Extending the double ramification cycle by resolving the Abel-Jacobi map

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to extend. We construct a `universal' resolution of the Abel-Jacobi map, and thereby extend the double ramification cycle to the whole of the moduli of stable curves. In the non-twisted case, we show that our extension coincides with the cycle constructed by Li, Graber, Vakil via a virtual fundamental class on a space of rubber maps.


Introduction
Fix integers g, n ≥ 0 satisfying 2g−2+n > 0, and integers a 1 . . . a n and k such that i a i = k(2g − 2). Over the moduli stack M g,n we have the universal curve C g,n with tautological sections x 1 , . . . , x n . Write J for the universal jacobian (an abelian scheme) and σ for the section of J given by the line bundle ω ⊗k (− i a i x i ) on C g,n . The pullback of the unit section along σ defines a codimension-g cycle on M g,n , the double ramification cycle (DRC). The problem of producing a 'reasonable' extension of the DRC to the Deligne-Mumford-Knudsen compactificationM g,n , and computing the class of the resulting cycle in the tautological ring, was proposed by Eliashberg.
In the case k = 0, an extension of the class to the whole ofM g,n was constructed by Li [Li01], [Li02], Graber, and Vakil [GV05]. This class was computed in the compact-type case by Hain [Hai13], and this was extended to treelike curves with one loop by Grushevsky and Zakharov [GZ14]. More recently, Janda, Pandharipande, Pixton and Zvonkine [JPPZ17] computed this class on the whole of M g,n , proving a conjecture of Pixton.
A construction for arbitrary k was proposed by Guéré [Gué16], but the situation here is more complicated. Pixton's conjecture makes sense for all k, but is purely combinatorial in origin. A more geometric conjecture is given by Faber and Pandharipande [FP05] for k = 1, generalised by Schmitt [Sch16] to all k ≥ 1; their formulae are moreover conjectured to coincide with Pixton's. However, it is at present unclear whether Guéré's construction is compatible with these conjectures.
In this paper we give a simple construction of an extension of an extension of the DRC for arbitrary k, and for k = 0 we verify that it agrees with the construction of Li, Graber and Vakil. Given the situation described above, it seems very interesting to know whether it coincides with the construction of Guéré when k is arbitrary; this is the subject of current work in progress.
We remark that Kass and Pagani [KP17] have recently constructed large numbers of extensions of the DRC (for every k ∈ Z); the relation to the known constructions remains to be understood.

Statement of main results
We write J/M g,n for the unique semiabelian extension of the universal jacobian (also denoted above by J), and ω for the relative dualising sheaf of the universal stable curve overM g,n . The section σ = ω ⊗k (− i a i x i ) does not in general extend over the whole ofM g,n . This can be partially resolved by blowing up M g,n . Let f : X →M g,n be a proper birational map from a normal stack (a 'normal blowup'). The section σ is then defined on some dense open of X. We writeX for the largest open of X on which this rational map can be extended to a morphism, and σ X :X → J for the extension.
We define the double ramification locus DRL X X to be the schematic pullback of the unit section of J along σ X , and the double ramification cycle DRC X to be the cycle-theoretic pullback, as a cycle supported on DRL X . Now the map X →M g,n is rarely proper, but somewhat miraculously the map DRL X →M g,n is quite often proper. More precisely, we have: Theorem 1.1 (theorem 6.3). In the directed system of all normal blowups ofM g,n , those X such that DRL X →M g,n is proper form a cofinal system. Now DRC X is supported on DRL X , so when the map DRL X →M g,n is proper, we can take the pushforward of DRC X toM g,n . Writing π X * DRC X for the resulting cycle onM g,n , we might hope that these cycles 'converge' in some way as we move up in the tower of blowups. In fact, this is true in a very strong sense: Theorem 1.2 (theorem 6.4). The net π X * DRC X is eventually constant in the Chow ring CH Q (M g,n ). We denote the limit byD RC. Theorem 1.3 (theorem 7.3). Suppose that k = 0. The classD RC coincides with the extension of the double ramification cycle defined by Li, Graber, and Vakil [GV05] (and computed by [JPPZ17]).
We are currently engaged (jointly with Alessandro Chiodo) in computing the class ofD RC in the tautological ring. When g = 1 and k = 0 this computation is very simple, and (reassuringly) some manipulation shows that our formula is equivalent to the one obtained by [JPPZ17] (which was already known to Hain [Hai13]).

Conjectural relationship to a cycle of Pixton
Given the data g, n, k and a, Pixton introduced a cycle P g,k g (A) in the tautological ring ofM g,n , given in terms of decorated graphs -here A = (a 1 + k, . . . , a n + k); the details of the construction can be found in [JPPZ17, section 1.1]. The main result of [JPPZ17] is that, when k = 0, there is an equality of cycles DRC a = 2 −g P g,0 g (A). Now that we have a construction ofD RC a valid for all k, it seems natural to propose Conjecture 1.4. For all k, there is an equality of cycles DRC a = 2 −g P g,k g (A) in CH g Q (M g,n ). Some evidence for this conjecture is given in the following section.

Multiplicativity of the double ramification cycle
In [HPS17] we will use these results to construct an extension of the double ramification cycle to the small b-Chow ring ofM g,n -this is the colimit of the Chow rings of the smooth blowups ofM g,n , with transition maps given by pulling back cycles. Given vectors a, b of ramification data, we will show that the basic mutiplicativity relation holds in the small b-Chow ring ofM g,n , but fails in the Chow ring ofM g,n .
One consequence of this is that, if conjecture 1.4 is true, the relation (1) should also hold for Pixton's cycles on the locus of compact type curves. This relation can be independently checked using known relations in the tautological ring (see [HPS17]), which may be seen as evidence for conjecture 1.4.

Comparison to the approach of Li, Graber and Vakil
The approach of Li [Li01], [Li02], Graber, and Vakil [GV05] when k = 0 is based on thinking of the DRC as the locus of curves admitting a map to P 1 with specified ramification over 0 and ∞. They define a stack of stable maps to 'rubber P 1 ', i.e. to [P 1 /G m ]. They then define the DRC as the pushforward of a virtual fundamental class from this stack of stable maps. This enables them to apply the well-developed machinery of virtual classes and spaces of stable maps. In contrast, our approach is in a sense very naive; using blowups to resolve the indeterminacies of rational maps goes back to classical algebraic geometry (and in the non-proper case to Raynaud and Gruson [RG71]). The more elementary nature of our approach makes it very easy to extend to the case k = 0, and we hope will allow further extensions; we are particularly interested in developing further the Gromov-Witten theory of BG m , extending the results of [FTT16] beyond the 'admissible' case.

Strategy of proof
Our proofs proceed by constructing a certain key normal blowup which is a 'universal resolution' of the Abel-Jacobi map -we denote it M g,n . We write M ♦ g,n :=M g,n . We will show a very powerful universal property of M ♦ g,n : a 'nondegenerate' map T →M g,n admits an extension of σ if and only if T →M g,n factors via M ♦ g,n (section 4). In the light of this, the morphism M ♦ g,n →M g,n may be viewed as a universal resolution of the indeterminacies of the Abel-Jacobi map. This solves a problem proposed by Gruschevzky and Zakharov [GZ14, Remark 6.3]. Together with our other results, it seems also to solve a problem of Cavalieri, Marcus and Wise proposed in [CMW12,Section 1.4].
Now if X →M g,n is any normal blowup which factors via x : X → M g,n , it follows that We will establish that DRL ♦ is proper overM g,n , whereupon the analogous properness result will hold for any normal blowup which factors via M g,n , establishing theorem 1.1. Theorem 1.2 will then follow fairly formally. Finally, when k = 0 we use the deformation-theretic tools of Marcus, Wise and Cavalieri to establish theorem 1.3.

Comparison to some other recent results
The preprint [KP17] of Kass and Pagani (posted on the same day as the first version of this preprint) gives a very different approach to resolving the Abel-Jacobi map, by studying families of stability conditions on the space of rank 1 torsion-free sheaves. Moving through a suitable family produces a series of flips of a certain compactified jacobian, after which the Abel-Jacobi map extends over M g,n (this series of flips depends on the ramification data). In essence, we modify the source of the Abel-Jacobi map, whereas Kass and Pagani modify the target. In this way, they produce a number of different extensions of the DRC; it remains to be understood how these extensions are related to one-another, and to the DRC of Li, Graber and Vakil.
More recently, Marcus and Wise [MW17] have given another approach to resolving the Abel-Jacobi map when k = 0, rather closer in spirit to the present preprint. They also use logarithmic geometry to modifyM g,n , but their construction is based on stacks of stable maps rather than a universal property as in the present preprint. We hope to understand the relation between these approaches more fully in future.
An extension of the Abel-Jacobi map over a large locus inM g,n (when k = 0) was produced some time ago by Bashar Dudin [Dud15]. His locus depended on a choice or ramification data, as in our construction. But he did not make blowups ofM g,n , and so was not able to extend over the whole of the boundary.
Smeets, Nicolo Sibilla, Mattia Talpo, Jonathan Wise, Paolo Rossi, and many others. I am also grateful to Dimitri Zvonkine for encouraging me to include the extension to the twisted case.

Notation and setup Base ring
We work over the fixed base ring Λ := Spec Z equipped with the trivial log structure. The reader who prefers to take Λ = Spec C can freely do so with no modifications to what follows, and a substantial simplification to the proof of lemma 6.1. All our constructions commute with arbitrary base-change over Λ.
It seems that the definition of the double ramification cycle given in [GV05] can readily be extended over Z, though we have not verified this. The computation of [JPPZ17] is carried out over C, and gives the class of the double ramification cycle as the value at zero of a polynomial P in a variable r. The value of P at a given value of r is computed using r-th roots of line bundles (cf. Chiodo's formulae [Chi08]), which may give problems in characteristic dividing r. But in fixed characteristic , the polynomial P is completely determined by its values on integers coprime to , so this problem can be circumvented (the author is grateful to Felix Janda for pointing this out). So it remains likely that the results of [JPPZ17] can be extended to arbitrary characteristic.

Stack of weighted stable curves
For us, 'curve' means proper, flat, finitely presented, with reduced connected nodal geometric fibres. Rather than treating eachM g,n and weighting a 1 , . . . , a n separately, we denote byM the stack of stable pointed curves together with a k-twisted integer weighting -in other words, points ofM consist of tuples (C, x 1 , . . . , x n , a 1 , . . . , a n , k) where the x i are the marked sections of our stable curve, and the a i and k are integers satisfying This stack is smooth over Λ, but is far from being connected -it is a countably infinite disjoint union of substacks, each proper over Λ.
We write C/M for the universal curve, and J = Pic 0 C/M for the universal jacobian (a semiabelian scheme, the fibrewise connected component of the identity in Pic C/M ). Let M denote the open substack ofM parametrising smooth curves.
We write x i for the tautological sections, and Σ for the Cartier divisor on C given by i a i x i . Then σ ∈ J M (M) denotes the tautological section given by ω ⊗k C (−Σ).

Log structures
We work with log structures in the sense of Fontaine-Illusie, using Olsson's generalisation to stacks [Ols01]. We put log structures on C/M following Kato [Kat96], and the log structure onM will be denoted αM : PM → OM, etc. If P is a (sheaf of) monoid(s), we writeP := P/P × ; this notation does not sit well with the notationM for the moduli stack of stable curves, but both are very standard, and there is no actual ambiguity.

Weightings on a graph
A graph consists of a finite set V of vertices, a finite set H of half-edges, a map 'end' from the half-edges to the vertices, an involution i on the half-edges, a genus g : V → Z ≥0 , and a twist k. Graphs are assumed connected unless stated otherwise, and the genus of a graph is its first Betti number plus the sum of the genera of the vertices.
Self-loops are when two distinct half-edges have the same associated vertex and are swapped by i. We define edges as sets {h, h } (of cardinality 2) with i(h) = h . Legs are fixed points of i. A directed edge h is a half-edge that is not a leg; we call end(h) its source and end(i(h)) its target, and sometimes write it as h : end(h) → end(i(h)). We write E = E(Γ) for the set of edges, and → E for the set of directed edges.
The valence val(v) of a vertex is the number of non-leg half-edges incident to it, and we define the canonical degree A closed walk in Γ is a sequence of directed edges so that the target of one is the source of the next, and which begins and ends at the same vertex. We call it a cycle if it does not repeat any vertices or (undirected) edges.
Definition 2.1. A G-weighting is a function w from the half-edges to a group G such that: When the twist k = 0, a G-weighting can be though of as a flow of an incompressible fluid around the graph.
A G-leg-weighted graph is a graph together with a function from the legs to G, such that the sum over all the legs is −k(2g(Γ) − 2). If Γ is a G-leg-weighted graph we write W (Γ) for the set of weights on Γ which restrict to the given values on the legs. It is easy to see that W (Γ) is never empty (this uses the running assumption of connectedness). After choosing an oriented basis of H 1 (Γ, Z), the set W (Γ) becomes a torsor under H 1 (Γ, G). In this article we will use only the cases G = Z and G = Q. We will refer to weightings taking values in Z just as weightings.

Combinatorial charts
If p : Spec k →M is a geometric point, we have an associated graph Γ p . If a node of the curve over p has local equation xy − r for some r ∈ O et M,p , then the image of r in the monoidPM ,p is independent of the choice of local equation. In this way we define a map : E(Γ p ) →P p , recalling that edges of the graph correspond to nodes of the curve. This is a logarithmic version of the labelling defined in [Hol17].
Given a leg-weighted graph Γ with edge set E, setM Γ = Spec Λ[N E ], equipped with the toric divisorial log structure. As usual we write α : PM Γ → OM Γ for the map from the sheaf of monoids to the structure sheaf.
To any point p inM Γ we associate the graph Γ p obtained from Γ by contracting exactly those edges e such that the corresponding basis elements of N E specialise to units at p. We define a map : E(Γ p ) →PM Γ ,p by sending an edge e to the image of the associated basis element of N E . The map naturally lifts to PM Γ ,p , and does not send any edge to a unit, by definition of Γ p .
A combinatorial chart ofM consists of a leg-weighted graph Γ and a diagram of log stacksM satisfying the following five conditions: 1. U is a connected log scheme 2. g : U →M Γ is strict and log smooth 3. f : U →M is strict and logétale 4. the image of g meets the minimal stratum ofM Γ .
Let p : Spec k → U be any geometric point, yielding natural maps 5. We require the existence of an isomorphism such that f ( (e)) = g ( (ϕ p (e)) for every edge e (which necessarily makes this ϕ p unique if it exists). Moreover, the map ϕ p sends the leg-weighting on Γ f •p coming from the −a i to the leg-weighting on Γ g•p coming from that on Γ.
It is clear from [Kat00] thatM can be covered by combinatorial charts. We will first construct M Γ and M ♦ Γ over theM Γ , and then descend these toM.

Construction of M ♦
The cone c w associated to a weighting For the remainder of this subsection we fix a combinatorial chart with leg-weighted graph Γ, writing E for the edge set. To a weighting w ∈ W (Γ) we will associate a rational polyhedral cone c w inside the positive orthant of , such a cone will induce an affine toric scheme overM Γ in the usual way, cf. [Ful93]. Such an object has a natural log structure. We will build M ♦ Γ by glueing together affine patches of this form. Fix a weighting w ∈ W (Γ) and let γ be an oriented cycle in Γ. If e is a directed edge appearing in γ, we define w γ (e) to be the value of w on the first half-edge of e -we might think of this as the flow along e in the direction given by γ.
Definition 3.1. Let t ∈ Q E ≥0 ; we refer to such an element as a thickness. We say t is compatible with w if for every cycle γ we have e∈γ w γ (e)t(e) = 0. (2) One checks easily that the set of all thicknesses t which are compatible with a given weighting w form a rational polyhedral cone in Q E ≥0 , which we denote by c w . Remark 3.2. Suppose w ∈ W (Γ) and that c w ∩ c w contains a thickness t which does not vanish on any edge. Then w = w . Definition 3.3. We write F Γ for the set of faces of the cones c w as w runs over weightings.
Remark 3.4 (Example: 2-gon, k = 0). Suppose the graph Γ has two edges and two (non-loop) vertices u and v. Suppose the leg weighting is +n at u and −n at v. Weightings consist of a flow of a along one edge from u to v, and n − a along the other (again from u to v), for a ∈ Z. The cone c w is non-zero if and only if both a and n − a are non-negative, and for such a the cone c w is the ray in Q 2 ≥0 through the point (n − a, a). Thus we get exactly n + 1 rays in the positive quadrant. This is the fan F Γ .
Remark 3.5 (Example: 3-edge banana, k = 0). Suppose again that the graph has two vertices u and v, but now three edges between them. Suppose that the weighting is +10 on u and −10 on v. Remark 3.5 shows the slice through the incomplete fan F Γ ⊆ Q 3 ≥0 where the sum of the values of the thickness on the edges is 1.
In the next subsection we will verify that F Γ is a finite fan (in the sense of toric geometry). The reader might prefer to skip the details, as they play little role in what follows.
F Γ is a finite fan Lemma 3.6. Let w 1 , w 2 ∈ W (Γ) be two weightings. Then the intersection of the cones c w 1 and c w 2 is a face of c w 1 .
Proof. Let t ∈ c w 1 ∩ c w 2 . For an edge e with t(e) = 0 we claim w 1 (e) = w 2 (e). To see this, let Γ t be the graph obtained from Γ by contracting those edges on which t vanishes. Then each w i | Γt is a weighting compatible with t| Γt , and the latter does not vanish on any edge, so w 1 | Γt = w 2 | Γt by remark 3.2. Define Writing Γ = for the graph obtained from Γ by contracting E = , so that w = := w 1 | Γ= = w 2 | Γ= . Define c = to be the cone in Q E= ≥0 corresponding to the weighting w 1 | Γ= = w 2 | Γ= . By the claim above we see that every t ∈ c w ∩ c w vanishes on every edge of E = .
I now claim that Now we know that any two c w intersect in a face. The following well-known lemma shows that that F Γ is a fan.
Lemma 3.7. Let Φ 0 be a set of cones in Q n , and assume that for all C, C ∈ Φ 0 , the intersection C ∩ C is a face of C. Let Φ be the set of all faces of cones in Φ 0 . Then Φ is a fan.
Proof. The proof is by induction on h 1 (Γ). Recalling that W is a torsor under H 1 (Γ, Z), we see that W is finite whenever h 1 (Γ) = 0, so in this case there is nothing to prove. In general, we say that a weighting w on Γ admits a positive cycle if there exists a cycle γ in Γ such that w(e) > 0 for every e ∈ γ. In the next two lemmas, we will show 1. For fixed Γ, all but finitely many w admit a positive cycle (lemma 3.9).
2. If γ is a positive cycle for w and Γ/γ is the graph obtained from Γ by contracting every edge in γ, then is the cone associated to the restricted weighting w| Γ/γ , and 0 is the zero vector in Q E(γ) ≥0 (lemma 3.10).
Now there are only finitely many cycles in Γ, and for every cycle γ we have that h 1 (Γ/γ) < h 1 (Γ), hence by our induction hypothesis we have that there are only finitely many cones for Γ/γ. Putting these ingredients together concludes the proof.
Lemma 3.9. For fixed Γ, all but finitely many w admit a positive cycle Proof.
Step 1: setup. Fix a weighting w, and fix a basis B of H 1 (Γ, Z) consisting of cycles. We can think of b ∈ B as a function from the set → E of directed edges of Γ, sending a directed edge e to 0 is e / ∈ b, and ±1 otherwise (depending on whether the orientation of e agrees with b). Given an element v ∈ Z B , we define cycle(v) to be the function → E → Z given by b v b b. This is somewhat clumsy notation, as it would be nicer just to think of v as an element in the cycle space, but distinguishing carefully between v and cycle(v) seems important to avoid confusion in this proof. In this way we see that every weighting on Γ is of the form w + cycle(v) for a unique v ∈ Z B .
Define recursively a function ϕ : Z ≥0 → Z by setting ϕ(0) = 1 and ϕ(n) = 0≤j<n ϕ(j). Write m := max e∈E(Γ) |w(e)|, and h := h 1 (Γ). Define N = mϕ(h) (this is rigged exactly to make step 3 of this argument work). Define a finite set. In the remainder of this argument, we will show that for every v ∈ Z B \ B(N ), the weighting w + cycle(v) admits a positive cycle. To this end, we fix for the remainder of the argument a v ∈ Z B \ B(N ).
Step 2: ordering the v b .
Up to changing the orientations of the elements of B, we may assume that all the integers v b are non-negative. Put an ordering on B so that the v b i are in increasing order: Step 3: choosing a critical b r . We now show by a small computation that there exists 1 ≤ r ≤ h such that Indeed, suppose no such r exists. Then for each 1 ≤ j ≤ h we have and induction on j yields v b j ≤ mϕ(j) for all 1 ≤ j ≤ h, contradicting our assumption that v b h > N = mϕ(h). From now on, we fix such an r.
Step 4: finding the positive cycle γ.
Define a function f : In the remainder of this step we will show that there exists a cycle γ with f (e) > 0 for every directed edge e ∈ γ. In step 5 we will see that any such γ is necessarily a positive cycle. First, because the b j are part of a basis, we see that f is not identical to zero. Hence there is a directed edge e with f (e) > 0. We build a path in Γ starting with e by the following procedure: whenever we hit a vertex v, choose an edge e v out of v such that f (e v ) > 0. Why is this always possible? Note that the sum over all edges e into v of f (e ) is necessarily zero, and since we arrived at v along an edge with f > 0, there must also be an edge leaving v with f > 0.
Since Γ is finite, this path must eventually meet itself, say at a vertex v 0 . Deleting the start of the path up to v 0 yields the cycle γ that we sought.
Step 5: showing that γ is indeed a positive cycle for the weighting w + cycle(v).
Choose any γ as in step 4. Define functions F , G : Observe that F + G = w + cycle(v) as functions → E → Z. Let e ∈ γ be a directed edge; we will show that F (e) + G(e) > 0 as required.
Lemma 3.10. Fir a weighting w. If γ is a positive cycle for w and Γ/γ is the graph obtained from Γ by contracting every edge in γ, then is the cone associated to the restricted weighting w| Γ/γ , and 0 is the zero vector in Q follows easily from the definition of the cone of a weighting, since every cycle in Γ/γ arises by restricting some cycle in Γ. We need to show the other inclusion, so let t ∈ c w be a thickness. The t satisfies the equation Putting together lemmas 3.6, 3.7, and 3.8 we immediately deduce: Corollary 3.11. The set of cones F Γ is a finite fan inside Q E ≥0 .
The construction of M ♦ Definition 3.12. We define M ♦ Γ to be the toric scheme overM Γ defined by the fan F Γ . We equip it with the toric log structure.
If w ∈ W (Γ) we define M ♦ w to be the affine toric variety associated to c w , an affine patch of M ♦ Γ .
Remark 3.13. It follows from [Ful93] that M ♦ Γ →M Γ is separated, of finite presentation, and normal.
Given a combinatorial chartM ← U →M Γ we define M ♦ U by pulling back M ♦ Γ fromM Γ . Such U form anétale cover ofM, and the collection of M ♦ U is easily upgraded to a descent datum.
Definition 3.14. We define π ♦ : M ♦ →M to be the algebraic space obtained by descending the M ♦ U .
Theorem 3.15. The stack M ♦ is normal, and the map π ♦ : M ♦ →M is separated, of finite presentation, relatively representable by algebraic spaces, and birational.
Note that π ♦ is almost never proper.
Proof. Toric varieties in this sense are always normal, see [Ful93]. The properties of π ♦ are all local on the target, so it is enough to check them for the M ♦ Γ → M Γ . Separatedness is automatic for toric varieties (in the sense of Fulton), finite presentation follows from the finiteness of the fans, see corollary 3.11. The maps are clearly isomorphisms over the locus M of smooth curves, hence are birational.

Universal property of M ♦
Definition 4.1. We say a stack T is locally desingularisable ifétale-locally it admits a proper surjective finitely presented map T → T with T regular and T → T inducing an isomorphism between some dense open substacks of T and T .
For example, this is true in characteristic zero, and for arithmetic surfaces and threefolds, and for log regular schemes. We can now give a more precise variant of the notion of a σ-extending morphism from the introduction. Recall that σ : M → J is the morphism given by ω ⊗k (− i a i x i ).
Definition 4.2. We say a map t : T →M is non-degenerate if T is normal and locally desingularisable, and t −1 M is dense in T . We say t is σ-extending if in addition the section t * σ ∈ t * J (t −1 M) admits a (necessarily unique) extension to t * J (T ).
For example, the open immersion M →M is clearly σ-extending, and in general the identity onM is not σ-extending. In this section we will show that M ♦ →M is σ-extending, and moreover that M ♦ is universal with respect to this property: that it is terminal in the 2-category of σ-extending morphisms (corollary 4.6). Our main technical result is the following: Lemma 4.3. Fix a combinatorial chartM ← U →M Γ , and let t : T → U be such that the composite T →M is non-degenerate. The following are equivalent: 1. Locally on T there exists a weighting w on Γ such that T →M Γ factors via M ♦ w →M Γ .

T →M is σ-extending.
Before giving the proof we set up a little notation, which will also be useful in section 5. For each edge e ∈ E we define (e) ∈ OM Γ (M Γ ) to be the image of the standard basis vector δ e ∈ N E under the log structure map. If w ∈ W (Γ) is a weighting and γ ⊆ Γ a cycle, we let where w γ (e) ∈ Z is the value of w on e in the direction dictated by γ. One easily verifies Lemma 4.4. The dual cone c ∨ w ⊆ Z E is the span of the positive orthant in Z E together with the δ γ for γ running over cycles in Γ.
Proof of lemma 4.3.
(1) =⇒ (2): We may assume T is local, since an extension is unique if it exists. Perhaps shrinking the combinatorial chart we may assume that Γ is the graph over the closed point of T .
The map T →M Γ corresponds to a map t # : Λ[N E ] → O T (T ), and each t # δ e is a unit on t −1 M. For each cycle γ we obtain an element t # δ γ ∈ Frac O T (T ), and the factorisation of t via M ♦ w says that t # δ γ ∈ O T (T ) ⊆ Frac O T (T ). If we write i(γ) for the cycle with the same edges as γ but in the reverse direction, we see that Since the product around each cycle in Γ of the t # δ wγ (e) e lies in O T (T ) × , we can choose elements r v ∈ Frac O T (T ) × for each vertex v of Γ so that for each directed edge e : u → v we have For a vertex v, write η v for the generic point of the component of the special fibre of C T corresponding to v. Now we define a Weil divisor Y on C T by specifying that Y is trivial over t −1 M, and that locally around η v it is cut out by r v . Then eq. (3) and a small computation implies that Y is actually a Cartier divisor. Now we claim that the line bundle ω ⊗k C (Σ + Y ) defines an extension of σ in t * J = Pic 0 C/T . Clearly it coincides with σ over t −1 M, so all we need to check is that Σ + Y has degree zero on every irreducible component of the central fibre of C T . Fix such a vertex v. We need to check that the degree of O C (Y ) on the component C v of the special fibre corresponding to v is exactly the sum of the weights of the non-leg half-edges out of v. After adjusting Y by the pullback of a divisor on T we may assume that r v = 1. Let e : u → v be an edge out of v. Then the completedétale local ring at the singular point corresponding to e is isomorphic to where we take x to be the coordinate vanishing on C v and y to be vanishing on C u . We may assume that r u is given by t # δ w(e) e (since we can ignore T -units), so Y is locally defined by y w(e) , and the order of vanishing on C v is exactly w(e) as required.
(2) =⇒ (1): Again, we may assume T is local. We consider first the case where T is regular. The argument above is almost reversible; we are assuming this extension of σ exists, and it is necessarily given by a line bundle L of degree 0 on every irreducible component of every fibre. Then By the regularity of T we can apply [Hol17, theorem 4.1] to see that for each directed edge e : u → v, an equation of the form holds for some a ∈ Z. Assigning to the directed edge e the integer a is easily verified to give a weighting w on Γ, and for each cycle γ we see , and we are done. It remains to reduce the general case to the case when T is regular. So assume T is local, normal and (locally) desingularisable, and let T → T be a desingularisation, so T is regular and T → T is proper, surjective, and birational. By Zariski's Main Theorem, the fibres of T → T are connected. Write t for the closed point of T .
Since we know the result in the regular case, we can apply this to T to find an open cover {V i } i∈I of T , and weightings w i on the V i , so that each V i →M Γ factors via M ♦ w i →M Γ . Adjusting the cover, we may assume that each V i is connected and meets the fibre T t of T over the closed point of T . If Γ is the graph over t, the weightings w i on Γ need not be unique, but their restrictions to the contracted graph Γ are unique. To simplify the notation we will assume Γ = Γ, since the value taken by the weightings on the contracted edges never plays any role. Now T t is connected and is covered by the V i ∩ T t , and the weightings w i must agree on overlaps of the V i ∩ T t , so we see that actually all the w i are equal. Write w for this weighting; we will show that T →M Γ factors via M ♦ w →M Γ . Note that each t # δ e is a regular element in O T (T ) since its restriction to t −1 M is invertible. We write div(t # δ e ) for the associated Cartier divisor on T .
Fix a directed loop γ in Γ. By a similar argument as in the regular case, to construct the map T → M ♦ w it is enough to show that, for each cycle γ we have e∈γ w γ (e) div(t # δ e ) = 0.
But we know that eq. (5) holds on each V i after pulling back (since we have maps Proof. Uniqueness is clear since the map is determined on M, whose pullback is dense in T . Existence then follows immediately from lemma 4.3, since M ♦ Γ is formed by glueing together the M ♦ w , and the uniqueness ensures that the maps we obtain glue on overlaps.
Corollary 4.6. M ♦ →M is the terminal object in the 2-category of σ-extending morphisms toM.
Proof. We know by theorem 3.15 that M ♦ is normal, and it is locally desingularisable by [Kat94] since it is locally toric (log regular). Applying lemma 4.3 we see that M ♦ →M is σ-extending, since M ♦ w →M Γ is clearly w-aligned. Corollary 4.5 then shows that it is terminal.

Properness of DRL ♦
From now on we will abuse notation by writing σ for the extension of the section σ over M ♦ . Write DRL ♦ for the schematic pullback of the unit section of the universal jacobian along σ, so DRL ♦ is a closed substack of M ♦ . We will show that DRL ♦ is proper overM (recalling that the map M ♦ →M is in general far from proper).
We will begin by describing a certain universal line bundle over M ♦ , which will play a crucial role in the proof. The proof itself is then mainly a matter of keeping careful track of isomorphisms and valuations. To simplify the notation, we write ω for the relative dualising sheaf of the universal curve overM.

The universal line bundle
Let p : Spec k → M ♦ be a geometric point, and write C p for the stable curve over k. The map σ : M ♦ → J determines an isomorphism class of line bundles on C p (necessarily with degree zero on every component). In this section we will give representatives of this isomorphism class.
Choose a combinatorial chartM Γ ← U →M containing p, and so that Γ is the graph of C p . Let w be the weighting on Γ such that p lies in M ♦ w . If v is a vertex of Γ, write C v for the corresponding irreducible component of C p , and define a line bundle F v on C v by the formula Here the sum runs over directed edges e out of v, and e v is the point on C v corresponding to the node e on C p . Now we need to glue the F v together along the nodes e to give a line bundle on C p . If e : u → v is an edge then we see By the deformation theory of stable curves the vector space O Cv ([e v ])| ev ⊗O Cu ([e u ])| eu is naturally a sub-space of the tangent space to p inM. The choice of combinatorial chart then yields a canonical generator of this summand of the tangent space, giving us a canonical isomorphism We can use this to explicitly describe how to glue the F v together to a line bundle on C p . The map p : Spec k → M ♦ w is a map p # : Λ[c ∨ w ] → k, and for each cycle γ we see p # δ γ ∈ k × . Choose a function λ : such that λ(i(e)) = λ(e) −1 and such that for every cycle γ we have e∈γ λ(e) w(e) = p # δ γ .
For an edge e : u → v, the −w(e)th power of the element λ(e) ∈ k × gives (via eq. (8) and eq. (7)) an isomorphism F v | ev ∼ −→ F u | eu . We use these isomorphisms to glue the F v to a line bundle on the whole of C p (cf. [Fer03]), which we denote F λ . Clearly F λ depends on the choice of λ, but a different choice of λ will yield an isomorphic F λ . Proof.
The map is clearly separated and of finite presentation since the same holds for M ♦ →M. We need to show that the dashed arrow in the following diagram can be filled in: where T is a strictly hensellian trait with generic point η and closed point p.
Choose a combinatorial chartM Γ ← U →M containing p, and such that Γ = Γ p is the graph of C p . We write Γ η for the graph over η, with edge set E η and vertex set V η , and similarly over p, so we have a contraction map Γ p → Γ η , and E η ⊆ E p . Given v ∈ V η we define C v to be the corresponding irreducible component of C η .
Let w η be a weighting on Γ η so that η lands in M ♦ wη .
Step 2: Extending the weighting to Γ p .
Applying proposition 5.1 over η we choose: • a line bundle L on C η ; • for each v ∈ V η an isomorphism • an isomorphism (the last is possible exactly because η lands in DRL ♦ ). Putting together eq. (12) and eq. (11) we obtain for each e : u → v an isomorphism and hence elements λ(e) ∈ O T (η) as in eq. (9). We writeC v for the closure of C v in the curve C T (so theC v are the irreducible components of C T ), and we writeF v for the line bundle onC v given bȳ where now we view e v as an element ofC v (T ) (cf. eq. (6)). Putting together eq. (12) and eq. (11) again we obtain trivialisations F v (−Σ) ∼ −→ O Cv , which we can think of as being trivialisations ofF v (−Σ) over η, which yield Cartier divisors Y v onC v supported on the special fibre.
If v ∈ V p is a vertex mapping to v ∈ V η , we define Y(v ) to be the multiplicity along η v of the divisor Y v ; this gives a function Y : One now checks easily that this w extends the weighting w η to a weighting w onto whole of Γ p .
where the isomorphisms a and b come from the definition of Y v . Now e ∈ E η lifts to a unique edge e ∈ E p , and we write v ∈ V p for the endpoint of e which maps to v, and similarly define u . A small computation then shows that Step 4: Constructing a lift T → M ♦ w . We want to construct a map Λ[c Step 5: Verifying the valuative criterion.
We have constructed a map T → M ♦ over t whose restriction to η is as in eq. (10). Since the inclusion DRL ♦ → M ♦ is proper it follows that T → M ♦ factors via DRL ♦ as required.
6 Proof of theorems 1.1 and 1.2 We now have all the tools to easily prove theorems 1.1 and 1.2. We begin by giving slightly more precise statements (in the introduction, we ignored the question of existence of resolutions of singularities, assuming that many readers will be mainly interested in characteristic zero).
An admissible blowup ofM is a morphism x : X →M satisfying: • x is proper and surjective; • x is birational (i.e. there exists a dense open U ⊆M such that x −1 U is dense in X and x −1 U → U is an isomorphism); • X is normal and locally desingularisable (definition 4.1).
Admissible blowups together with maps overM form a directed system. We check that M ♦ can be compactified to an admissible blowup (perhaps after blowup of M ♦ itself). Proof. We begin by giving a simple argument in characteristic zero. First construct some compactification M of M ♦ →M following [Ryd11,§6], then apply Hironaka [Hir64] to see that M is desingularisable hence M →M is an admissible blowup.
In arbitrary characteristic the proof is slightly more involved. It is enough to treat the connected components ofM separately, so we fix one such component; abusing notation, we will still write it asM, and similarly write M ♦ →M.
Choose a finite cover U = {U i →M} by combinatorial charts (each U i having graph Γ i ), then choose a finite cover V = {V j → U ×M U} of U ×M U by combinatorial charts, each V j having graph Γ j . For each U i we have a fan F i in Q E(Γ i ) from definition 3.3, and similarly for each V j a fan F j in Q E(Γ j ) .
Fix a V j . Recalling that combinatorial charts are by definition connected, composing the map V j → U×M U with one of the projections yields a map V j → U i for some i. This yields maps Γ i → Γ j , E(Γ j ) → E(Γ i ), and Q E(Γ j ) → Q E(Γ i ) , and F i necessarily pulls back to F j (we say the F i are 'compatible on overlaps').
For each i we can choose a finite refinementF i of F i which is complete in the sense that it fills the positive orthant. Hence if M i /U i is the toric variety associated toF i then the map M i → U i is proper. If we writeF i for the fan obtained by restrictingF i to the support of F i then the associated toric varietỹ M ♦ i over U i is a blowup of M ♦ U i and has a natural open immersion to M i . Since toric varieties are desingularisable we have proven the lemma 'locally onM'.
Since the transition maps Q E(Γ j ) → Q E(Γ i ) are just inclusions of coordinate subspaces, it is not hard to check that theF i can be chosen to be compatible on overlaps, whence they will glue to give a global construction.
both conditions clearly hold. More generally, the finiteness of normalisation holds