tropical geometry. Our analysis of the logarithmic Picard group, and our construction of the covers invoked in Theorems A and C, are direct: we do not rely on general representability criteria, nor the theory of logarithmic 1-motifs. Instead, our main tool is the intimate connection between algebraic, logarithmic, and tropical geometry. This connection with tropical geometry is in our view a significant advantage, and perhaps the central point of this paper. From a strictly algebraic perspective $\operatorname {\mathbf {Log\,Pic}}$ and $\operatorname {Log\,Pic}$ may be mysterious objects, lying outside the province of algebraic geometry. However, the logarithmic perspective affords them a modular description and a tropicalization – which has a modular description of its own – that precisely control and explain our transgression beyond the boundaries of algebraic geometry.

We are not the first to observe a connection between the logarithmic Picard group and tropical geometry: indeed, Foster, Ranganathan, Talpo, and Ulirsch observed that the geometry of the logarithmic Picard group is intimately tied up with the geometry of the tropical Picard group [Reference Foster, Ranganathan, Talpo and UlirschFRTU16], and the connection can also be seen in Kajiwara's work [Reference KajiwaraKaj93], albeit without explicit mention of tropical geometry. Our main contribution here is perhaps to extend the connection to a family of logarithmic curves over an arbitrary base.

A tropical curve is simply a metric graph. Baker and Norine introduced the tropical Jacobian as a quotient of tropical divisors by linear equivalence [Reference Baker and NorineBN07]. At first, the tropical Jacobian of a fixed graph (not yet metrized) was only a finite set, but subdivision of the graph suggests the presence of a finer geometric structure. This was explained by Gathmann and Kerber [Reference Gathmann and KerberGK08], who extended Baker and Norine's results to metric graphs, and Amini and Caporaso added a vertex weighting [Reference Amini and CaporasoAC13]. Mikhalkin and Zharkov defined tropical line bundles as torsors under a suitably defined sheaf of linear functions on a tropical curve [Reference Mikhalkin and ZharkovMZ08, Definition 4.5]. They gave a separate definition of the tropical Jacobian as a quotient of a vector space by a lattice [Reference Mikhalkin and ZharkovMZ08, § 6.1], and proved an analogue of the Abel–Jacobi theorem, showing that the tropical Jacobian parameterizes tropical line bundles of degree $0$. We will recover this result in Corollary 3.4.8.

In order to relate the tropical Picard group and tropical Jacobian to their logarithmic analogues, we require a formalism by which tropical data may vary over a logarithmic base scheme. This formalism is supplied by Cavalieri, Chan, Ulirsch, and the second author [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, § 5], who allow an arbitrary partially ordered abelian group to stand in for the real numbers in the definition of a tropical curve as a metric graph. Logarithmic schemes come equipped with sheaves of partially ordered abelian groups and one can therefore speak of tropical curves over logarithmic base schemes. We summarize these ideas in §§ 2.3.1–2.3.3. In a nutshell, given a logarithmic curve $X \rightarrow S$, we obtain a family of tropical curves $\mathscr X$ over $S$, whose fiber over a geometric point $s \in S$ is the dual graph of $X_s$, metrized by the characteristic monoid $\overline {M}_{S,s}^{\rm gp}$ of the logarithmic structure at $s$. We call the family $\mathscr X$ the tropicalization of $X/S$. The family $\mathscr X$, although an object over an arbitrary logarithmic scheme $S$, is essentially a combinatorial object: the logarithmic scheme $S$ has a stratification on which the characteristic monoid $\overline {M}_S$ is constant, and the fibers of $\mathscr X$ are constant on each stratum. The tropicalization $\mathscr X$ can thus be thought of as a combinatorial shadow of $X/S$, which remembers the combinatorics of the irreducible components of each fiber, and how the nodes of each fiber deform as one moves along strata of $S$.

However, the connection of $X/S$ with $\mathscr X$ only comes to life after we begin doing some geometry on $\mathscr X$. Indeed, two of the most crucial constructions of the paper occur in § 3, where we define a topology, and sheaves $\mathsf {PL}$ and $\mathsf {L}$ of piecewise linear and linear functions respectively on $\mathscr X$, over an arbitrary base $S$. This allows us, so to speak, to do some honest tropical geometry on $\mathscr X$.

Our sheaf $\mathsf {PL}$ is different from the sheaves of piecewise linear functions that are usually encountered in tropical geometry in that our piecewise linear functions are allowed to take values in a group $\overline {M}^{\rm gp}$ of arbitrary finite rank instead of the integers or real numbers. The groups $\overline {M}^{\rm gp}$ that appear vary over points of $S$, but are essentially the groups of sections of $\overline {M}_S^{\rm gp}$ over appropriately small neighborhoods around each point of $S$. This is the formalism that allows us to capture the fact that $X$ varies over a logarithmic base scheme $S$ instead of being the total space of a one-parameter degeneration. The sheaf of linear functions is built from $\mathsf {PL}$ by imposing the analogue of the balancing condition that is ubiquitous in tropical geometry.

We use the sheaf $\mathsf {L}$ to define the tropical Picard group $\operatorname {Tro\,Pic}(\mathscr X/S)$ and Picard stack $\operatorname {\mathbf {Tro\,Pic}}(\mathscr X/S)$ over logarithmic bases. We define them as the sheaf or stack of tropical line bundles, which are the bounded monodromy torsors under $\mathsf {L}$. The sheaf $\operatorname {Tro\,Pic}(\mathscr X/S)$ and stack $\operatorname {\mathbf {Tro\,Pic}}(\mathscr X/S)$ are combinatorial objects, which in practice are simple to compute. For example, one still has a formula for the tropical Jacobian analogous to the formula of [Reference Mikhalkin and ZharkovMZ08], which, over a point $s \in S$, takes the form

(1.1)\begin{equation} \operatorname{Tro\,Jac}(\mathscr X_s/s) = \operatorname{Hom} (H_1(\mathscr X_s, \overline{M}_{S,s}^{gp}))^{{\dagger}}/H_1(\mathscr X_s). \end{equation}

Here the ${\dagger}$ symbol indicates the bounded monodromy condition, which has a simple description in terms of the above formula: every loop $\gamma$ in the homology of the tropical curve $\mathscr X_s$ has a length $\ell (\gamma )$ valued in the monoid $\overline {M}_{S,s}$, and a homomorphism in $\operatorname {Hom} (H_1(\mathscr X_s, \overline {M}_{S,s}^{gp}))$ has bounded monodromy if it sends every loop to an element of $\overline {M}_{S,s}^{\rm gp}$ that is bounded by some multiple of the length of the loop. This has the effect that if one generizes from a point $s$ to a point $t$, smoothing some of the nodes of the curve $X_s$, and therefore contracting some edges in $\mathscr X_s$, the homomorphism descends to be well defined on the homology of $\mathscr X_t$. We refer the reader to § 3 for a thorough explanation of this phenomenon and the rest of the terms appearing in the definition of $\operatorname {Tro\,Jac}(\mathscr X/S)$. We note also that when we are working with a one-parameter degeneration of a curve, the group $\overline {M}_{S,s}^{\rm gp}$ reduces to $\mathbf {R}$, the bounded monodromy condition is automatic, and our formula recovers the [Reference Mikhalkin and ZharkovMZ08] formula.

The connection with the theory of tropical divisors is also simple: the sheaves of linear and piecewise linear functions fit into an exact sequence

(1.2)\begin{equation} 0 \to \mathsf{L} \to \mathsf{PL} \to \mathsf V \to 0 \end{equation}

with $\mathsf V$ the sheaf of tropical divisors. Thus, each tropical divisor $D$ determines a tropical line bundle $\mathsf {L}(D)$, which is the $\mathsf {L}$-torsor describing the obstruction of lifting $D$ to a piecewise linear function whose bend locus is $D$. It is shown in § 3.5 that, if the base $S$ is a valuation ring of arbitrary rank, $\operatorname {Tro\,Jac}(\mathscr X/S)$ is precisely the group of tropical divisors on all semistable models (that is, subdivisions) of $\mathscr X$, up to piecewise linear functions. This gives another interpretation of the bounded monodromy condition in the valuative case, as those torsors that can be represented by a divisor on a semistable model; but for general $S$, the group $\operatorname {Tro\,Jac}(\mathscr X/S)$ may be larger, with additional torsors that correspond to divisors on semistable models of $\mathscr X$ over logarithmic modifications of $S$ as well.

The presentation (1.1), and the exact sequence (1.2), describe two different means of producing tropical line bundles: from local systems and from tropical divisors. The relationship between these is encoded in diagram (3.4.1). The referee pointed out to us that this gives a compelling third method of producing tropical line bundles, from a labeling of the edges of the tropical curve by integers (see § 4.5).

The connection of the tropical picture with logarithmic geometry is obtained through the process of tropicalization. We do not attempt to explain this in the introduction, but we mention that the germ of the idea is elementary, utilizing the following formula, which is valid for every geometric point $s \in S$:

\[ H^i(X_s,\overline{M}_{X,s}) = H^i(\mathscr X_s,\mathsf{PL}_{\mathscr X_s}). \]

Thus, in a sense, $\mathscr X$ and $\mathsf {PL}$ together capture all information of $X/S$ that is reflected in $\overline {M}_X$.

The connection between $M_X^{\rm gp}$ and the sheaf of linear functions, $\mathsf {L}$, and the connection between logarithmic line bundles and tropical line bundles, are more subtle. In § 4.14 we observe that the sheaf $\mathsf V$ is the tropicalization of the Néron–Severi group: while the Néron–Severi group is only a presheaf on $X$, it descends to a sheaf on the tropicalization $\mathscr X$. We obtain a tropicalization map $\operatorname {\mathbf {Log\,Pic}}(X/S) \to \operatorname {\mathbf {Tro\,Pic}}(\mathscr X/S)$ from a morphism of complexes of presheaves $M_X^{\rm gp} \to [ \overline {M}_X^{\rm gp} \to \operatorname {NS} ]$ that is derived from the fundamental exact sequence of logarithmic geometry:

(1.3)\begin{equation} 0 \rightarrow \mathcal{O}_{X}^* \rightarrow M_{X}^{\rm gp} \rightarrow \overline{M}_{X}^{\rm gp} \rightarrow 0 \end{equation}

Theorem D Let $X$ be a proper, vertical logarithmic curve over $S$ and let $\mathscr X$ be its tropicalization. There is an exact sequence

\[ 0 \to \operatorname{Pic}^{[0]}(X/S) \to \operatorname{Log\,Pic}(X/S) \to \operatorname{Tro\,Pic}(\mathscr X/S) \to 0 \]

There is also an exact sequence of group stacks

\[ 0 \to \operatorname{\mathbf{Pic}}^{[0]}(X/S) \to \operatorname{\mathbf{Log\,Pic}}(X/S) \to \operatorname{\mathbf{Tro\,Pic}}(\mathscr X/S) \to 0 \]

Here the symbol $[0]$ denotes the multidegree-0 part of $\operatorname {Pic}$. This is an instance where the connection between algebraic, logarithmic, and tropical geometry becomes exceptionally transparent. The tropicalization morphism $\operatorname {Log\,Pic}(X/S) \to \operatorname {Tro\,Pic}(\mathscr X/S)$ allows us to understand $\operatorname {Log\,Pic}(X/S)$ in terms of a simple combinatorial object and a classical algebro-geometric object, the semiabelian scheme $\operatorname {Pic}^{[0]}(X/S)$. The tropicalization morphism also allows us to identify the combinatorial data associated with $\operatorname {Tro\,Pic}(\mathscr X/S)$ necessary to construct proper, schematic compactifications of the Picard group, following Kajiwara, Kato, and Nakayama [Reference KajiwaraKaj93, Reference Kajiwara, Kato and NakayamaKKN15], in § 4.17.

The exact sequences of Theorem D are also needed in our demonstrations of Theorems A and C, particularly in the demonstration of the boundedness of $\operatorname {\mathbf {Log\,Pic}}$ and its diagonal. The tropical boundedness statements, proved in §§ 3.10 and 3.11, are surely the most technical parts of the paper, and were the most difficult parts for us. We rely on what might be called an ‘arithmetic $\epsilon$–$\delta$’ formalism, in which $\epsilon$ and $\delta$ take values in a monoid; one cannot simply ‘choose $\delta > 0$’ but must choose it from the monoid of available positive elements of the monoid.

Invariance properties and construction of the cover. Polyhedral subdivisions of $\operatorname {Tro\,Jac}(\mathscr X/S)$ yield toroidal compactifications of $\operatorname {Pic}^{[0]}(X/S)$, which can in turn be interpreted as logarithmic modifications of $\operatorname {Log\,Pic}(X/S)$. Logarithmic modifications, together with logarithmic root stacks, are purely combinatorial operations yielding proper monomorphisms of logarithmic schemes. Together with étale maps, the two operations generate the (full) logarithmic étale topology of a logarithmic scheme. Logarithmic modifications, and similarly roots, form an inverse system, where $f_2: X_2 \to X$ can be considered finer than $f_1: X_1 \to X$ if $f_2$ factors as $g \circ f_1$ for a log modification $g:X_2 \to X_1$. Thus, Theorem E can be seen as a heuristic ‘formula’:

\[ \operatorname{Log\,Pic}^0(X/S) = \varinjlim \{\text{toroidal compactifications of }\operatorname{Pic}^{[0]}(X/S) \}. \]

Of course, this colimit does not exist as a scheme, but logarithmically it expresses $\operatorname {Log\,Pic}^0(X/S)$ as the colimit of all its logarithmic modifications – the minimal toroidal compactification of $\operatorname {Pic}^{[0]}(X/S)$, so to speak.

One of the remarkable properties of $\operatorname {Log\,Pic}(X/S)$ is that it is often invariant under both logarithmic modifications and root constructions: suppose that $S$ is logarithmically flat and that $f: T \rightarrow S$ is a logarithmic modification or a root $S$, and $Y$ is a logarithmic modification or root of the pullback of $X$ on $T$, for which $Y \rightarrow T$ is a logarithmic curve. Then $f^*\operatorname {Log\,Pic}(X/S) = \operatorname {Log\,Pic}(Y/T)$. In particular, $\operatorname {Log\,Pic}(X/S)$ forms a sheaf for the (small) full logarithmic étale topology on $S$: see Corollary 4.4.14.2. This invariance, together with the fundamental exact sequence, allows us to relate $\operatorname {Log\,Pic}(X/S)$ with all Picard groups $\operatorname {Pic}(Y/T)$ of all semistable models $Y$ of $X$ over logarithmic modifications and roots $T$ of $S$, by combining the natural map $\operatorname {Pic}(Y/T) \to \operatorname {Log\,Pic}(Y/T)$ with the isomorphism to $\operatorname {Log\,Pic}(X/S)$. In fact, it is this collection of spaces $\operatorname {Pic}(Y/T)$ that provides the cover in Theorem A. As the kernel of the map $\operatorname {Pic}(Y/T) \to \operatorname {Log\,Pic}(Y/T)$ is precisely the group of piecewise linear functions on the tropicalization of $Y/T$, we can write another heuristic ‘formula’,

\[ \operatorname{Log\,Pic}(X/S) = \varinjlim \operatorname{Pic}(Y)/\mathsf{PL}(\mathscr Y), \]

with $\mathsf {PL}(\mathscr Y)$ denoting the piecewise linear functions on the tropicalization of $Y$. The map $\mathsf {PL}(\mathscr Y) \to \operatorname {Pic}(Y)$ comes from the fundamental exact sequence (1.3).

With the benefit of hindsight, this formula could have been used as a definition of $\operatorname {Log\,Pic}(X/S)$ (at least over a logarithmically flat base): as line bundles on semistable models of $X/S$, up to the equivalence relation generated by pulling back to a further semistable model and the action of piecewise linear functions. In our point of view, this presentation of $\operatorname {Log\,Pic}(X/S)$ is an extrinsic presentation, whereas the definition we have chosen, in terms of torsors, is intrinsic.

This intrinsic/extrinsic interplay is now seen in multiple places in logarithmic geometry. For example, it is observed in logarithmic Gromov–Witten theory, where an ‘intrinsic’ definition of logarithmic stable maps is given by [Reference ChenChe14, Reference Abramovich and ChenAC14, Reference Gross and SiebertGS13], whereas an extrinsic definition is given in the work of [Reference LiLi01, Reference LiLi02, Reference KimKim10, Reference RanganathanRan20]. In the stable map setting, the different definitions produce different spaces; yet, their Gromov–Witten invariants coincide. This is an incarnation of the principle that logarithmic geometry captures the geometry of the interior of a space, and not of the specific logarithmic compactification chosen. Remarkably, for $\operatorname {Log\,Pic}$, both intrinsic and extrinsic approaches yield the same space in many cases (rather than the same invariants of the space). This property has proved to be very useful in the study of $\operatorname {Log\,Pic}$; it is, for example, key in the construction of a principal polarization, or in the study of Néron models via $\operatorname {Log\,Pic}$. In recent years, various central problems in logarithmic geometry have been studied, using either an intrinsic or extrinsic approach – for example, Chow theory for logarithmic schemes (by Barrott extrinsically [Reference BarrottBar20], or Herr intrinsically [Reference HerrHer19]) or Donaldson–Thomas theory ([Reference Maulik and RanganathanMR20], extrinsically). We expect that understanding the connection between the dual approaches in any given problem will prove to be very fruitful.

Future work. The Jacobian (and even the Picard stack) is equipped with a canonical principal polarization. We are mute about the logarithmic analogue in this paper, but we will construct it in a subsequent one.

Our results are limited to relative dimension $1$ because we do not yet have the means to study families of tropical varieties of higher dimension over logarithmic bases. We also do not yet understand the higher-dimensional analogue of the bounded monodromy condition.

Neither have we addressed any algebraicity properties of the tropical Picard group in a systematic way. It follows from our results that the tropical Picard group has a logarithmically étale cover by a Kato fan, but it is less clear how one should characterize its diagonal (we prove only that it is quasicompact here), or whether one should demand further properties of a purely tropical cover.

In § 4.17, we indicate how the tropical Picard group can be used to construct proper schematic models of the logarithmic Picard group over a local base. Recent work of Abreu and Pacini describes polyhedral subdivisions of $\operatorname {Tro\,Pic}(\mathscr X/S)$ when $\mathscr X$ is the universal curve over the moduli space of $1$-pointed tropical curves (and, for certain degrees, over the moduli space of unpointed tropical curves) [Reference Abreu and PaciniAP20]. They show that the corresponding compactification of the Picard group coincides with Esteves's compactification [Reference EstevesEst01]. We are pursuing a global construction of more general toroidal compactifications over the moduli space of stable curves in collaboration with Melo, Ulirsch, and Viviani.

The tropicalization method used in § 4.14 appears to generalize well to higher-dimensional logarithmic varieties. We hope to make further use of this construction in the future.

Conventions. Let $X$ be a curve over $S$. We use the term ‘Picard group’ to refer to the sheaf on $S$ of isomorphism classes of line bundles on $X$, up to isomorphism, and denote it $\operatorname {Pic}(X/S)$. The stack of $\mathbf {G}_m$-torsors on $X$ is denoted in boldface: $\operatorname {\mathbf {Pic}}(X/S)$. We use a superscript to denote a restriction on degree, and we refer to $\operatorname {Pic}^0(X/S)$ as the Jacobian of $X$. We apply similar terminology when $X$ is a logarithmic curve or tropical curve over a logarithmic base $S$.

Throughout, we consider a logarithmic curve $X$ over $S$. We regularly use $\pi : X \to S$ to denote the projection.