Models of Jacobians of curves

We show that the Jacobians of prestable curves over toroidal varieties always admit N\'eron models. These models are rarely quasi-compact or separated, but we also give a complete classification of quasi-compact separated group-models of such Jacobians. In particular we show the existence of a maximal quasi-compact separated group model, which we call the saturated model, which has the extension property for all torsion sections. The N\'eron model and the saturated model coincide over a Dedekind base, so the saturated model gives an alternative generalisation of the classical notion of N\'eron models to higher-dimensional bases; in the general case we give necessary and sufficient conditions for the N\'eron model and saturated model to coincide. The key result, from which most others descend, is that the logarithmic Jacobian of \cite{Molcho2018The-logarithmic} is a log Neron model of the Jacobian.

1. Introduction 1.1. Néron models. Let X ! S be a prestable curve (this means that X=S is proper, flat, finitely presented, and the geometric fibers are reduced and connected of pure dimension 1 and with at worst ordinary double point singularities; for example, a stable curve) over a scheme S , smooth over a schematically dense open subscheme U S . The Jacobian Pic 0 X U is then an abelian scheme over U which (in general) admits no extension to an abelian scheme over all of S . Néron suggested that one should look instead for a Néron model of Pic 0 X U ; a smooth algebraic space N over S, such that Theorem 1.2 (Proposition 7.3, Corollary 7.10). The map ‰ 7 ! G .‰/ induces a bijection of partially ordered sets from the set of quasi-finite open subgroups of sTPic 0 X=S to the set of smooth, separated, quasi-compact S -group models of the Jacobian Pic 0 X U =U .
As an application, we consider the case S D M g;n the moduli stack of stable curves, and we take X=S the universal curve. One shows that the strict tropical Jacobian is torsion-free in this case; an immediate consequence of Theorem 1.2 is: (Lemma 5.5). The universal Jacobian Pic 0 X=M g;n admits a unique smooth, separated group model over M g;n , namely the generalized Jacobian Pic 0 X=M g;n parametrizing line bundles of multidegree .0; 0; : : : ; 0/.
In particular, this shows the non-existence of a smooth separated group scheme (or space) G over M g;n whose every fiber G s is isomorphic to the special fiber of the Néron model of the Jacobian of a regular 1-parameter smoothing of X s , 2) as such a space would be a model of Pic 0 X=M g;n . A question about the existence of such "universal Néron models of the Jacobian" over a compactification of M g was asked by Chiodo in the introduction of [8]. Caporaso investigates in [6] the analogous problem where, instead of Jacobians, one tries to fit into a universal family the special fibers of the Néron models of the Pic d Y s =T s , where Y s =T s is a regular 1-parameter smoothing of X s .
1.3. The saturated model. We define the saturated model of the Jacobian Pic 0 X U =U to be the smooth separated quasi-compact group model which is maximal for the relation of inclusion.
Theorem 1.6 (Theorem 8.5). Let S be a toroidal variety with U S the open complement of the boundary divisor, and let X=S be prestable curve, smooth over U (or more generally, a log smooth curve over a log regular base). Then the following are equivalent: (i) The strict tropical Jacobian sTPic 0 X=S is quasi-finite over S . (ii) The Néron model of Pic 0 X U is separated over S. (iii) The Néron model of Pic 0 X U is quasi-compact and separated over S . (iv) The saturated model and the Néron model of Pic 0 X U are equal.
1.5. Log geometric interpretation. Although many of our results concern classical algebraic geometry, they become more natural in the context of logarithmic geometry. To explain the connection, suppose that X ! S is a family of logarithmic curves (see Definition 2.1). In [26], following ideas of Illusie and Kato, the authors constructed the analogue of the Picard scheme in the category of logarithmic schemes, the logarithmic Picard group LogPic X=S . This is the sheaf of isomorphism classes of the stack which parametrizes the logarithmic line bundles alluded to above, that is, certain 3) torsors under the associated group M gp X of the log structure.
The logarithmic Picard group is a group, is log smooth and proper over S, and on the locus U of S where the log structure is trivial it coincides with the ordinary Picard group Pic X U . Furthermore, logarithmic line bundles have a natural notion of degree, extending the notion of degree for ordinary line bundles, and LogPic X=S splits into connected components according to degree. Thus, the logarithmic Jacobian LogPic 0 X=S provides a "best possible" extension of the Jacobian Pic 0 X U =U . The caveat is that the logarithmic Jacobian is a sheaf on the category of log schemes, not schemes, and it is in general not algebraic -i.e., it is not representable by an algebraic space with a log structure. In fact, it is "log algebraic", that is, it satisfies the analogous properties that algebraic spaces enjoy, but only in the category of log schemes; for instance, it has a logarithmically étale cover by a log scheme. See Example 3.60 for the case of the Tate curve.
Nevertheless, properness of LogPic 0 X=S suggests that it is close to a Néron model for Pic 0 X U =U . For example, in the simplest case when S is a trait 4) , the valuative criterion tells us that every line bundle L on X U extends uniquely to a log line bundle on X. In fact, this "limit bundle" is simply the pushforward j L of the O -torsor associated to L along the inclusion j W X U ,! X . Remarkably, the description of the limit goes through whenever S is a log regular scheme, showing that the logarithmic Jacobian satisfies the logarithmic version of the Néron mapping property: Theorem 1.7 (Theorem 6.11). Let S be a log regular scheme. Then LogPic 0 X=S satisfies the Néron mapping property for log smooth morphisms.
In the case where S is Dedekind this answers positively for Jacobians of curves a question of Eriksson, Halle, and Nicaise in [9], who asked for the existence of a log Néron model.
To connect Theorem 1.7 with classical algebraic geometry, we have to bring the problem back from the category of log schemes to the category of schemes. There is a standard proce-dure to do so: the category Sch=S embeds into LSch=S by giving T ! S its pullback ("strict") log structure, and we may thus restrict the functor LogPic 0 X=S to Sch=S. We denote the resulting functor, the "strict" log Jacobian, by sLPic 0 X=S . It is an immediate consequence that the strict logarithmic Jacobian sLPic 0 X=S satisfies the classical Néron mapping property. The functors LogPic 0 X=S and sLPic 0 X=S are very different in nature: good properties of LogPic 0 X=S such as properness, or even quasi-compactness, are generally lost in passing to sLPic 0 X=S . This is however compensated by the following positive result: Theorem 1.8 (Theorem 4.4). Let X=S be a vertical log curve. The functor sLPic 0 X=S is representable by a quasi-separated, smooth algebraic space over S.
If S is log regular (e.g., a toroidal variety with divisorial log structure), then sLPic 0 X=S is the Néron model of Pic 0 X U =U .
The strict tropical Jacobian sTPic 0 X=S , defined above as a quotient, also has a natural log geometric interpretation. The log Picard group LogPic X=S has a "tropicalization" TroPic X=S , an essentially combinatorial object which determines the features of LogPic X=S which are not present in the Jacobian Pic 0 X=S of X=S. Restricting the tropical Jacobian TroPic 0 X=S -that is, the degree 0 part of TroPic X=S -to schemes by giving a scheme its pullback log structure, as before, produces sTPic 0 X=S . The tropical Jacobian plays an important role in the theory of compactifications of the universal Jacobian. It was essentially shown in [16,17] that subdivisions of TroPic 0 X=S correspond to toroidal compactifications of Pic 0 X=S . Theorem 1.2 provides a complementary view of the role of TroPic X=S : the quasi-finite open subgroups of its strict locus determine the quasi-compact, smooth, separated group models of Pic 0 X=S .

Background
Here we collect for the convenience of the reader the necessary facts that we will use, especially from the paper [26].
2.1. Log schemes. All our log schemes are fine and saturated. For a log scheme S we denote by S the underlying scheme. We denote by LSch=S the category of log schemes over S, and by .LSch=S / K et the (big) strict étale site over S ; the small strict étale site is denoted S K et . We write M S for the log structure of a log scheme S, M S for its characteristic monoid M S =O S (these are sheaves on S K et ). For a map of log schemes f W X ! S we denote by M X=S the relative characteristic monoid M X =f M S D M X =f 1 M S . Let S be a log scheme; we define the logarithmic and tropical multiplicative groups on S to be the sheaves of abelian groups on .LSch=S / K et given by The underlying morphism of schemes X ! S is a prestable curve as in [37,Tag 0E6T]. Our definition is the same as that of [20] except that we have added the assumption that the morphism be vertical; this means that the characteristic sheaf M X=S is a sheaf of groups, or equivalently that it is supported exactly on the non-smooth locus X nsm of X over S .
2.3. Sites, constructibility and representability. We leave for a moment the category of log schemes. For a scheme X , we write X K et for the small étale site and .Sch=X / K et for the big étale site.
There is a morphisms of sites i W .Sch=X / K et ! X K et given by the inclusion of categories We have functors between the categories of sheaves Representability reduces to local constructibility via the following well-known lemma: The first forgets the log structure; the second endows an S -scheme with the strict (pullback) log structure from S . The functor f is the left adjoint of s: for X 2 LSch=S and Y 2 Sch=S, we have Hom.X; Y / D Hom.X; .Y; g M S //: We write Sh.LSch=S / K et (resp. Sh.Sch=S/ K et ) for the category of sheaves on the strict étale site on LSch=S (resp. the étale site on Sch=S). The functors f and s give rise to pushforward functors f and s on the categories of sheaves: The functor s takes étale coverings to strict étale coverings, and commutes with fibered products. Therefore, s admits a left adjoint The tropical multiplicative group satisfies the following useful property (not shared by its logarithmic counterpart G

Tropical notions on the big site
In this section we develop the necessary tools to introduce the tropical and logarithmic Jacobian of [26]. We carefully define the sheaf of lattices H 1;X=S of first Betti homologies of the dual graphs of the fibers. Then we construct the tropical Jacobian as a global quotient of a tropical torus Hom.H 1;X=S ; G trop m;S / by the sheaf of lattices H 1;X=S ; this will facilitate the proof of the representability of the strict tropical Jacobian in Section 4. The original definition of Molcho and Wise is slightly different but we show it to be equivalent to ours.

Tropicalization.
A graph consists of finite sets V of vertices and H of half-edges, with an 'attachment' map r W H ! V from the half-edges to the vertices, and an 'opposite end' involution Ã W H ! H on the half-edges. To be consistent with our convention that log curves are vertical, we require this involution Ã to have no fixed points; because our curves have connected geometric fibers we also require our graphs to be connected. An edge is an unordered pair of half-edges interchanged by the involution, and we denote the set of them by E.
Let M be a sharp monoid. A tropical curve metrized by M is a graph .V; H; r; i / together with a function`W E ! M n ¹0º.
Let S be a geometric logarithmic point 5) , and let X=S be a log curve. The associated tropical curve (tropicalization) X of X has as underlying graph the usual dual graph of X (with a vertex for each irreducible component and a half-edge for each branch at each singular point).
To define the labelling`W H ! M S .S / we recall from [20] that the stalk of the log structure at a singular point x has characteristic monoid M X;x Š M S .S /˚N N 2 , where the coproduct is over the diagonal map N ! N 2 , and a map N ! M S .S /. If e is the edge corresponding to x then we set`.e/ to be the image of 1 in M S .S /; this is independent of the choice of presentation.
A map of monoids ' W M ! N determines a "contracted" tropical curve X ' , whose graph is obtained from the graph underlying X by contracting all edges e whose length`.e/ maps to 0 via ', and with length of the remaining edges induced by '. Molcho and Wise define a tropical curve over an arbitrary log scheme S as the data of a tropical curve for each geometric point of S , together with contraction maps between them compatible with geometric specializations, but we will not use this notion.
If S is a log scheme (but not necessarily a log point) and X=S a log curve, the edgelabellings of the tropicalization of X at various geometric points of S vary nicely in families, as a consequence of the following proposition: Proposition 3.1. Let W X ! S be a log curve and˛W X nsm ! S the structure morphism of the non-smooth locus of . The category of cartesian squares where X 0 ! S 0 is a log curve and S ! S 0 induces the identity on underlying schemes (with obvious morphisms) has a terminal object where the log structure M # S on S # satisfies M # S D˛ N (and N denotes the constant sheaf with value N on X nsm ). 5) A log scheme whose underlying scheme is the spectrum of an algebraically closed field.
Proof. This is [29,Theorem 2.7], combined with the observation that X=S is special in the sense of [29,Definition 2.6] if and only if M S D˛ N.
3.2. Subdivisions. Given a tropical curve X D .V; H; r; i;`/ with edges marked by a monoid M , and an edge e D ¹h 1 ; h 2 º, one can define a new tropical curve as follows: V 0 is obtained from V by adjoining a new vertex v, H 0 is obtained by adding two half-edges 1 ; 2 with r. 1 / D r. 2 / D v. Then we set i 0 .h 1 / D 1 and i 0 .h 2 / D 2 . Finally, we choose lengths`0. 1 / and`0. 2 / so that their sum is`.h 1 /. On the remaining edges and vertices r 0 ; i 0 ;`0 are set to agree with r; i;`.
Definition 3.2. A tropical curve X 0 constructed from X as in the paragraph above is called a basic subdivision of X. A subdivision of X is a tropical curve obtained by composing a finite number of basic subdivisions.
We state a few simple facts regarding the behavior of subdivisions with respect to contractions.
Fact 3.3. Let X be a tropical curve metrized by a monoid M , let ' W M ! N be a map of monoids, and let X ' be the contraction of X induced by ' as in Section 3.1. Let Y be a subdivision of X. Then the contraction Y ' is a subdivision of X ' . Fact 3.4. Let S be a log scheme with log structure M S and X=S a log curve. Let . N Á/ ! .N s/ be an étale specialization of geometric points, with induced morphism of characteristic monoids ' W M N s ! M N Á (see [7, Appendix A] for the notion of étale specialization). Let Fact 3.5. Let X ! S be a log curve, and Y ! X a logarithmic modification such that Y ! S is a log curve. For every log geometric point t ! S, the tropicalization of Y t is a subdivision of the tropicalization of X t . Moreover, for an étale specialization . N Á/ ! .N s/ as above, the subdivision of X N Á is Y ' , where Y is the subdivision of X N s over s. Lemma 3.6 ([26,2.4.3]). Let N s ! S be a geometric point and Y a subdivision of X N s . Then there exist an étale neighborhood V of N s, a log curve Y =V and a logarithmic modification 3.3. The tropical Jacobian over a point. Let X be a tropical curve over a monoid M . After choosing an orientation on the edges we have a boundary map Z E ! Z V , whose kernel is the first homology group H 1 .X; Z/. Molcho and Wise define an intersection pairing For a log curve X=S, Molcho and Wise then define the tropical Jacobian to be the data of the Jacobians of the tropicalizations of X s for every geometric point of s, together with their étale specialization maps (see [7,Appendix A]). However, for our purposes it is important to upgrade the groups Z E and Z V , and thereby H 1 .X; Z/ and the tropical Jacobian, to sheaves of abelian groups on the big étale site of S. The definitions become slightly intricate, because a choice of orientation on the tropicalization does not exist in families, and because locally constructible sheaves of abelian groups are not isomorphic to their duals. Definition 3.8. We define E (the sheaf of edges) to be the sheaf on .Sch=S/ K et represented by the non-smooth locus˛W X nsm ! S. We will use the notations E and X nsm interchangeably, depending on context.
If we denote by ¹?º the final object in the category of sheaves of sets on .Sch=X nsm / K et , then E is˛Š¹?º. Then Definition 3.9. The base change X 0 WD X S X nsm ! X nsm along the finite unramified morphism X nsm ! S admits a natural section a W X nsm ! X S X nsm (a closed immersion). Let Y denote the blowup of X 0 along a, and define the scheme of branches The projection X br ! X nsm is finite étale of degree 2; in fact, it is a Z=2Z-torsor.
(ii) The projection X br ! S is finite unramified; in particular, after replacing S by an étale cover, both X nsm ! S and X br ! S are disjoint unions of closed immersions, so that locally on S we have X br Š X nsm t X nsm as schemes over X.
Definition 3.11. We say that X ! S has split branches if X nsm ! S is a disjoint union of closed immersions, and X br Š X nsm t X nsm over X.
Remark 3.12. A choice of section of the torsor X br ! X nsm is equivalent to a choice of compatible orientations of the tropicalizations of the fibers of X ! S . Example 3.13. Let k be a field and let X be the prestable irreducible curve X WD Proj kOEx; y; z=.y 2 z x 3 x 2 z/; with the involution W X ! X given by y 7 ! y. Consider the prestable curve Y WD X k X ! X : Let P be a Z=2Z-torsor on X, and consider the twisted form of Y given by where Z=2Z acts on Y =X by . Then the scheme of branches of Y P =X is canonically isomorphic to the torsor P ! X. In particular, for P whose class in H 1 .X K et ; Z=2Z/ D Z=2Z is non-zero, the curve Y P does not have split branches.
Definition 3.14. We define the sheaf of half-edges H 2 Sh..Sch=X nsm / K et / to be X br . We will use the notations H and X br interchangeably. We will use the same notation for the pushforward (left adjoint to pullback) to Sh..Sch=S / K et /.
The sheaf H comes with a natural involution Ã over X nsm and hence over S . Composition induces an involution Ã on Z H WD Hom S .H ; Z/, and we find that:  The involution Ã induces an automorphism Ã Wˇ Z X br !ˇ Z X br . We then define a map M gp X=S !ˇ Z X br sending m to .ˇ m/. Now Ã. .ˇ m// D .ˇ m/, so this map factors via the inclusion .ˇ Z X br / Ã ,!ˇ Z X br . Locally on X nsm the induced map M gp X=S ! .ˇ Z X br / Ã is a map of free rank-1 Z-modules, and is easily checked to be an isomorphism, from which the second equality is immediate. Lemma 3.16. The sheaves Z E and Z E are isomorphic étale locally on S.
Proof. Étale locally on S the curve X has split branches, so we conclude by (3.15).
3.4.2. Representability. Lemma 3.17. The sheaf Z E on .Sch=S / K et is representable by a quasi-separated étale group algebraic space on S .
Proof. Working étale locally on S, we reduce to the case where E ! S is a finite disjoint union of closed immersions. We may then assume i W E ! S is a closed immersion with E connected, and with complement j W U ! S. The constant sheaf Z S is representable by an étale algebraic space, and so is its open subgroup j Š Z U . So the quotient Hom S .E; Z/ D i Z E is representable by an étale algebraic space. For quasi-separatedness, we may reduce to S locally noetherian because E is finitely presented, and then use Remark 2.4.
Combining Lemmas 3.16 and 3.17 yields: Lemma 3.18. The sheaf Z E is representable by a quasi-separated étale group algebraic space over S. Remark 3.20. In [36], Romagny studies relative irreducible components in more detail and generality. He shows that Irr X=S is representable by a finitely presented étale algebraic space over S.
Remark 3.22. The sheaf Z V is the Néron-Severi group of X =S. If S is a geometric point, then there is a natural isomorphism Z V ! Z V , motivating the notation.
Remark 3.23. We emphasize that the sheaf Z V should not be confused with the sheaf Z Irr X=S D Hom.Irr X=S ; Z/ on .Sch=S / K et . For example, if S is a strictly henselian discrete valuation ring and X=S a prestable curve, smooth over the generic point, and whose special fiber has two irreducible components, then the global sections of ZOEIrr X=S form a free abelian group of rank 2, and those of Z Irr X=S a free abelian group of rank 1. Lemma 3.24. The sheaf Z V on .Sch=S / K et is representable by a quasi-separated étale group algebraic space over S.
Proof. As Irr X=S is of finite presentation, we may assume S is locally Noetherian. By Lemma 2.3 it suffices to prove that Z V is locally constructible. From Remark 3.20 we know that Irr X=S is locally constructible, in other words is of the form i J for some sheaf J on the small étale site of S. Then where for the last equality we use that the functors i and ZOE commute, which follows in turn from the easy observation that their right adjoints (i and the forgetful functor from sheaves of abelian groups to sheaves of sets) commute. Proof. It suffices to prove Irr is generated by global sections étale-locally on S, which follows from [10, Lemma 18].
3.6. The tropical boundary map. If X is a tropical curve with a choice of orientation on the edges, there is a boundary map sending an edge E to its endpoint minus its startpoint. We will define an analogous map ı W Z E ! Z V , independent of choices. Since both Z E and Z V are locally constructible, it suffices to define ı on the small étale site of S . In what follows, we work exclusively on small étale sites, implicitly applying the functor i wherever necessary. Working locally on S K et , we may assume that X=S has split branches.
Write X nsm =S as a disjoint union of closed immersions F i Z i˛i ! S , and for each i put where Z E i is the subgroup of . Ã/-invariants of .˛i / Z H i . We will construct natural maps Z E i ! Z V , and sum them. Pick some i and put Z WD Z i ,˛WD˛i . Here˛is a closed immer- 6) For a sheaf F on S, this means that every geometric point of S admits an étale neighborhood U such that the natural map from the constant sheaf on F .U / to the restriction F jU is surjective. sion, and we use the notation from [24, Proposition II.3.14] for pushing and pulling along of sheaves on the small étale site. Denote by j the open immersion U WD S n Z ! S . By [24, Proposition II.3.14],˛ is left adjoint to the functor˛Š W Ab.S K et / ! Ab.Z K et / taking a sheaf to (the pullback of) its subsheaf of sections supported on Z. Explicitly, for a sheaf F on S , There is a natural map of S-algebraic spaces sending a point to the irreducible component of the normalization of its fiber on which it lies. WritingˇW Z br ! Z, the map b induces a global section ofˇ ˛ Z V , i.e., a morphism Sinceˇis a disjoint union of isomorphisms, it follows thatˇ is simultaneously the left and right adjoint ofˇ . By adjunction and after restricting to the . Ã/-invariants, we get a map B 2 The fact that ı does not depend on the chosen expression of X nsm ! S as a disjoint union of closed immersions is clear from the construction.
Remark 3.28. If S is a log point we have a canonical isomorphism Z V D Z V (cf. Remark 3.22). A choice of orientations of the edges of the tropicalization of X=S provides an isomorphism Z E D Z E , which identifies ı with the boundary map of (3.6.1).
Definition 3.29. We denote by H 1;X=S the kernel of ı. It is a sheaf of abelian groups whose stalks are free and finitely generated. Its value at a geometric point s of S is isomorphic to the first homology group H 1 .X s ; Z/, cf. Remark 3.28.
Remark 3.30. Prompted by a request from an anonymous referee, we give an alternative interpretation of the map ı, via the interpretation of Z V as the Néron-Severi group of X=S (see Remark 3.22). For this remark, we work in the sites .Sch=X / K et and .Sch=S/ K et . Let X st be X with the log structure pulled back from S . We restrict sheaves naturally defined on et without additional decorations, to avoid overcrowding the notation. Recalling that Z E D G trop m;X=S , the cohomology of the short exact sequence gives a canonical map We also have an exact sequence which splits after base changing to a strict étale cover S 0 ! S. Choosing a splitting gives a map G log m;X st S 0 ! G m;X S 0 , and taking cohomology gives a map This map depends on the choice of splitting, but we claim that the composite with the natural map to the Néron-Severi group Pic X S 0 =S 0 = Pic 0 X S 0 =S 0 does not depend on this choice. In particular, the composite descends to a canonical map The claim may be proven assuming S 0 D S . The difference between two choices of splittings is measured by a map which by adjunction is the same as a map G W G trop m;S ! G m;X D G m;S (the latter equality since X=S is proper with reduced and connected geometric fibers). Then we have a map (3.6.10) and composing with the canonical map 1 G m;S ! G m;X yields a map (3.6.11) we must show that this map factors via Pic 0 X=S . WritingˇW 1 G m;S ! G m;X for the natural inclusion, we find that F Dˇı 1 F , and so (3.6.12) and factors through Pic 0 X=S . Indeed, the fiber of T WD R 1 1 G m;S over a geometric point s ! S is the torus part of Pic 0 X s =s . This shows the existence of a slight refinement of ı; we defined it as a map from Z E to the Néron-Severi group Z V D Pic X=S = Pic 0 X=S , but it comes naturally from a map to the quotient Pic X=S =T by the "torus part of Pic X=S ". We do not know if this lift has any application. We will define an analogous map where f is the functor Ab.LSch=S / K et ! Ab.Sch=S/ K et introduced in Section 2.4. Let X=S be a log curve. We write˛W X nsm ! S and ' W X br ! X nsm . On .Sch=X nsm / K et , we have a sheaf of abelian groups ' Z X br D Hom X nsm .X br ; Z/ which is locally free of rank 2 and endowed with the involution Ã. We write .' Z X br / Ã for the invariants under Ã; this is a locally free rank 1 sheaf of abelian groups, hence self-dual. Pushing forward the natural pairing where the last equality is Lemma 2.7.   Remark 3.37. Suppose S is a log point, and fix an orientation of the edges of the tropicalization X of X at s (i.e., an isomorphism Z E D Z E ). Then we recover the usual notions of length and bounded monodromy on the tropical curve X. As explained in [26], for Á a generization of s, any homomorphism of bounded monodromy induces a unique homomorphism (of bounded monodromy) S;Á I moreover, an orientation of X s induces a unique orientation of its contraction X Á . There is therefore an induced generization map [26] is slightly different than the one we gave and relies on the generization maps defined above. The rest of this section is devoted to verifying that the two definitions are actually equivalent.
3.10. Equivalence with the definition of Molcho and Wise. All our log schemes are fine and saturated, so in particular they have étale-local charts by finitely presented monoids. This allows for the following convenient definitions of finiteness conditions for log schemes.
Definition 3.40. We say that a log scheme is of finite type, resp. locally of finite type, resp. Noetherian, resp. locally Noetherian if its underlying scheme is.
We say that a morphism of log schemes is locally of finite type, resp. of finite type, resp. locally of finite presentation, resp. of finite presentation if the underlying scheme map is.
Remark 3.41. Any log curve is of finite presentation. In particular, any log curve is the base change of a log curve over a log scheme locally of finite type.
Definition 3.42. Let W X ! S be a log curve. We say that S is nuclear (with respect to ) if: (1) the stratification of S induced by the log structure M S has only one closed stratum Z, Z is connected, and every connected component of every stratum specializes to Z, where J is a set), (3) X=S has split branches, (4) the sheaf Z V is generated by global sections.
We say that S is pre-nuclear if it satisfies (2), (3) and (4). We say that S is a nuclear neighborhood of a geometric point t of S if S is nuclear and t maps to the closed stratum.
Remark 3.43. Conditions (2), (3) and (4) are stable under strict étale base change. In particular, so is pre-nuclearity. Lemma 3.45. If X=S is a log curve, then S admits a strict étale cover by pre-nuclear schemes. If in addition S is locally Noetherian, then any geometric point has a nuclear strict étale neighborhood.
Proof. First we show that S has a cover by pre-nuclear schemes. By Remark 3.43, it suffices to show that conditions (2), (3) and (4) individually hold locally on S for the strict étale topology. We can assume that S meets condition (3) by Remark 3.10; condition (4) by Lemma 3.25; and condition (2) by the existence of étale local charts for log schemes. Now, assume S is locally Noetherian and let t be a geometric point of S. We will show that t has a nuclear strict étale neighborhood. Shrinking, we may assume that S is pre-nuclear and Noetherian. In particular, the connected components of strata (of the stratification induced by M S ) form a partition of S into finitely many locally closed subschemes. If the closure Y of a piece Y does not meet t , take out Y from S . The resulting S is a nuclear neighborhood of t . Proof. The sheaf M S is constant on Z by Remark 3.44. Since Z is locally Noetherian, t and t 0 can be joined by a sequence of geometric points where the squiggly arrows denote étale specialization, and all points land in Z. Every specialization Á i t j induces an edge contraction X t j ! X Á i , which is an isomorphism of tropical curves over M S .Z/. The automorphism of X t induced by any such étale path from t to itself must be trivial: it sends every vertex to itself since Irr X=S is generated by global sections, and every half-edge to itself since X=S has split branches. Therefore, the isomorphism X t ! X t 0 is independent of the choice of étale path.  Proof. Since E D X nsm ! S is a disjoint union of closed immersions, and s lies in the unique closed stratum, the map Z E .S / ! Z E .s/ is an isomorphism. Because X=S has split branches, we can make a choice of isomorphism W Z E Š Z E , so the restriction map   with S 0 locally of finite type and define F X=S as the pullback of F X 0 =S 0 . This is independent from the choice of cartesian square, and the formation of F X=S commutes with base change.
When there is no ambiguity, we will write F instead of F X=S .
This is the way in which the tropical Jacobian is defined in [26]. Proof. If S is locally of finite type and T is a S-log scheme of finite type, there is a natural mapˆ.
T / W TroPic 0 .T / ! F .T / taking˛2 TroPic 0 .T / to the system ¹i ˛2 TroPic 0 .t /º i Wt!T , where i W t ! T are all geometric points of T . Now, let T be an arbitrary S -log scheme. Since H 1;X=S is locally free and finite over S, we find that locally on T , TroPic 0 X=S .T / is the colimit of the TroPic 0 X=S .T 0 /, taken over all T ! T 0 ! S with T 0 of finite type. Combining this with the fact that the formation of TroPic 0 X=S commutes with strict base change, we get a morphism of abelian sheaveŝ W TroPic 0 X=S ! F for any log curve X=S. We will show thatˆis an isomorphism. We may do so assuming that S is of finite type. Then, working locally, it suffices to show thatˆ.T / is an isomorphism when T is nuclear.
Let x be a geometric point of T landing in the closed stratum Z. Consider the natural map W F .T / ! TroPic 0 .x/: The composition ıˆ.T / is the restriction map TroPic 0 .T / ! TroPic 0 .x/, which is an isomorphism by Lemma 3.48. It suffices therefore to show that is injective (in order to conclude that it is an isomorphism, and that thereforeˆ.T / is an isomorphism as well).
The argument is essentially the same as the end of the proof of Lemma 3.48. Let y ! T be any geometric point, W 0 the stratum containing y, and W the connected component of W 0 containing y. By condition (1) in the definition of nuclearity there exists an étale specialization Á with Á in W and in Z. By Corollary 3.47, we obtain a canonical map The compatibility condition forces all elements of the system F .T / to be determined by the element belonging to TroPic 0 .x/. This proves the injectivity. The first equality is because M gp is torsion-free. Surjectivity is due to the fact that M gp X=S is supported on X nsm and therefore H 1 .X; M gp X=S / D 0. We say that L has bounded monodromy if some (equivalently, any) preimage of L in Hom.H 1 .X/; M gp / has bounded monodromy as in Definition 3.33.
Let now W X ! S be a log curve over a general log base; we say that an M gp X -torsor L has bounded monodromy, if for every strict geometric point s ! S , the restriction L s has bounded monodromy. Such a torsor is called a logarithmic line bundle.  Consider the diagram where Pic X=S is the Picard stack, the left vertical map is the multidegree map, † is the sum, and the top horizontal map associates to a O X -torsor L the log line bundle L˝O X M gp X . It is shown in [26, 4.5] that there is a unique degree map making the diagram commute.
Definition 3.53. We define LogPic 0 X=S to be the substack of LogPic X=S of logarithmic line bundles of degree zero, and similarly for the sheaf LogPic 0 X=S LogPic X=S , which is called logarithmic Jacobian.
For the convenience of the reader we recall from [26] a list of properties of the logarithmic Picard and Jacobian that we will use. the "tropicalization", which fits into a short exact sequence where Pic 0 X=S is the generalized Jacobian, i.e., the sheaf of line bundles of degree zero on every irreducible component, which is representable by a semiabelian scheme on S. Moreover, f Pic 0 X =S is representable by a semiabelian scheme with pullback log structure from S .  . If Y ! X is a log modification such that Y =S is a log curve, the induced maps LogPic 0 X=S ! LogPic 0 Y =S and TroPic 0 X=S ! TroPic 0 Y =S are isomorphisms. Although the statement for TroPic 0 is not explicitly stated in [26], this follows from [26, Corollary 4.4.14.1] together with the fact that Pic 0 X=S ! Pic 0 Y =S is an isomorphism and the exact sequence (3.11.1).  . The stack LogPic X=S satisfies the valuative criterion for properness for log schemes: it has the unique lifting property with respect to valuation rings R whose log structure is the direct image of a valuative log structure on the fraction field. In general, LogPic X=S (resp. LogPic X=S ) is not representable by a log algebraic stack (resp. log algebraic space), as the following example shows: Example 3.60 ([17, Section I]). Let K be a complete discrete valuation field, with ring of integers O K , 2 m K a uniformizer, q D 2 and E q the corresponding Tate elliptic curve. Any proper log smooth model E q of E q over O K gives the same logarithmic Jacobian LogPic 0 E q , and comes by Theorem 1.7 with a unique (birational) morphism E q ! LogPic 0 E q . The minimal regular model E mi n q has two irreducible components over the residue field; blowing down any of them to a point gives two log smooth models E 1 q ; E 2 q . As both E 1 q and E 2 q map to LogPic 0 X=S , we see that if the latter were a log algebraic space, its closed fiber would have to be a point, contradicting the log smoothness.   [26] is by no means the first time the term "logarithmic Jacobian" appears in the literature. To our knowledge the earliest appearances of the term are in [15,16,19]; a notion of logarithmic Jacobian was then studied, among others, by Olsson in [30] and Bellardini in [4]. Their notion is from the beginning a notion for schemes, and thus closer to our strict log Jacobian than to the log Jacobian of [26]. However, even so there are differences -for instance, in Olsson's and Bellardini's work the log structure on S is restricted, and the subset of M gp X torsors they consider is different from the subgroup of all bounded monodromy torsors. In particular, the objects constructed by Bellardini and Olsson will in general not satisfy the Néron mapping property over higher-dimensional bases.

Strict log and tropical Jacobian and representability
There is an obvious modular interpretation for sLPic 0 X=S , as the sheafification of the functor associating to a map of schemes a W T ! S the set ¹log line bundles L on X S sT of degree zero º=Š: By exactness of the functor s , the exact sequence (3.11.1) yields an exact sequence  Recall that the stack LogPic 0 X=S is proper and log smooth over S (Property 3.58 and Property 3.59), but in general not algebraic. The main purpose of this section is to prove the following: Proof. Part (ii) of the theorem is immediate from part (i) and the exact sequence (4.0.1), as it realizes sLPic 0 as a Pic 0 -torsor over sTPic 0 , which makes it representable by a smooth separated algebraic space over sTPic 0 . Let us prove part (i). Representability by a quasi-separated algebraic space is étale local on the target, and X=S is of finite presentation, so we reduce via Lemma 4.3 to the case where S is a log scheme of finite presentation (in particular Noetherian).
We reduce by virtue of Lemma 2.3 to checking that the sheaf F D TroPic 0 X=S is locally constructible, i.e., that the canonical morphism˛W i i F ! F is an isomorphism.
It suffices to show that for any morphism T ! S of schemes, the restriction of˛to the small étale site over T is an isomorphism. That is, that for any T ! S and any geometric point t of T , the map colim is an isomorphism, where the colimits are over factorizations t ! V ! T with V ! T étale. The right-hand side is F .t /, by Lemma 3.48 and the fact that factorizations such that V (with pullback log structure from S) is a nuclear neighborhood of t form a cofinal system. The left hand side becomes colim where V ! T and W ! S are étale. This can in turn be replaced by the colimit of F .W / over the diagrams of the form t V T

W S
with V ! T and W ! S étale, where t ! T and T ! S are fixed and the remainder is allowed to vary. But this is simply the colimit over the factorizations t ! W ! S with W ! S étale. Since those factorizations with W a nuclear neighborhood of t in S form a cofinal system, by Lemma 3.48 the colimit is equal to F .t /.

Examples of strict logarithmic Jacobians.
Example 4.5. Let S be the spectrum of a discrete valuation ring with divisorial log structure, and X=S a log curve. Call Á the generic point of S . The closed fiber of the strict tropical Jacobian is identified with the finite étale group scheme of components of the Néron model of Pic 0 X Á =Á , and sLPic 0 X=S is the Néron model itself. This will be shown in greater generality in Corollary 6.13. Another explicit description of the Néron model is known in this case: after a finite sequence of log blowups of X , we obtain a new log curve X 0 =S whose nodes all have length 1 2 N D M S .S /. By [5,9.5,Theorem 4], the Néron model of Pic 0 X Á =Á is the quotient of the Picard group of degree 0 line bundles Pic tot0 X 0 =S by the closure of its unit section. The relation between sLPic 0 X=S and quotients of Picard spaces is explored further in Section 9. Let E=S be the degenerate elliptic curve in P 2 S with equation We make E ! S into a log smooth morphism by putting the log structures associated to the divisors D and E S D onto S and E respectively. It follows that sTPic 0 X=S is an étale group space with fiber Z at the closed point of S and 0 everywhere else. In particular, sLPic 0 X=S is not quasi-compact. It will follow from Proposition 7.3 that it is not separated either.
Example 4.7. Note that the tropical Jacobian of a log curve depends on the log structures on C =S, and not only on the underlying scheme map C =S . Keeping the notations of Example 4.6, E=S comes via base-change from a log curve E 0 =S 0 , where S 0 is S with log structure given by .kOEOEu; v/ ˚Nuv ! kOEOEu; v: The tropical Jacobian of E=S at the closed point is a free abelian group of rank 1, while that of E 0 =S 0 is trivial. In this example, E 0 ! S 0 coincides with the log curve E # ! S # provided by Proposition 3.1.

The saturation of Pic 0 in sLPic 0
We have seen in Example 4.6 that the étale algebraic space sTPic 0 X=S does not in general have finite fibers, and that consequentially sLPic 0 X=S is not in general quasi-compact. In this section we introduce a new quasi-compact quasi-separated (qcqs) smooth algebraic space naturally associated to the log curve X=S and sitting in between Remark 5.2. The strict saturated Jacobian has the following modular interpretation: it is the sheafification of the presheaf of abelian groups on .Sch=S/ K et whose T -sections are the isomorphism classes of log line bundles L of degree zero on X S sT such that some positive power of L is a line bundle. This line of thought is pursued further in [14]. Lemma  For quasi-compactness, the question being local on the base, we assume that S is affine and that M S has a global chart from an fs monoid. Then the stratification of S induced by M gp S has only finitely many strata Z i , and these strata are affine. Therefore, it suffices to show that for each i , the preimage of Z i in sTPic tor X Z i =Z i is quasi-compact. On each stratum Z, the sheaf Hom.H 1;X Z =Z ; s G trop m;Z / is locally constant by Corollary 3.47, so sTPic 0 X Z =Z is locally constant as well. Its torsion part sTPic tor X Z =Z is then a locally constant sheaf of finite abelian groups and in particular a finite étale scheme over Z. It is therefore quasi-compact. Proof. The first part of the statement follows from Lemma 5.3 and base change. The representability then follows from Theorem 4.4.We deduce that sPic sat X=S is qcqs because it is a torsor over the quasi-finite and quasi-separated sTPic tor X=S , under the quasi-compact separated group scheme Pic 0 X=S .
Lemma 5.5. Let S D M g;n , the moduli stack of genus g stable curves with n marked points. Let X g;n be the universal curve. Endow both stacks with the divisorial log structure so as to make the universal curve into a log curve. Then sPic sat X g;n D Pic 0 X g;n .
Proof. For s a geometric point of M g;n , the tropicalization X of the fiber has each edge labelled by a distinct base element of the free monoid M S;s . It follows that TroPic 0 .X/ is torsion-free.

The Néron mapping property of LogPic
In this section we prove the Néron mapping property for the logarithmic Jacobian, under the assumption that the base is log regular.
6.1. Classical and log Néron models. For comparison, we briefly recall the definitions of classical, non-logarithmic Néron models. Definition 6.1. Let S be a scheme, U S a schematically dense open, and N =S a category fibered in groupoids over Sch=S. We say that N =S has the Néron mapping property with respect to U S if for every smooth morphism of schemes T ! S, the restriction map N .T / ! N .T S U / is an equivalence.
If in addition N is an algebraic stack and is smooth over S , we say N is a Néron model of its restriction N U =U . Remark 6.2. Note that the definition of Néron models in [12] and the classical one over Dedekind schemes require them to be separated. When the base is a Dedekind scheme and the generic fiber is a group scheme this is automatic by [5, Theorem 7.1.1]. However, over higher-dimensional bases this is not true, and non-separated Néron models exist in much greater generality than separated ones (the author of [12] did not realize this at the time). From this perspective, [12] should be seen as investigating the existence of separated Néron models.
This definition extends naturally into the logarithmic setting: Definition 6.3. Let S be a log scheme, U S strict schematically dense open, and N =S a category fibered in groupoids over LSch=S. We say that N =S has the Néron mapping property with respect to U S if for every log smooth morphism T ! S, the map N .T / ! N .T S U / is an equivalence. Remark 6.4. In order to qualify as a Néron model, the functor N should not only satisfy the Néron mapping property, but also be a sheaf for a suitable topology, and be representable in some suitable sense. Requiring representability as an algebraic space or stack with log structure is too restrictive; the log Jacobian does not satisfy these criteria, and in general the log Néron model will not exist as an algebraic space or stack with log structure. The most appropriate notions of representability are perhaps: (i) for the functor, that it is a sheaf for the strict log étale topology, has diagonal representable by log schemes, and admits a log étale cover by a log scheme, (ii) for the stack, that it is a sheaf for the strict log smooth topology, it has diagonal representable in the sense of (i), and admits a log smooth cover by a log scheme.
The Log Picard space and stack do satisfy these additional conditions. This is shown in [26], except for verifying that LogPic is a stack for the log smooth topology ( [26] only verify it for the log étale topology), but this will be proven in the forthcoming [25]). One consequence of imposing the above assumptions is that Néron models are unique, 7) by analogous descent arguments to those for the classical case. For example, suppose that N and N 0 are log Néron models, and let V ! N be a log étale cover by a log scheme; then the map V U ! N U ! N 0 U extends uniquely to a map V ! N 0 by the universal property of the latter, and a further application of uniqueness and the sheaf property shows that this descends to a map N ! N 0 . 7) Either up to unique isomorphism, for the functor, or up to 1-isomorphism which is itself unique up to a unique 2-isomorphism, in the stack case. Remark 6.5. In classical algebraic geometry the distinction between the smooth and étale topologies is often not so important since every smooth cover can be refined to an étale cover (see [37,Tag 055V]). In the logarithmic setting this is no longer true; for example the map where G m ; A 1 have their toric log structure is a log smooth cover, but does not admit a refinement by a log étale cover in characteristic p. It does admit a refinement by a Kummer cover, and this is in fact a general phenomenon; one can show using the results of [1] that every log smooth cover can be refined by a cover that is a composite of log étale covers and Kummer maps.
Remark 6.6. Suppose N =S is a functor on .LSch=S / with the log Néron mapping property with respect to U S . Then s N has the Néron mapping property with respect to U S since, for every smooth morphism of schemes g W T ! S , the morphism .T; g M S / ! S is log smooth.
6.2. Log regularity. We will show that the logarithmic Jacobian of X=S is a log Néron model when S is log regular, a notion which we now recall. We say that S is log regular if it is log regular at all geometric points. Remark 6.8. Any toric or toroidal variety with its natural log structure is log regular. In particular, a regular scheme equipped with the log structure from a normal crossings divisor is log regular. If S is log regular then the locus on which the log structure is trivial is schematically 8) dense open. Lemma 6.9 ([28, Lemma 5.2]). Let S be a log regular scheme. Then the underlying scheme of S is regular if and only if the characteristic sheaf M S is locally free. In this case, the log structure on S is the log structure associated to a normal crossings divisor on S.
We want to prove that the map given by P 7 ! j P is an equivalence. We start with full faithfulness: let P; Q be log line bundles on X S T . It suffices to show that the map Isom.P; Q/ ! j Isom.j P; j Q/ of isomorphism sheaves is itself an isomorphism. The natural map M gp X T ! j O X V is an isomorphism by log regularity of X T and Lemma 6.10. Hence the natural map P ! j j P is a map of M gp X T -torsors, hence is an isomorphism (and similarly for Q). The map Aut .P / ! j Aut .j P / is simply the natural map M gp X T ! j O X V , hence is also an isomorphism. The map Isom.P; Q/ ! j Isom.j P; j Q/ is then a map of M gp X T -pseudotorsors, so it suffices to show that Isom.P; Q/ is non-empty whenever j Isom.j P; j Q/ is. But given an isomorphism j P ! j Q, taking j gives an isomorphism P D j j P ! j j Q D Q (the above argument can be summarized by saying that j is a quasi-inverse to j ). We now prove essential surjectivity. Letting L be a, O X V -torsor on X V , we will exhibit an M gp X T -torsor P of bounded monodromy such that j P D L. The torsor P will simply be j L; as a priori this is only a pseudotorsor under j O X V D M gp X T , and, if it is a torsor, it is not obvious that it has bounded monodromy, we will give a geometric construction of P , using the invariance of LogPic under log blowups and taking root stacks. Let D be a Cartier divisor on X V representing L. By [1,Theorem 4.5], there is a cover u W T 0 ! T , which is a composition of a log blowup and a root stack, such that u restricts to an isomorphism over V , and a log modification p W X 0 ! X T 0 D X T T 0 with X 0 ! T 0 semistable ([1, Definition 2.2]). In particular, X 0 ! T 0 is a log curve, and the total space of X 0 is regular. The schematic closure D of D in X 0 is therefore a Cartier divisor, and O.D/ is a line bundle whose associated O X 0 -torsor extends L. By Property 3.56, it follows that there is a M gp X T 0 -torsor P 0 of bounded monodromy on X T 0 such that j P 0 D L. Then, by the full faithfulness of (6.3.1) and the fact LogPic X=S is a stack for the log étale topology (Property 3.57), this P 0 descends to an M gp X T -torsor P with bounded monodromy. Remark 6.12. It will be shown in the work [25] that, when S is logarithmically smooth, LogPic X=S is a stack, not only for the log étale topology, but for the topology generated by all logarithmically flat morphisms, from which it follows that LogPic X=S and LogPic 0 X=S are in fact the Néron models of their restrictions to U . Corollary 6.13. Let X=S be a log curve over a log regular scheme. The smooth algebraic spaces sLPic X=S and sLPic 0 X=S are Néron models of Pic X=S and Pic 0 X=S respectively.
Proof. This follows from Theorem 6.11 together with Remark 6.6.
Corollary 6.14. The strict saturated Jacobian sPic sat X=S satisfies the Néron mapping property for torsion sections. That is, for every T ! S smooth and L W T U ! Pic 0 X=S of finite order, there exists a unique extension to a map T ! sPic sat X=S . In particular, for every prime l T l sPic sat X=S D j T l Pic 0 X U =U as sheaves on the étale site over S , where j W U ! S is the inclusion and T l indicates the l-adic Tate module.
Proof. By Theorem 6.11, L extends uniquely to L in sLPic 0 X=S .T /. As the image of L in sTPic 0 X=S .T / is torsion, L actually lies in sPic sat X=S .

Models of Pic 0
We saw in Section 6 that the logarithmic Jacobian and its strict version are Néron models. However, while LogPic 0 X=S is proper (in a suitable sense), its strict version sLPic 0 X=S is in general neither quasi-compact, separated, nor universally closed.
In this section we undertake the study of smooth separated group-models of Pic 0 X U . We establish a precise correspondence between such models and subgroups of the strict tropical Jacobian sTPic 0 X=S .

A tropical criterion for separatedness.
We start by considering a tropical curve X D .V; H; r; i;`/ metrized by a sharp monoid M . For a given a monoid homomorphism ' W M ! N we denote by X ' the induced tropical curve metrized by N . We recall that by assumption all monoids we work with are fine and saturated. Lemma 7.2. Let X be a tropical curve metrized by a sharp monoid M , let ‰ be an abelian group, and let˛W ‰ ! TroPic 0 .X/ be a homomorphism. The following are equivalent: (i)˛is injective and ‰ is finite.
(ii) For every monoid P and homomorphism ' W M ! P not contracting any edge of X, the composition ‰ ! TroPic 0 .X/ ! TroPic 0 .X ' / is injective.
.ii/ ) .iii/ is clear. .iii/ ) .i/ We first establish the existence of a monoid homomorphism ' W M ! N not contracting any edge of X: since M is fine and saturated, it is the intersection of a cone P gp˝R with P gp for a finitely generated lattice P gp . Then any integral element in the interior of the dual cone _ Hom.P gp ; R/ will not contract any non-zero element of M and thus gives rise to such a map. Injectivity of˛thus follows. Now TroPic 0 .X ' / is finite since it is the cokernel of the homomorphism H 1 .X ' / ! Hom.H 1 .X ' /; Z/ induced by the monodromy pairing, which is injective by [26,Corollary 3.4.8]. Hence, ‰ is finite as well.
We fix a log curve X=S with S locally noetherian. We introduce the category Et whose objects are pairs .‰;˛/ of an étale group algebraic space ‰=S and a homomorphism W ‰ ! sTPic 0 X=S : Similarly, we let Sm be the category of pairs .G ;˛/ of a smooth group algebraic space G =S and a homomorphism˛W G ! sLPic 0 X=S . There is an obvious base change functor Recall the functor f introduced in Section 2.4. Notice that for an object .‰;˛/ of Et, we have a natural map f ‰ ! f sTPic 0 ! TroPic 0 and similarly for an object of Sm. The next proposition is the key statement of the section. Remark 7.4. The restriction of sLPic 0 X=S to U is naturally identified with Pic 0 X U =U , and therefore sTPic 0 X=S restricts to ¹0º. The fact that ‰ U D ¹0º implies that G U D Pic 0 X U =U .
Proof. We prove that (i) implies (ii). Since S is locally noetherian, it follows that ‰=S is quasi-separated, hence so is its base change G =S. By [37, Tag 0ARI] we may check the valuative criterion for a strictly henselian discrete valuation ring V ; we write Á for its generic point and s for its closed point. Fix a map a W Á ! G and two lifts to b; b 0 W V ! G ; using the group structure of G , we may assume b 0 D 0 and therefore a D 0.
Denote by d the map from V to S ; we endow Á with the log structure M Á pulled back from S , and we write M V for the maximal extension of M Á to a log structure on V . We obtain a commutative diagram Combining the assumption that (i) holds with Lemma 7.2 yields that the composition We move on to the next part of the statement, so from now on we suppose that S is log regular and that ‰ U D ¹0º. This in particular implies that G U D Pic 0 X U =U . It is clear that (iii) implies (ii); if we show that (ii) implies (i), then we immediately obtain (ii) ) (iii). Indeed, quasi-finiteness of ‰ together with the fact that Pic 0 X=S is quasi-compact, implies that G =S is quasi-compact.
It remains to prove that (ii) implies (i). Write K for the kernel of ‰ ! sTPic 0 X=S . It is étale, and since ‰ U D 0, K also vanishes over the open dense U S . Moreover, K is identified with the kernel of G ! sLPic 0 X=S ; as G =S is separated, so is K, hence K is trivial. This shows that ‰ ! sTPic 0 X=S is an open immersion. Now let t ! S be a geometric point, with image s 2 S . Because S is locally noetherian, by a special case of [11, 7.1.9] there exists a morphism Z ! S from the spectrum of a discrete valuation ring such that the closed point is mapped to s and the generic point to U . Now consider the composition ! W Z sh ! Z ! S with the strict henselization induced by t ! s. G is a smooth, separated model of Pic 0 X U and therefore ! G is a Z sh -smooth separated model of ! Pic 0 X U . The restriction of the latter to the generic point of Z sh is an abelian variety, which therefore admits a Néron model of finite type N =Z sh by [5,Corollary 1.3.2]. We claim that the natural mapˆW ! G ! N is an open immersion. In the case of schemes this would follow immediately from [5,Proposition 7.4.3]. The same proof shows for algebraic spaces that the natural map on identity components ! G 0 ! N 0 is an isomorphism. This implies thatˆis flat, hence is an open immersion by Lemma 7.5. In particular, the map of fibers G t ! N t is an open immersion. As N t is of finite type (and S is locally noetherian), so is G t . Then by descent ‰ t is of finite type as well; as it is moreover étale, it is finite over k.t /. In particular, the map ‰ t ! sTPic 0 t factors via sTPic tor t (which is quasi-finite by Lemma 5.3); it follows that the open immersion ‰ ! sTPic 0 factors via the open immersion sTPic tor ! sTPic 0 . The resulting open immersion ‰ ! sTPic tor is quasi-compact (indeed sTPic tor is locally noetherian since S is). This proves that ‰=S is quasifinite. The image of sTPic tor X=S under the functor F is sPic sat X=S , yielding: Corollary 7.6. Let X=S be a log curve with S locally noetherian. Then sPic sat X=S is separated.
7.2. Equivalence of categories. From this point until the end of the section we will assume that S is log regular and denote by U S the open dense where M S is trivial. Recall that the étale algebraic space sTPic 0 X=S has trivial restriction to U while sLPic 0 X=S is a smooth group space whose restriction to U is naturally identified with Pic 0 X U =U . We consider the full subcategory of Et Et ı WD ¹.‰;˛/ with ‰ ! S an étale group-algebraic space such that ‰ U D ¹0º;˛W ‰ ! sTPic 0 X=S a homomorphismº: and the full subcategory of Sm Sm ı WD ¹.G ;ˇ/ with G ! S a smooth quasi-separated group algebraic space; with fiberwise-connected component of identity G 0 separated over S; is an isomorphismº: The fiberwise connected-component of identity G 0 is an open subgroup space of G containing the identity section and whose geometric fibers G 0 s are the connected component of identity of G s . See Appendix A for details on G 0 . Proof. Over U there is already a natural identification G U D G 0 U D Pic 0 X U . For every point s of S of codimension 1, the restriction of Pic 0 X=S to O S;s is the identity component of its own Néron model. By [5, 7.4.3], the same holds for G 0 . Now by [35,XI,1.15], the isomorphism G 0 U ! Pic 0 X U extends uniquely to an isomorphism G 0 ! Pic 0 X=S .
Because of the Néron mapping property of sLPic 0 X=S (Corollary 6.13) there is a natural equivalence between Sm ı and the category with objects ¹.G ; '/ with G ! S a smooth quasi-separated group-algebraic space; with fiberwise-connected component of identity G 0 separated over S; ' W G U ! Pic 0 X U =U an isomorphismº; so we will not distinguish between the two. We obtain by restriction of the functor F of (7.1.1) a functor We are going to construct a quasi-inverse to F ı . We denote by 0 .G / the étale algebraic space G =G 0 of Definition A.2. By the universal property of G =G 0 for maps to étale spaces (Lemma A.4), together with the fact that 0 .sLPic 0 X=S / D sTPic 0 X=S (Lemma A.5), we obtain a functor Lemma 7.8. The functor F ı W Et ı ! Sm ı is an equivalence with … 0 as a quasi-inverse.
Proof. Let ‰ ! sTPic 0 X=S be in Et ı , with image G ! sLPic 0 X=S via F ı . The surjective map G ! ‰ factors by the universal property via an étale surjective map 0 .G / ! ‰. It remains to show that its kernel K vanishes. We obtain a commutative diagram of exact sequences Lemma 7.7, the left vertical map is an isomorphism and we conclude.
Conversely, let G 2 Sm ı . Because the functor F ı is defined as a fiber product there is a natural map f W G ! F ı .… 0 .G // D 0 .G / sTPic 0 sLPic 0 , and we obtain a commutative diagram of exact sequences where the rightmost vertical map is the identity and the leftmost vertical map h is the induced map ker.p ı f / ! ker.p/. Since f restricts over U to the identity of Pic 0 X U , so does h. It follows that h is the isomorphism of Lemma 7.7, and that f is an isomorphism as well.
As a corollary of Proposition 7.3 we refine the equivalence F ı . Definition 7.9. We let Et qf;mono to be the full subcategory of Et ı of those .‰;˛/ with a monomorphism (i.e., an open immersion) and ‰=S quasi-finite. We let Sm qc;sep be the full subcategory of Sm ı of those .G ; '/ with G ! S separated and quasi-compact.
Both Et qf;mono and Sm qc;sep are equivalent to partially ordered sets. For Et qf;mono this is clear, and for Sm qc;sep we observe that, for an object .G ; ' W G ! sLPic 0 X=S / of Sm qc;sep , ' is the base change of 0 .G / ! sTPic 0 X=S , by Lemma 7.8. The latter is an open immersion by Proposition 7.3, so ' is an open immersion.
The following corollary allows us to describe all possible smooth separated group models of Pic 0 X U in terms of open subgroups of the strict tropical Jacobian.
Corollary 7.10. The equivalence F ı W Et ı ! Sm ı restricts to an order-preserving bijection F W Et qf;mono ! Sm qc;sep .
Proof. This is immediate by Proposition 7.3.
The partially ordered set Et qf;mono has a maximal element, namely the quasi-finite étale group space sTPic tor X=S representing the torsion part of the sheaf sTPic 0 X=S . From Corollary 7.10 we deduce: Theorem 7.11. Let X=S be a log curve over a log regular base S , and U S the open where the log structure is trivial. The partially ordered set of smooth separated group-S -models of finite type of Pic 0 X U has F .sTPic tor X=S / D sPic sat X=S as maximum element. Namely, any other such model has a unique open immersion to sPic sat X=S .
7.3. Possible extensions to the case of log abelian varieties. It is natural to ask which of the results of this paper remain valid when the Jacobian of a curve is replaced by an arbitrary abelian variety. Suppose that we have a log regular log scheme S, and a log abelian variety A log =S (which is necessarily an abelian variety A U over the open locus U Â S on which the log structure is trivial). Now A log is a sheaf on .LSch=S / ét which has a tropicalization A trop over S . We can restrict A log and A trop to sheaves on the strict étale site .Sch=S/ ét to obtain algebraic spaces sA log and sA trop . The tropicalization sA trop can equivalently be defined as the quotient sA trop D sA log =sA 0 of sA log by the (semiabelian) fiberwise connected component of the identity sA 0 , and, as described in detail in [18, 4.1.2] also has an explicit combinatorial description étale locally in S as Hom.X; M gp S / .Y / =Y for lattices X and Y ; the subscript .Y / indicates a subgroup of Hom.X; M gp S / (see [18, 3.1]), analogous to the bounded monodromy subgroup of the Jacobian.
Conjecture 7.12. The log abelian variety A log has the log Néron mapping property (Definition 6.3) with respect to U . Conjecture 7.12 would in particular imply that sA log is always a Néron model for A U , but is rarely separated: it is separated if and only if sA trop is finite. At the moment, we do not have a proof of Conjecture 7.12. Our proof for the Jacobian uses the geometry of the curve to produce the extension. On the other hand, the proof that abelian varieties are their own Néron models goes by extending line bundles on the dual abelian variety. This argument would extend to the case of logarithmic abelian varieties if we had a theory of log Picard functors for higher dimensional logarithmic schemes which satisfies analogues of Property 3.55, Property 3.56, Property 3.57, and the usual duality axioms.
Conditional on Conjecture 7.12, our proof of Theorem 1.2 goes through verbatim to show that there is a bijection between quasi-finite open subgroups of sA trop and smooth, separated, quasi-compact S -group models of A U .

Alignment and separatedness of strict log Pic
For X=S a log curve over a log regular base S with U S the largest open where the log structure is trivial, we have shown in Corollary 6.13 that sLPic 0 X=S is the Néron model of Pic 0 X U =U . It is worth stressing the fact that classically, the term Néron model is reserved for separated, quasi-compact models satisfying the Néron mapping property. The strict logarithmic Jacobian fails in general to satisfy these properties, as observed in Example 4.6.
In the papers [12,31,33], several criteria were introduced for the Jacobian of a prestable curve X=S (or for an abelian variety in [32]) to admit a separated, quasi-compact Néron model. They are all closely related to the general notion of log alignment that we introduce here: Definition 8.2. A cycle in a graph is a path that begins and ends at the same vertex, and which otherwise repeats no vertices. A subset S of the edges of a graph is called cycleconnected if for every pair e, e 0 2 S of distinct edges there exists a cycle in S containing e and e 0 . It is shown in [13, Lemma 7.2] that the maximal cycle-connected subsets (which are there called circuit-connected) form a partition of the edges of the graph. Definition 8.3. We say that a tropical curve X metrized by a sharp fs monoid M is log aligned when for every cycle in X, all lengths of edges of lie on the same extreme ray of M . Let X ! S be a log curve. We say that X=S is log aligned at a geometric point N s of S when the tropicalization of X at N s is log aligned. We say that X=S is log aligned if it is log aligned at every geometric point of S . Lemma 8.4. Let M be a sharp fs monoid, X a tropical curve metrized by M and Y a subdivision of X. Then Y is log aligned if and only if X is.
Proof. It suffices to treat the case where Y is a basic subdivision of X. Suppose it is, and call e the subdivided edge: it is replaced in Y by a chain of two edges e 1 ; e 2 of the same total length. There is a canonical bijection between cycles of X and of Y. where the first equality holds since bounded monodromy can be checked separately on each X i by Remark 3.34. Quotienting by H 1 .X/, we obtain The right-hand side is finite, as the rank of Hom.H 1 .X i /; Z i / is equal to the rank of H 1 .X i /.
For the reverse implication, suppose X=S is not log aligned; we will show As the extreme rays of M span M over Q, and we are free to replace M by finite index extensions, we can assume that each length`.e/ can be written as a sum of elements in M that lie on the extreme rays of M . We may then subdivide X so that each edge in the subdivision has length along the extreme rays of M . Using Lemma 8.4 and the invariance of the tropical Jacobian under subdivisions of X, we may then assume that each edge of X has length which lies along an extreme ray of M . Pick a spanning tree T of X.
The edges e 1 ; : : : ; e r not in T correspond to cycles 1 ; : : : ; r forming a basis of H 1 .X/. By hypothesis, X is not log aligned, so one of the i , for example 1 , has length not belonging to an extremal ray of M gp . Therefore, there exists an edge e in 1 of length along an extreme ray of M different than the ray containing the length of e 1 . We claim that the intersection pairings of the family .e; e 1 ; : : : ; e r / are independent bounded monodromy maps H 1 .X/ ! M gp . The fact that intersection pairing with an edge has bounded monodromy is general: for any edge e, and any cycle 2 H 1 .X/, the intersection pairing e: evidently has length bounded by the length of . To see that the pairings are independent, notice that e i : j is ı ij`. e i / where ı ij is the Kronecker delta. Consider a linear combination b D ae C P a i e i with coefficients in Z, and suppose the intersection pairing of b is trivial. Then a`.e/ C a 1`. e 1 / D b: 1 D 0, combined with the fact e and e 1 have independent lengths, yields a D a 1 D 0. Hence for j > 1 we have a j`. e j / D b: j D 0, from which we deduce a j D 0.

The strict logarithmic Jacobian and the Picard space
Over a Dedekind base S, Raynaud constructed the Néron model of the Jacobian of a curve X=S as the quotient of the relative Picard space by the closure N e of the unit section (see [5,Section 9.5]). When dim S > 1, the closure N e is in general neither S -flat nor a subgroup, and so this quotient is not representable. In [12] and [31], necessary and sufficient conditions for the flatness of N e are given, proving the existence of separated Néron models when these conditions hold. In this section we show that Raynaud's approach can be extended over higher-dimensional bases even when N e is not S -flat, simply by replacing N e by its largest open subspace N e K et which is étale over S . This allows us to describe the Néron model (constructed above as the algebraic space sLPic 0 ) as the quotient of the Picard space by N e K et , under somewhat more restrictive assumptions on X=S. The construction of the Néron model in [33] is done by gluing local models of this form.
Letting W X ! S be a log curve, we write Pic tot0 for the kernel of the composition There is a canonical map Pic tot0 X=S ! sLPic 0 X=S taking a line bundle to the associated log line bundle.
Write N e for the schematic closure of the unit section in Pic tot0 X=S . If U , V ,! N e are open immersions étale over S , then the same is true of their union. Hence N e has a largest open subscheme which is étale over S , which we denote by N e K et , a locally closed subscheme of Pic tot0 X=S .
Theorem 9.1. Suppose that S is log regular. Then: (i) The map f W Pic tot0 X=S ! sLPic 0 X=S has kernel N e K et .
(ii) If in addition X is regular, then f is surjective.
Remark 9.2. By Property 3.56, a log modification X 0 ! X induces an isomorphism sLPic 0 X=S ! sLPic 0 X 0 =S . On the other hand Pic tot0 X=S ! Pic tot0 X 0 =S is an open immersion but in general not an isomorphism. When S is log regular and regular, we can always find, étale locally on S , a log modification of X=S with regular total space. To see this, note that since S is locally Noetherian, it has an étale cover by nuclear schemes by Lemma 3.45, so we can assume that S is nuclear. Let X denote the tropicalization of X over the closed stratum. As S is log regular and regular, every edge of X is marked by an element of a free monoid N r . Put X 0 D X. As long as X 0 has an edge whose length is not one of the generators of N r , replace X 0 by any basic subdivision at that edge. The process terminates, and provides a maximal subdivision of X. This subdivision lifts to a log modification X 0 ! X whose total space is regular: X 0 is evidently log regular, and the log structure around a node of length a generator of N r is isomorphic to N r˚N N 2 Š N rC1 . Combining this observation with Theorem 9.1, we obtain a local description of sLPic 0 X=S as a quotient of a Picard space. where K WD ker f is equal to ker g by the snake lemma. The map g is a map of étale group spaces (Theorem 4.4), hence its kernel is étale, i.e., K is étale. Note that the locus U ,! S over which the morphism X ! S is smooth is schematically dense by our log regularity assumption. One checks immediately that K is trivial over U , hence the map K ! Pic tot0 X=S factors via the closure N e of the unit section (since K ! S is étale, so the pullback of U Â S is schematically dense in K). The map K ! N e is a quasi-compact immersion. To show that it is open, we may pick a geometric point p 2 K and restrict to the strict henselization of S at p. There is a unique section S ! K through p, which by Lemma 9.4 is open in N e. Thus K ! N e is open, and factors through an open immersion K Â N e K et . The reverse inclusion is easier: the map N e K et ! sLPic 0 X=S is zero when restricted to U , so by the Néron mapping property of sLPic 0 X=S (Theorem 6.11), the map N e K et ! sLPic 0 X=S is zero and therefore N e K et K. (ii) Note that since S is log regular, so is X ; thus, if in addition X is regular, its log structure is locally free. Since the log structure of S is isomorphic to the log structure of X away from the singular points of the fibers, the log structure on S must therefore also be locally free, and thus S must also be regular. As the Néron model N D sLPic 0 X=S of Pic 0 X U =U is smooth over S, it follows that X N is regular as well. The canonical isomorphism N U D Pic 0 X U =U corresponds to a line bundle L on X N U , represented by a Cartier divisor D. The schemetheoretical closure D of D in X N is Cartier by regularity of the latter. Thus the line bundle O.D/ provides a lift of L under the natural morphism Pic tot0 X=S ! N , which is therefore surjective. Lemma 9.4. Let S be the spectrum of a local ring, let U ,! S be open, and let X ! S be a morphism such that X U ! U is an isomorphism, and X U is schematically dense in X . Let W S ! X be a section. Then is open.
Proof. Writing s for the closed point of S , let .s/ 2 V ,! X be an affine open neighborhood; then factors via V (the preimage of V via is open and contains s). We write 0 W S ! V for the factored map.
Since V ! S is separated, the map 0 W S ! V is a closed immersion. On the other hand, its image contains the schematically dense V U ,! V , hence 0 is an isomorphism.
A. The functor of connected components of a smooth quasi-separated group algebraic space Throughout this appendix, S denotes a scheme and G=S a group algebraic space with unit section e 2 G.S /. We extend some results of Romagny [36] to the case where G=S is smooth and quasi-separated, avoiding the quasi-compactness assumptions of [36].
Lemma A.1. Suppose that G=S is quasi-separated, flat, locally of finite presentation, and has reduced geometric fibers. Then there is a unique open subspace G 0 of G such that each fiber G s;0 of G 0 =S is the connected component of G s containing e.s/. Moreover, G 0 is a subgroup of G.