A N\'eron model of the universal jacobian

The jacobian of the universal curve over $\mathcal{M}_{g,n}$ is an abelian scheme over $\mathcal{M}_{g,n}$. Our main result is the construction of an algebraic space $\beta\colon \tilde{\mathcal{M}}_{g,n} \rightarrow \bar{\mathcal{M}}_{g,n}$ over which this jacobian admits a N\'eron model, and such that $\beta$ is universal with respect to this property. We prove certain basic properties, for example that $\beta$ is separated, locally of finite presentation, and satisfies a certain restricted form of the valuative criterion for properness. In general, $\beta$ is not quasi-compact. We relate our construction to Caporaso's balanced Picard stack $\mathcal{P}_{d,g}$.


Introduction
This paper concerns the existence or otherwise of Néron models for the jacobians of 'universal curves'.We write M g,n for the Deligne-Mumford stack of smooth proper curves of genus g with n disjoint ordered marked sections, and M g,n for its Deligne-Kundsen-Mumford compactification.The universal curve over M g,n is smooth and proper, and hence has a jacobian, an abelian scheme, which we shall denote J g,n /M g,n .One can ask whether this abelian scheme admits a Néron model over M g,n .We need a definition: Definition 1.1.Let S be an algebraic stack and U ⊆ S a scheme-theoreticallydense open substack 1 .Let A/U be an abelian scheme.A Néron model for A over S is a smooth separated group algebraic space N/S together with an isomorphism A → N × S U satisfying the following universal property: let T → S be a smooth morphism representable by algebraic spaces, and f : T U → A any U-morphism.Then there exists an S-morphism F : T → N such that F | T U = f , and moreover this morphism F is unique up to unique isomorphism. 2  The answer to our question is then 'no' in general; this follows immediately from the non-existence results in [Hol14].More precisely, a Néron model will exist if and only if M g,n has dimension 1 or g = 0.
Perhaps things can be improved by 'blowing up the boundary' of M g,n , or more generally allowing an alteration of M g,n ?Again, the answer is no, and again this follows from [Hol14].On the other hand, we deduce from [Hol14, corollary 1.3] that there is an open substack U of M g,n whose complement has codimension 2 and over which a (finite type) Néron model exists, and we also know that if T is any trait 3 and f : T → M g,n is any map sending the generic point of T to a point in M g,n , then f * J g,n admits a (finite type) Néron model.It may then seem reasonable to ask for the 'best' morphism to M g,n such that the pullback of the universal Jacobian J g,n admits a Néron model.To make this precise, we use a universal property: Theorem 1.3 (corollary 11.3, proposition 11.4).A universal Néron-model-admitting morphism β : M g,n → M g,n exists.The morphism β is separated, locally of finite presentation, and relatively representable by algebraic spaces.Moreover, β satisfies the valuative criterion for properness for morphisms from traits to M g,n which map the generic point of the trait to a point in M g,n .The Néron model of J g,n over M g,n is of finite type over M g,n , and its fibrewise-connected-component-of-identity is semiabelian.
Note that β is in general not quasi-compact (in fact, it is quasi-compact if and only if it is an isomorphism, if and only if M g,n has dimension 1 or g = 0).The statement about the valuative criterion for properness for certain traits can be thought of as saying 'suitable test curves in M g,n extend to M g,n '.One might reasonably ask whether the same result holds true if M g,n+1 /M g,n is replaced by some other family of (semi)stable curves.In general the answer seems to be no; we make essential use of the precise structure of the labelled graphs arising from M g,n+1 /M g,n .The most general situation to which our results apply is described in theorem 11.2.

Comparison to Caporaso's balanced Picard stack
In [Cap08], Caporaso constructs a 'balanced Picard stack' P d,g → M g , and shows that it acts as a 'universal Néron model' for smooth test curves f : T → M g,n such that the corresponding stable curve X/T is regular.More precisely, for any such test curve, the scheme f * P d,g is canonically isomorphic to the Néron model of the jacobian of the generic fibre of X.Note that this canonical isomorphism is not a morphism of group spaces.
Our Néron model N g,n of the universal jacobian J g,n satisfies a similar property; it follows easily from the definition that for any test curve f : T → M g,n such that the given curve X/T is regular, the map f factors (uniquely up to unique isomorphism) via a map g : T → M g,n and moreover we have that g * N g,n is canonically isomorphic as a group space to the Néron model of the jacobian of the generic fibre of X → T .
These properties may seem rather similar, but there are important differences: 1. Caporaso's P d,g is a scheme over M g , whereas our Néron model N g,n can only exist over M g,n ; 2. The scheme P d,g can never admit a group structure compatible with that on J g , whereas our Néron model N g,n is automatically a group space; The scheme P d,g has a good universal property for test curves f : T → M g such that the pullback of the universal curve to T is regular -roughly speaking, the test curve is 'sufficiently transverse to the boundary'.In contrast, our construction has a good universal property for all morphisms from test curves mapping the generic point into M g,n , and even for test object of higher dimensions.Indeed, test curves f : T → M g such that the pullback of the universal curve to T is regular factor through a quasi-finite part of M g,n -a precise description of this quasi-finite part is given in section 12.
In summary, it seems reasonable to say that our Néron model N g,n over M g,n has a stronger universal property than Caporaso's P d,g , but this comes at the price of replacing M g,n by M g,n .

Outline of the paper
In section 2 we briefly recall from [Hol14] the definition of when a semistable curve is aligned, and also introduce a slight variant, being 'regularly-aligned' (definition 2.4).Recall from [Hol14, theorem 1.2] that there is a close connection between a curve being (regularly-)aligned and its jacobian admitting a Néron model.
In sections 3 to 7 we will construct a 'universal regularly aligning morphism' for a semistable curve (definition 3.1).The construction proceeds via an intermediate object, the 'universal aligning morphism' (definition 7.1) for a 'correlation collection' (definition 6.1).The motivation for this intermediate object is that, in order to construct a universal aligning morphism for a semistable curve, one does not in fact need all the data of the curve.Indeed, it simplifies things not to carry all of it around.Instead we define the 'correlation collection' for a curve (definition 6.1.7),which contains just enough data to allow us to define and construct a 'universal aligning morphism for a correlation collection' which is analogous to the universal regularly aligning morphism for a curve.Indeed, under suitable hypotheses the universal aligning morphism for a correlation collection for a curve C will in fact be equal to the universal regularly aligning morphism for C itself.This machinery will enable us to prove theorem 7.9, the existence of universal regularly aligning morphisms.To facilitate this, we construct in section 5 'specialisation maps' between labelled graphs on suitable semistable curves.
In section 8 we show that, under suitable hypotheses, the universal aligning morphism for a correlation collection is a morphism from a regular scheme.From this we deduce in section 9 an analogous statement for semistable curves (this is mainly a matter of setting up the definitions).Then in section 10 we show that, under rather restrictive hypotheses on C/S, the pullback of C along the universal regularly aligning morphism admits a regular aligned semistable model; we do this by explicitly resolving singularities, in a fashion analogous to [dJ96, §5].In section 11 we collect together all our earlier results to prove the main theorem; this consists mainly of descent arguments and checking that M g,n+1 /M g,n satisfies our hypotheses.In section 12, we relate our results to some constructions of Caporaso from [Cap08].
In this paper, 'algebraic stack' means 'algebraic stack in the sense of [Sta13, Tag 026O]'.

Acknowledgements
The author would like to thank Owen Biesel, José Burgos Gil, Bas Edixhoven, Robin de Jong and Morihiko Saito for helpful comments and discussions.

Definition and basic properties of aligned curves
In this section we briefly recall the definitions from [Hol14], where more precise statements can be found.For us 'monoid' means 'commutative monoid with zero', and a graph has finitely many edges and vertices, and is allowed loops and multiple edges between vertices.We are interested in graphs whose edges are labelled by elements of a monoid.Definition 2.1.Let L be a monoid, and (H, ℓ) a 2-vertex-connected4 graph labelled by L (so ℓ is a map from the set of edges of H to L).We say (H, ℓ) is aligned if for all pairs of edges e, e ′ there exist positive integers n, n ′ such that ℓ(e) n = ℓ(e ′ ) n ′ .
We say (H, ℓ) is regularly aligned if there exists l ∈ L and positive integers n(e) for each edge e such that for all edges e we have ℓ(e) = l n(e) .
Let L be a monoid, and (G, ℓ) a graph labelled by L. We say G is (regularly) aligned if every 2-vertex connected subgraph of G is (regularly) aligned.
1.If G is regularly aligned then it is aligned.
2. If L is a free monoid then the converse also holds.Now we turn to semistable curves (by which we mean proper, flat curves whose fibres have at worst ordinary double point singularities).We recall from [Hol14, propositions 2.5 and 2.9] a result on the local structure of such curves: 1.Let f : T → S be a morphism of algebraic stacks.We say that f is nondegenerate if U × S T is scheme-theoretically-dense in T .
2. Let f : T → S be a non-degenerate morphism of locally-noetherian stacks.We say f is (regularly) aligning if C × S T → T is (regularly) aligned.
3. The category of (regularly) aligning morphisms is defined as the full sub-2category of stacks over S whose objects are (regularly) aligning morphisms.
4. A universal (regularly) aligning morphism is a terminal object in the 2category of (regularly) aligning morphisms over S.
1.A large part of the paper is taken up with proving the existence (and certain properties) of a universal regularly aligning morphism.
2. The existence of a universal aligning morphism can be shown in a similar way, but it seems to be a less interesting object (in general it is not regular, so we cannot construct a Néron model over it), so we will not give details.
3. We will show that the universal regularly aligning morphism is in fact a separated algebraic space locally of finite type over S.
6 Sketch proof for fppf-descent of regular alignment: Let R → R ′ be a faithfully flat ring map, and let r 1 , r 2 ∈ R and a, u 1 , u 2 ∈ R ′ with u i units and r i = u i a ni for some ) becomes an isomorphism after base-change to R ′ , so by faithful flatness it was an isomorphism.

Quasisplit curves
In order to construct the universal regularly aligning morphism by descent, we will make use of the notion of a 'quasisplit' semistable curve.This differs from (though is somewhat similar to) the notion of a split semistable curve in [dJ96]; neither is stronger than the other.Definition 4.1.Let S be a scheme, and let C/S be a semistable curve.As usual, we write Sing(C/S) for the closed subscheme of C where C → S is not smooth (more precisely, it is the closed subscheme cut out by the first fitting ideal of the sheaf of relative 1-forms of C/S).We say C/S is quasisplit if the following two conditions hold: 4.1.1 the morphism Sing(C/S) → S is an immersion Zariski-locally on the source (for example, a disjoint union of closed immersions); Summarising, given a quasisplit curve C/S and a point s ∈ S, it makes sense to talk about the labelled dual graph of the fibre of C over s, and the labels live in the Zariski local ring at s.For the remainder of this paper, we will do this without further comment.
Lemma 4.3.Let S be a locally noetherian scheme and C/S a semistable curve.Then there exists an étale cover f : Proof.First we construct an étale cover over which the non-smooth locus is a disjoint union of closed immersions.Note first that this holds after base-change to the étale local ring at any geometric point, by the structure of finite modules over strictly henselian local rings [Sta13, Tag 04GL].Since C/S is of finite presentation, one can show by a standard argument that the same statement holds on some étale cover.
The non-smooth locus being a union of closed immersions is a property stable under base-change, so we now construct a further étale cover over which the irreducible components of fibres become geometrically irreducible.This is easy: if C → S admits a section through the smooth locus of a given irreducible component of a field-valued fibre, then that component is geometrically irreducible.Moreover, the smooth locus of C/S is dense in every fibre.As such, it is enough to show that, étale-locally, there is a section through the smooth locus of every component of every field-valued fibre.This holds since a smooth morphism admits étale-local sections through every point.

Specialisation maps between labelled graphs for quasisplit curves
Given a quasisplit semistable curve C/S, and two points s, ζ of S with s in the closure of ζ, we will show that there is a well-defined 'specialisation map' Γ s → Γ ζ on the labelled graphs.First, we need to define a morphism of labelled graphs.
Definition 5.1.A morphism of graphs sends vertices to vertices, and sends edges to either edges or vertices (thinking of the latter as 'contracting an edge'), such that the obvious compatibility conditions hold.
A morphism of edge-labelled graphs is a pair consisting of a morphism of graphs Γ → Γ ′ and a morphism of monoids from L to L ′ such that the labellings on non-contracted edges match up.An isomorphism is a morphism with a two-sided inverse.
Definition 5.2.Let S be a locally noetherian scheme, and let C/S be a quasisplit semistable curve.Let s, ζ ∈ S be two points such that ζ specialises to s (i.e.s ∈ {ζ}).Write Γ s and Γ ζ for the corresponding labelled graphs -recall from remark 4.2 that this makes sense without choosing separable closures of the residue fields, and moreover that the labels of Γ s and Γ ζ may be taken to lie in the Zariski local rings at s and ζ respectively.Write for the canonical (injective) map.We will define a map of labelled graphs writing ϕ V for the map on vertices and ϕ E for the map on edges.First we define the map on vertices.
One checks easily that each irreducible component of C s arrises in this way from exactly one vertex of Γ ζ , so we obtain a well-defined map ϕ V : Next we define the map on edges.Write Z = {ζ} ⊆ S for the closure of ζ.Let e ∈ E s be an edge of Γ s .Then there are exactly two possibilities 7 : Case 1) there exists an open neighbourhood s ∈ Z 0 ⊆ Z and a unique section ẽ : Case 2) case 1 does not hold and (writing v 1 , v 2 for the endpoints of e) we have that Then map e to ϕ V (v 1 ).
In fact, case 1 holds if and only if the label l(e) is a unit at ζ.
A more intuitive description of the specialisation morphism ϕ : Γ s → Γ ζ on labelled graphs may be given as follows: starting with Γ s , first replace each label by its image under sp.Then contract every edge whose label is a unit.This is exactly the labelled graph Γ ζ .Given that such a simple description is available, why did we give the long-winded definition above?Essentially this is because it is otherwise not a-priori clear that the labelled graph resulting from this simple description is (naturally) isomorphic to the labelled graph Γ ζ .

Correlation collections
When constructing universal regularly aligning morphisms for semistable curves, it turns out not to be necessary to carry around all the data of the curve C/S.In this section, we will associate a 'correlation collection' to a semistable curve (after suitable étale base-change), which will carry just enough data to allow us to construct a universal regularly aligning morphism.This is technically a matter notation -we could in principle work with curves throughout, without defining correlation collections etc. at all.However, in practice this would become very unwieldy, and moreover it is perhaps easier to see exactly what properties of C/S are required for a 'nice' universal regularly aligning morphism by abstracting them in this way.Definition 6.1. 7To see this, note first that a section in C T /T as in case 1 is unique if it exists.Suppose we have ϕ In this situation observe that e ∈ T 1 ∩ T 2 , and (by considering the local structure of the singularities of a quasisplit semistable curve) we find that T 1 ∩ T 2 is locally on Z a union of sections, so case 1 must hold.
6.1.1Let S be a scheme.A correlation set on S is a finite non-empty subset K ⊆ O S (S).The non-degeneracy locus U K of K is defined to be the largest open subscheme of S on which all elements of K are units (elements of 6.1.2Let S be a scheme.A correlation collection on S is a finite multi-set8 K = {K 1 , • • • , K n } whose elements are correlation sets.The non-degeneracy locus U K of K is defined to be the intersection of all the non-degeneracy loci U K i .
6.1.3Given a morphism of schemes f : T → S and a correlation set K on S, we define the pullback f * K by pulling back all its elements.Given a correlation collection K on S, we define the pullback f * K by pulling back each element of the collection.
6.1.4We say a correlation set K on a scheme S is strictly aligned if there exists an element σ ∈ O S (S) such that for every section k ∈ K there exists an integer n ≥ 0 such that k and σ n differ by multiplication by an element of O S (S) × .
6.1.5We say a correlation set K on a scheme S is aligned if there exists a Zariski (equivalently, fppf) cover {f i : S i → S} such that each f * i K is strictly aligned.We say a correlation collection K on a scheme S is aligned if every K ∈ K is aligned.
6.1.6Let (S, s) be a local scheme, and C/S a quasisplit semistable curve.Write Γ s for the labelled dual graph of C at s, and write Γ {2} s for the set of maximal 2-vertex-connected subgraphs of Γ s .Let K be a correlation collection on S. We say K is a correlation collection for C/S if there exists an injective map such that (a) for every H ∈ Γ {2} s , we have that the set of labels ℓ(H) appearing on H is exactly equal to the set of ideals is a correlation collection for C/S locally at s.It remains to show that, after shrinking S further, this correlation collection K is in fact a correlation collection for C/S (this fact is clear at the point s, but we need it to hold locally at every point in S).
Let Σ denote the smallest collection of (reduced) closed subsets of S which is closed under: • pairwise intersections; • taking irreducible components; and which contains the image in S of Sing(C/S).

Now set
the union of all elements of Σ which don't contain s.Note this is a closed subset since Σ is finite.Replace S by the complement of Z in S. Now let t ∈ S be any point.We will show that K is a correlation collection for C/S at t. Let σ ∈ Σ be the smallest element of Σ which contains t (note that σ contains s by construction), and let η be the generic point of σ.We have two 'specialisation' maps on labelled graphs: Both maps are surjective on the underlying graphs by construction.The key point is that the map sp t is in fact an isomorphism on the underlying graphs.This is easy to see: the map sp t simply contracts every edge of Γ t whose label is a unit at η.Well, any label l on an edge of Γ t which becomes a unit at η will cut out a proper closed subscheme of σ containing t, but this is impossible by the definition of σ.
Now let H be any maximal 2-vertex-connected subgraph of Γ t .Let H ′ be the corresponding maximal 2-vertex-connected subgraph of Γ η .Now since the map sp s simply contracts some edges, we see that there exists a 2-vertex-connected subgraph H ′′ of Γ s such that H ′ is contained in its image.Then the corresponding element K(H ′′ ) of K contains generators for each of the labels of H. Conversely, given an element K = K(H) of K, either all elements of K(H) are units at t (in which case the corresponding component H of Γ s is contracted by sp s ), or the image of H in Γ t is a 2-vertex-connected component whose labels are the pullback of K(H) to O S,t .Remark 6.3.Let C/S be a semistable curve over a locally noetherian base.Given a geometric point s of S, it is clear that whether C/S is aligned at s depends only on the image of s in S. As such, given a point s ∈ S, it makes sense to ask whether C is aligned at s.We write S ali for the subset of S at which C/S is aligned.
Suppose now that C/S admits a correlation collection.Then it is clear that S ali is a constructible subset of S.Moreover, the specialisation morphisms defined above make it clear that the complement S \ S ali is closed under specialisation.From this we see that S ali is an open subset of S. By étale descent, the same conclusion holds without the assumption that C/S admit a correlation collection.Proposition 6.4.Let C/S be semistable with correlation collection K. Then 1. for any morphism f : T → S, we have that f * K is a correlation collection for f * C/T ; 2. the curve C/S is regularly aligned if and only if K is aligned.
Proof. 1) Being quasisplit is preserved under pullback.Let t ∈ T be any point, and set s = f (t).Then Γ t has the same underlying graph as Γ s , and the labels on Γ t are simply the pull-backs of the corresponding edges of Γ s .The result is clear.
2) Clear from the definitions.

Universal aligning morphisms for correlation collections
For almost the entirety of this section, we will forget about semistable curves, and focus on constructing universal aligning morphisms for correlation collections.At the end of the section, we will use our results to draw conclusions about semistable curves.
Definition 7.1.Let S be a scheme, and let K be a correlation collection on S with non-degeneracy locus U K .
2. A K-non-degenerate morphism f : T → S is an aligning morphism for K if the pullback f * K is aligned.
3. A morphism β : S → S is a universal aligning morphism for K if every aligning morphism for K factors uniquely via β.
A universal aligning morphism is unique up to unique isomorphism if it exists.If K is aligned and the non-degeneracy locus U K is dense in S then id : S → S is a universal aligning morphism for K. Now we will prove existence of universal aligning morphisms, and later some properties.3. Let S be a scheme with a correlation set K, let f : T → S be a non-degenerate morphism, and let t ∈ O T (T ).We say that f is M-aligning by t if for aligning by t for some t.Note that the property of being M-aligning is fppflocal on the target.If f is M-aligning then an aligning element for f is an element t ∈ O T (T ) such that f is M-aligning by t.Since K is by assumption non-empty, an aligning element is unique up to multiplication by units in O T (T ) × ; the aligning ideal is defined as (t), and is unique.
4. Given a correlation collection K for a scheme S, we write , let f : T → S be a non-degenerate morphism, and let t = (t 1 , • • • , t r ) ∈ O T (T ) r .We say f is M-aligning by t if f is M i -aligning by t i for every 1 ≤ i ≤ r.We say f is M-aligning if it is M-aligning by t for some t ∈ O T (T ) r .If f is M-aligning then an aligning sequence for f is an element t ∈ O T (T ) r such that f is M-aligning by t, and the aligning ideal sequence is the sequence ((t 1 ), • • • , (t r )).
6.We define universal M-aligning schemes for elements M of M(K) and M(K) in analogous ways to those in definition 7.1.
Definition 7.3.Let S = Spec R be an affine scheme, and . This depends on the n i (so the notation is not good).However, this dependance will turn out not to matter (see proposition 7.5).To simplify the notation, we will assume that a choice of n i has been made once-and-for-all for every collection of m i .
We have a natural map to the coordinate ring of the non-degeneracy locus: (1) We call the element a ∈ R ′ M the special aligning element of R ′ M .Definition 7.4.
By the universal property of the tensor product we get a canonical map ), and we define R M to be the image of R ′ M under the above map.Set S M = Spec R M (in other words, S M is the closure of the image of U K in Spec R ′ M under the given map).Write a i for the image in R M of the special aligning element in R ′ M i , then set a := ((a 1 ), • • • , (a r )).
Proposition 7.5.Given M ∈ M(K) as above, the natural map S M → S is a universal M-aligning morphism.The sequence a is the aligning ideal sequence.
Proof.It is clear that S M → S is M-aligning and that a is the aligning ideal sequence.Let f : T → S be any M-aligning morphism.The uniqueness of a factorisation of f via S M is clear as f is non-degenerate and S M → S is affine and hence separated.As such, it remains to construct a factorisation of f via T → S M .To simplify the notation we will treat only the case where K has exactly 1 element (write K = {K}); the general case follows by combining the argument below with the universal property of the fibre product. Write By definition of being M-aligning, we have an element t ∈ O T (T ) such that t M (x j ) ∼ T ′ f * x j .Then we define an R-algebra map (2) This yields a map T → Spec R ′ M , which in fact factors uniquely via S M as f is non-degenerate.Proof.The map β : S → S is quasi-separated because S is locally noetherian, so it is enough to check the valuative criterion for separatedness.Let V denote the spectrum of a valuation ring, with generic point η and closed point v. Let f, g : V → S be morphisms which agree on η and such that the composites with the canonical map S → S agree.We will show that f = g.
If the images of f and g are contained in the same S M for some M ∈ M(K) then the result is clear since S M → S is affine and hence separated.Thus we may as well assume that the image of f is in S M and the image of g in S N for some M = N.Our aim now is to show that the image of f is in S M,N and the image of g is in S N,M , so that f = g by separatedness of S M,N = S N,M over S.
Since the composites of f and g with the projections to S agree, we find that f * x j = g * x j for every j.If we can show that g * x j is a unit on V for every j then we have that g factors via S M,N (by applying the same argument to every K i ∈ K with M i = N i ), and the same holds for f by symmetry, so we are done.
Since we have gcd j (m j ) = 1, gcd j (n j ) = 1 and for some j we have m j = n j , we can find c 1 , • • • , c r ∈ Z such that j c j m j = 0 and d := j c j n j = 0.
We know that f * x j ∼ a m j and g * x j ∼ b n j for some elements a, b ∈ O V (V ).Then we find and since the left hand sides are equal this implies b d ∼ 1, so (as d = 0) we get that b is a unit on V , and so all the g * x j are units on V as desired.
Proposition 7.8.The map β : S → S is a universal aligning scheme for K.It is separated and locally of finite presentation.
Proof.Separatedness is proposition 7.7, and local-finite-presentation is clear as the same holds for the S M by construction, and local-finite-presentation descends.Let f : T → S be any aligning morphism.We need to show that f factors uniquely via β.Uniqueness is clear from non-degeneracy of T and separatedness of S and of S → S; we must check existence of a factorisation.Let T ′ → T be an fppf cover as in definition 6.1.5.We easily get a morphism ϕ : T ′ → S. It remains to descend this to a morphism T → S. We will in fact descend the morphism id ×ϕ : T ′ → T ′ × S S to a morphism T → T × S S, along the fppf cover T ′ → T .We immediately obtain covering data for T ′ and T ′ × S S over T ; in order for id ×ϕ to descend, we just need that id ×ϕ is compatible with the covering data.This reduces to checking that a certain square of T ′ × T T ′ -morphisms with T ′ × T (T ′ × S S) in the bottom right corner commutes.However, I claim the only T ′ × T T ′ -morphisms from T ′ × T (T ′ × S S) to itself are the identity.Indeed, such a map must be the identity over U since SU = U.Moreover, U is schemetheoretically dense in S, so T ′ × T T ′ × S U is scheme-theoretically dense in T ′ × T T ′ .As such, a T ′ × T T ′ morphism from T ′ × T (T ′ × S S) to itself is the identity over U and is determined entirely by what it does over U (by separatedness), so must in fact be the identity.
Theorem 7.9.Let C/S a generically-smooth semistable curve over a separated locally noetherian algebraic stack.Then a universal regularly aligning morphism for C/S exists, and is a separated algebraic space locally of finite presentation over S.
Proof.The stack S admits a smooth cover by a scheme, and a universal aligning morphism will descend along a smooth cover (as will the property of being an algebraic space) since it is defined by a universal property.As such, it is enough to consider the case where S is a scheme.
After étale cover S ′ → S, the curve C ′ = C × S S ′ /S ′ has a correlation sequence K by proposition 6.2.By proposition 7.8, this has a universal aligning morphism β ′ : S′ → S ′ .I claim that this is a universal regularly aligning scheme for C ′ /S ′ .By proposition 6.4 we see that β ′ is regularly aligning.Let f : T → S ′ be a stack over S such that f is regularly aligning.Let T ′ → T be a smooth cover by a scheme, and f ′ : T ′ → S ′ the evident composite map.Then f ′ is regularly aligning (since being regularly aligning is fppf-local on target), so applying proposition 6.4 again we see that f ′ is aligning for K, and hence factors via β ′ .Using that morphisms to separated schemes are determined by what they do on scheme-theoreticallydense opens, this descend to give a factorisation of f via β ′ .Uniqueness of the factorisation holds by a similar argument.
Finally, we want to descend β ′ to a universal aligning morphism for C/S.Note that S′ is a scheme over S ′ , in particular an algebraic space.As usual the universal-property construction and separatedness gives us a descent datum, and descent data for algebraic spaces are always effective.Separatedness and local-finite-presentation follow from the same properties for the universal aligning scheme for correlation collections (proposition 7.8 again).

Regularity and normal crossings for correlation collections
Definition 8.1.Let (S, s) be a local scheme, and K a correlation collection on S. Let Σ = K∈K K. We say K has normal crossings at s if the following conditions hold: 1. the sets K are all disjoint; 2. for all subsets J = {j 1 , • • • , j n } ⊆ Σ, the scheme is regular and has dimension at most (dim S − #J) (in particular, S itself is regular).
Given a scheme S and K a correlation collection on S, we say K has normal crossings if for all s ∈ S, the pullback of K to Spec O S,s has normal crossings at s.Note that having normal crossings is smooth-local on the target.Lemma 8.2.Let S = Spec A be an affine scheme, and K be a correlation set such that K := {K} has normal crossings.Let M ∈ M(K).Then 1. the ring A ′ M (as in definition 7.3) is regular; 2. the quotient of A ′ M by the special aligning element is regular.
(strictly speaking, we chose the n i in definition 7.3.The choice made does not matter).Then recall from definition 7.3 that .
At this point it is clear that the quotient by the special uniformiser a is regular; it is even smooth over A/(x i : m i > 0), which is regular by our normal-crossings assumptions.We still need to show A ′ M itself is regular; this will take more care, since it is not in general smooth over its image -it resembles an affine patch of a blowup.
Let p ∈ Spec A ′ M be any point, and write q for the image of p in Spec A. Localising A at q, we may assume that A is local, with closed point q.Re-ordering the x i , we may assume that x 1 , • • • , x e ∈ q and x e+1 , • • • , x d / ∈ q for some 0 ≤ e ≤ d.Writing D = dim A, our normal crossing assumptions imply that there exist Now if e = 0 then the result is clear since A ′ M is smooth over A (by the differential criterion).Hence we may assume e ≥ 1.It then follows that m i = 0 for every e < i ≤ d, otherwise A ′ M /qA ′ M is empty, contradicting the existence of p. Again reordering, we may assume that 1 ≤ m 1 ≤ m 2 ≤ • • • ≤ m e .We find that .
We then see that is regular and of dimension e − 1.From this we deduce that there exist elements so p can be generated by D elements.Now it is clear that every irreducible component of A ′ M has dimension at least D (count generators and relations), and hence it follows that A ′ M is regular at p and has pure dimension D.
Lemma 8.3.Let S = Spec R be an affine scheme, and let for the pullback of K to Spec R t , and set K t,0 = {a 1 , • • • , a t }, where a i is the special uniformiser in R ′ M (cf.definition 7.3).Then the multiset K t := {K t,0 , K ′ t+1 , • • • , K ′ r } has normal crossings.Proof.By induction on t; we may assume the result for every t ′ < t (and the result for t = 0 is obvious).It follows that K t−1 has normal crossings.Let J ⊆ K∈Kt K be any subset.We split into two cases, depending on whether or not a t ∈ J.
Case 1: a t / ∈ J. Then there exists a subset which is regular by our induction assumption.Then is exactly the ring (R J 0 ) ′ Mt , which is regular by applying the first part of lemma 8.2 to A = R J 0 Case 2: a t ∈ J. Then write J = {a t } ∪ J, and let J 0 ⊆ K t−1 be a set which pulls back to J .Again set is exactly the ring (R J 0 ) ′ Mt /(a t ), which is regular by applying the second part of lemma 8.2 to A = R J 0 Theorem 8.4.Let S be a scheme and K a correlation collection with normal crossings.Let M = (M 1 , • • • , M r ) ∈ M(K).Write S M for the universal Maligning scheme, and A = (A 1 , • • • , A r ) for the aligning ideal sequence on S M .Then A defines a normal crossings divisor in S M .
Proof.The corresponding statement for R ′ M is given exactly by lemma 8.3, so it remains to deduce from this the statement for R M .Note that the map S M → S becomes an isomorphism after base-change over S to U K , which implies that the closure of the image of M is regular, this closure must in fact be a connected component, and must itself be regular.This closure is S M by definition.
Given J ⊆ {A 1 , • • • , A r }, the subscheme V (j : j ∈ J) ⊆ S m is a union of connected components of a subscheme of Spec R ′ M cut out by a collection of elements of K r,0 (in the notation of lemma 8.3); in particular, it is again regular since K r,0 defines a normal crossings divisor (by the same lemma).9 Regularity and normal crossings for semistable curves Definition 9.1.Let C/S be a semistable curve over a scheme, and let s ∈ S be a geometric point.Write G 1 , • • • , G n for the maximal 2-vertex-connected subgraphs of Γ s, and write L i for the set of labels appearing on edges in We say C/S has normal crossing singularities at s if all of the following hold: 1. the sets L i are pairwise disjoint; 2. for every subset 10 Resolving singularities over the universal aligning scheme Lemma 10.2.Let S = Spec R be a regular affine scheme, and let C/S be a regular semistable curve with normal crossing singularities, and a correlation collection Then the pull-back of C U to S M := Spec R M has a semistable regular aligned model.
Proof.Let f : S M → S be the structure map.Note that S M is regular by theorem 9.5.Write C 0 = C × S S M , by assumption this is an aligned semistable curve.We will resolve the singularities of C 0 by blowing up, taking care to preserve alignment as we do so.
For i = 0, 1, • • • , we define Z i ⊆ C i to be the reduced closed subscheme where C i is not regular, and then define C i+1 to be the blowup of C i at Z i .It is enough to show: 1.1 some C N is regular; 1.2 each C i is aligned.
Write P for the (finite) set of generic points of the union of the V (f * x) as x runs over elements of elements of K.Note that each p ∈ P is a codimension 1 point on the regular scheme S M ; write ord p for the corresponding (Z∪{∞})-valued discrete valuation on the local ring at p. Given i ≥ 0, define a non-negative integer lying over p (ord p ℓ(q) − 1); here ℓ(q) denotes the label of the edge of the dual graph Γ p corresponding to q; it lives in the étale local ring at p.It is enough to show: We first show item 2.1.Let i be such that δ i > 0. Let p ∈ P , and let q ∈ C i be a non-smooth point lying over p, with ord p ℓ(q) = t.Let p be a generator for p.The completed local ring on C i at q is given by Assume t ≥ 2 (this holds for some p and q, otherwise δ i = 0).The blowup of O C i ,q at q = (u, v, p) has three affine patches: 2.1.1 'u = 1', given by: ) which is regular (a calculation, or cf.[dJ96, 5.4]); 2.1.2'v = 1', which is also regular by symmetry; .
This patch is empty and hence regular if t = 2.If t > 2 then the patch is regular except at q ′ := (u 1 , v 1 , p).In the latter case we see that ord p q ′ = t−2; it has dropped by 2.
As such, we see that at most one non-regular point q ′ of C i+1 maps to q, and if q ′ exists we have ord p q ′ = ord p q − 2. This shows that δ i+1 < δ i .Next we show item 2.2.Let i be such that δ i = 0. Let c ∈ C i lie over s ∈ S M .We will show C i is regular at c.If C i → S M is smooth at c we are done, so assume this is not the case.Then the completed local ring of C i at c is given by where ℓ c ∈ O S,s is an element which (by definition) generates the label of the graph Γ s at the edge e c corresponding to the point c.
Let p ∈ P be such that ℓ c ∈ p (such p exists by construction).The specialisation map Γ s → Γ p does not contract e c ; rather it sends it to an edge labelled by the ideal generated by sp(ℓ c ), where sp : is the specialisation map.By our assumption that δ i = 0, it follows that (sp ℓ c ) = p.By theorem 9.5, we know that C 0 is strongly aligned, and hence the subscheme V (p) is in fact a regular subscheme of S M .Thus we see that the closed subscheme V (ℓ c ) ⊆ Spec O S M ,s is regular.Write dim s S M = d.By the above regularity statement, we can find elements Hence the ideal corresponding to c can be generated by in other words it can be generated by d + 1 elements.Since dim c C M = d + 1, this proves that C M is regular at c. Finally, we show item 2.3.We proceed by induction on i.For i = 0 the result holds by definition of the universal aligning scheme.Let i ≥ 1, and assume the result for i − 1.Let s ∈ S M be any point, and for each j let Γ j s be the graph of C j over s.Then the labelled graph Γ i s can be constructed from the labelled graph Γ i−1 s by the following recipe: 1. for each edge e with n(e) = 2, replace e by two edges both with label a; 2. for each edge e with n(e) > 2, replace e by three edges with labels a, a n(e)−2 and a in that order; 3. delete any edge labelled by a unit.
In pictures: It is clear that applying this procedure to a regularly aligned graph will yield a regularly aligned graph.
11 Existence of Néron models over universal aligning schemes We define the category of Néron-model-admitting morphisms as the full sub-2-category of the 2-category of stacks over S whose objects are Néron-modeladmitting morphisms.
A universal Néron-model-admitting morphism for C/S is a terminal object in the 2-category9 of Néron-model-admitting morphisms; it is unique up to isomorphism unique up to unique 2-isomorphism if it exists.We write β : S → S for a universal Néron-model-admitting morphism, and N / S for the Néron model of J.
Theorem 11.2.Let C/S be a regular semistable curve over a separated algebraic stack such that C has normal-crossing singularities (in particular, C is smooth over some dense open substack U ⊆ S).Let β : S → S denote the universal regular aligning morphism (note β is an isomorphism over U).Then S is a universal Néron-model-admitting morphism for C/S, and moreover the Néron model N is of finite type over S, and its fibrewise-connected-component-of-identity is semiabelian.
Note that the universal aligning morphism exists as an algebraic space over S by theorem 7.9.
Proof.The fact that C is regular and has normal-crossing singularities descends along smooth surjective base-change over S. The universal regularly aligning morphism descends along fppf morphisms, and regularity of it descends along smooth surjective base-change over S.Moreover, Néron models descend along smooth surjective separated morphisms over separated bases by [Hol14, lemma 6.1].Since every algebraic stack has by definition a smooth cover by a scheme, it suffices to prove the result in the case where S is an affine scheme.
Next, we will show that S → S is a Néron-model-admitting morphism.By theorem 9.5 we know that S is regular.Combining the same descent arguments as above with proposition 6.2, we may assume C/S has a correlation collection K. Then S has a Zariski cover by opens S M as M runs over M(K).Again using that Néron models descend along smooth (and hence Zariski) covers, it is enough to construct a Néron model for the pullback of J to S M for each M. Fix an M ∈ M(K).By lemma 10.2 we know that the pullback of C U to S M admits a semistable regular aligned model.Then by [Hol14, theorem 1.2] we know that the pullback of J to S M admits a finite-type Néron model as desired.
Finally, we need to show that any other Néron-model-admitting morphism factors via S. Let f : T → S be a Néron-model-admitting morphism.Since T is reduced and S separated, a factorisation of f via β will descend along an fppf-cover of T ; as such, (and using that a scheme smooth over a regular base is regular) we may assume T is a scheme.Then by [Hol14] and the fact that f * C U has a Néron model over T , we know that f * C/T is aligned.Since T is regular, this implies that f * C/T is regularly-aligned, and hence it factors uniquely via the universal regularly aligning morphism as required.
From now on, we work relative to a fixed base scheme Λ which we assume to be regular -the basic examples to keep in mind are Spec Z and the spectrum of a field.Let g, n be non-negative integers such that 2g − 2 + n > 0. We write M g,n for the modui stack of smooth proper connected n-pointed curves of genus g over Λ, and M g,n for its Deligne-Mumford-Knudsen compactification.By [Knu83, theorem 2.7] we know that M g,n is a smooth proper Deligne-Mumford stack over Λ, and the boundary M g,n \M g,n is a divisor with normal crossings relative to Λ in the sense of [DM69, definition 1.8].We write J g,n for the jacobian of the universal curve M g,n+1 × Mg,n M g,n over M g,n ; this is an abelian scheme over M g,n .
Corollary 11.3.A universal Néron-model-admitting morphism for M g,n exists and is a separated locally-finity-type algebraic space over M g,n .The Néron model over it is of finite type, and its fibrewise-connected-component-of-identity is semiabelian.
Proof.Since M g,n and M g,n+1 are smooth over Λ, they are in particular regular.By theorem 11.2, it is enough to show that M g,n+1 /M g,n has normal crossing singularities (definition 9.1).Let s be a geometric point of M g,n , and let S be an étale scheme-neighbourhood of s, and write Γ s for the labelled dual graph of the fibre over s of the semistable curve C := M g,n+1 × Mg,n S. By the discussion after the proof of [DM69, proposition 1.5], all the labels of Γ s are distinct, and they form a relative normal crossing divisor over Λ.Since Λ is regular, these two properties together imply that C/S has normal crossing singularities at s. Applying this argument to each geometric point of M g,n , and using that 'having normal-crossing-singularities' is smooth-local on the target, we are done.'Finite type' descends from the same property in theorem 11.2.Proposition 11.4.These objects also satisfy the following properties: 11.4.1 the morphism β : M g,n → M g,n is separated and locally of finite presentation; 2. the corresponding stable curve X/T is regular.
Remark 12.2.Let e ≥ 0 be an integer.A combinatorial definition of M ≤e g,n can be given.Let S be a scheme.Given a correlation set K on S, write M ≤e (K) for the subset of maps M : K → Z ≥0 in M(K) whose image is contained in {0, • • • , e − 1, e} ⊆ Z ≥0 .Given a correlation collection K = {K 1 , • • • , K n } on S, let Now define S≤e to be the open subscheme of S covered by the S M as M runs over M ≤e (K).Note that S≤e → S is flat if and only if it an open immersion if and only if e = 0 or K is aligned.
Now the stack M g,n is constructed, after a sequence of base-change and descent operations, from a scheme S coming from a correlation collection.If the same procedure is applied to S≤e in place of S, then the resulting algebraic space over M g,n will be exactly M ≤e g,n .The next proposition shows that the restriction of the Néron model of the universal jacobian to the open substack M ≤1 g,n is given by the pullback of Caporaso's construction.
Proposition 12.3.Let g ≥ 3, n ≥ 1 and d be integers such that gcd(d − g + 1, 2g − 2) = 1.Write P for the pullback of P d,g to M g,n via the forgetful map, and view P as a model of the universal jacobian J g,n over M g,n via the first of the n marked sections.Write P for the pullback of P to M ≤1 g,n , and write N g,n for the Néron model of J g,n over M ≤1 g,n .Then the canonical map h : P → N g,n given by the Néron mapping property is an isomorphism.
Proof.It is enough to check the map h is an isomorphism on every geometric fibre over M ≤1 g,n .Let p be a geometric point of M ≤1 g,n .Then there exists a trait T with geometric closed point t, and a morphism g : T → M g,n such that 1. the given stable curve X → T is regular; 2. the map g factors via a map g : T → M ≤1 g,n ; 3. this factorisation g maps t to p. Now since X is regular, we find that g * N g,n is the Néron model of the jacobian of the generic fibre of X → T , and the same holds for g * P = g * P. In particular, this shows that the fibres of N g,n and of P over p are isomorphic.Moreover, the given map between them is an isomorphism; this is true because it is so over the generic point of T (apply the uniqueness part to the Néron mapping property).
In particular, this shows that, after pullback along the morphism M ≤1 g → M g , the stack P d,g admits a group structure extending that over M g,n .
a scheme S and elements a, b ∈ O S (S), we write a ∼ b (or a ∼ S b) if there exists a unit u ∈ O S (S) × such that au = b.2.Given a scheme S and a correlation setK = {k 1 , • • • , k d }, we write M(K)for the set of maps of setsM : K → Z ≥0with the property that the image of M has no common factor (i.e.gcd(M(k 1 ), • • • , M(k d )) = 1).
the set of elements at which M and N take different values.DefineS M,N = Spec R M [x −1 : x ∈ δ M,N ],an open subscheme of S M .Define S N,M ⊆ S N similarly.Note that S M,N is Naligning, so we obtain a map S M,N → S N , and this map clearly factors via S N,M .The symmetry of the situation then yields a canonical isomorphism S M,N ∼ → S N,M .Define β : S → S to be the result of glueing together all the S M : M ∈ M along the open subschemes S M,N .Proposition 7.7.The map β : S → S is separated.

Definition 10. 1 .
Let S be a scheme, U ⊆ S a scheme-theoretically-dense open, and C/U a smooth proper curve.A model for C/U is a proper flat morphism C → S together with an isomorphism C × S U ∼ −→ C.This isomorphism will often be suppressed in the notation.
Proposition 2.3.Let S be a locally noetherian scheme, C/S a semistable curve, s a geometric point of S, and c a geometric point of C lying over s.Then there exists an element α in the maximal ideal of the étale local ring O et The element α is not in general unique, but, the ideal αO et S,s ⊳ O et S,s is unique.We call it the singular ideal of c.If C/S is smooth over a scheme-theoretically-dense open of S then the singular ideal is never a zero-divisor.Definition 2.4.Let S be a locally noetherian scheme and C/S a semistable curve.Let s ∈ S be a geometric point, and write Γ for the dual graph 5 of the fibre C s .Let 4.1.2for every field-valued fibre C k of C/S, every irreducible component of C k is geometrically irreducible.Remark 4.2.Given a quasisplit semistable curve C/S and a geometric point s of S with image s ∈ S, the graph Γ s depends only on s and not on s.As such, it makes sense to talk about the dual graph of C s for s ∈ S a point in.Moreover, the labels on such a graph, which a-priori live in the étale local ring at s, are easily seen to live in the henselisation of the Zariski local ring, by condition 4.1.2.Applying condition 4.1.1,we even see that the labels in fact live in the Zariski local ring at s.
only units on S.6.1.7LetC/Sbea semistable curve.We say a correlation collection K on S is a correlation collection for C/S if C/S is quasisplit and for every s ∈ S we have that the pullback of K to O S,s is a correlation collection for the base-changeC O S,s / Spec O S,s .Proposition 6.2.Let C/S be a semistable curve over a noetherian base.Then there exists an étale cover S ′ → S such that C × S S ′ → S ′ has a correlation collection.Proof.By lemma 4.3, we may assume C/S is quasisplit.It is enough to show that every point s ∈ S has an étale neighbourhood for which a correlation collection exists.Let s ∈ S be a point.Shrinking S, we may assume that every label on the graph Γ s is generated by an element of O S (S) (i.e.these locally-principal ideals are principal).Write Γ {2} s , and for each label I appearing on an edge in H, choose a generator of I. Let K(H) denote the set of chosen generators of labels appearing on H. Then the multiset