Infinitesimal structure of the pluricanonical double ramification locus

We prove that a formula for the `pluricanonical' double ramification cycle proposed by Janda, Pandharipande, Pixton, Zvonkine, and the second-named author is in fact the class of a cycle constructed geometrically by the first-named author. Our proof proceeds by a detailed explicit analysis of the deformation theory of the double ramification cycle, both to first and to higher order.


Introduction
Inside the moduli space M g,n of smooth pointed curves (C, p 1 , . . . , p n ) there are natural closed subsets H k g (m) = (C, p 1 , . . . , p n ) : ω ⊗k where m = (m 1 , . . . , m n ) ∈ Z n is a vector of integers summing to k(2g − 2). Since the above isomorphism of line bundles is equivalent to the existence of a meromorphic k-differential on C with zeros and poles at the points p i with specified orders m i , these subsets are called strata of meromorphic k-differentials. These strata appear naturally in algebraic geometry, the theory of flat surfaces and Teichmüller dynamics and have been studied intensively in the past, see the surveys [Zor06,Wri15,Che17] and the references therein. Motivated by problems in symplectic geometry, Eliashberg asked whether there was a natural way to extend these strata and their fundamental classes to the Deligne-Mumford-Knudsen compactification M g,n and how to compute the resulting cycle class. For k = 0 there are two geometric avenues to defining such an extension. The first is via relative Gromov-Witten theory and the space of rubber maps to P 1 ([LR01, Li02,Li01,GV05]). This is based on the observation that for a smooth curve C, a meromorphic 0-differential on C as above corresponds to a morphism C → P 1 with given ramification profiles over 0, ∞. The second series of approaches, viable for any k 0, uses that H k g (m) can be obtained by pulling back the zero section e of the universal Jacobian J → M g,n via the Abel-Jacobi section σ : M g,n → J , (C, p 1 , . . . , p n ) → ω k − n i=1 m i p i .
The map σ does not extend naturally to M g,n , but various geometric extensions of its domain and target have been proposed that yield cycles on M g,n ([KP19, HKP18, MW20, Hol19, AP19]). These constructions all produce the same cycle class on M g,n , which we denote DRC; an overview of one construction is given in section 1.2.
Pixton [Pix14] defined a class P g,k g (m) in the tautological ring of M g,n . The equality DRC = 2 −g P g,k g (m) (1.0.2) was conjectured by Pixton for k = 0, and in [Hol19] for all k. An introduction to Pixton's formula in the case k = 0 can be found in [Pan18, Section 6.4], and in the general case in [JPPZ17]. The conjectured equality 2 −g P g,k g (m) = DRC for k = 0 was proven in [JPPZ17]. Since this preprint was posted, the equality 2 −g P g,k g (m) = DRC for all k has been established in [BHP + 20]. A new geometric approach to extending the cycle appears for k 1: assuming that one of the integers m i is negative or not divisible by k, the papers [FP16,Sch18] define a cycle H k g,m obtained as a weighted 1 fundamental class of an explicit closed subset H k g (m) ⊂ M g,n extending H k g (m), and propose Conjecture A ( [FP16,Sch18]) Let k 1 and m = (m 1 , . . . , m n ) ∈ Z n with m 1 + . . . + m n = k(2g − 2). Assume that one of the m i is negative or not divisible by k and letm = (m 1 + k, . . . , m n + k). Then H k g,m = 2 −g P g,k g (m) ∈ A g (M g,n ). At the time these papers were written the geometric class DRC had not been defined for k > 0; from our current perspective it seems most natural simply to conjecture that all three classes (DRC, 2 −g P g,k g (m) and H k g,m ) are equal whenever they are defined. The main result of our paper is the following. Combined with the recent proof of 2 −g P g,k g (m) = DRC in [BHP + 20], this yields a proof of Conjecture A.
In fact, not only do we prove the equality (1.0.3) of cycle classes, but as a byproduct of our proof we demonstrate how the weights in the weighted fundamental class H k g,m arise from intersection multiplicities of the Abel-Jacobi section with the zero section in the construction of [Hol19]. Further, with a little extra work our method allows us to compute not only the multiplicities of the cycle, but even give a presentation for the Artin local rings at generic points of the double ramification locus (theorem 5.6).
In the remainder of this introduction, we recall the definition of H k g,m and the construction of DRC from [Hol19] before stating a more refined version of our main result in section 1.3. We give a sketch of the proof in section 1.4 and discuss some future research directions in section 1.5. We finish by giving a more detailed overview of the relations between the various approaches for defining the extended cycles that we discussed before.

The moduli space of twisted k-differentials
A first idea for extending the stratum H k g (m) of k-differentials is to consider its closure H k g (m) ⊂ M g,n . Stable curves (C, p 1 , . . . , p n ) in this closure have been characterized in [BCG + 19a,BCG + 18] in terms of existence of k-differentials on the components of C satisfying certain residue conditions. For k = 1 and all m i 0, the closure H 1 g (m) is of pure codimension g − 1, and [RPZ19] gives a conjectural relation of the fundamental class of this closure to Witten's r-spin classes.
A larger compactification containing the closure H 1 g (m), the moduli space of twisted kdifferentials H k g (m), has been proposed by Farkas and Pandharipande in [FP16]. The idea here is that as the curve C becomes reducible, it is no longer reasonable to ask for an isomorphism of line bundles ω ⊗k C ∼ = O C ( n i=1 m i p i ), since these line bundles will have different degrees on the various components of C. However, these multidegrees can be balanced out by twisting the line bundles by (preimages of) the nodes of C. The way this balancing happens is encoded in a twist on the stable graph Γ of C. This is a map I from the set of half-edges of Γ to the integers, satisfying I(h) = −I(h ′ ) if (h, h ′ ) form an edge, together with a further combinatorial condition (see definition 1.6 for details). Given a twist I on the dual graph Γ of a stable curve C, let ν I : C I → C be the map normalizing the nodes q ∈ C belonging to edges (h, h ′ ) with I(h) = 0. Let q h , q h ′ ∈ C I be the corresponding preimages of q under ν I . Then the curve (C, p 1 , . . . , p n ) is contained in H k g (m) if and only if there exists a twist I on its stable graph, such that we have an isomorphism of line bundles ω ⊗k on C I . This corresponds to requiring the existence of a k-differential on the components of the partial normalization C I of C with zeros and poles at markings and preimages of nodes, where the multiplicities at the node preimages are dictated by the twist I.
The space H k g (m) is a closed subset of M g,n containing H k g (m) but possibly having additional components supported in the boundary of M g,n . It turns out that these extra components are essential when trying to associate a natural cycle class to the extension of the strata of kdifferentials. Assume we are in the case that k 1 and that at least one of the m i is either negative or not divisible by k. Then it is shown in [FP16] (for k = 1) and [Sch18] (for k > 1) that H k g (m) ⊂ M g,n has pure codimension g. In this situation, instead of studying the fundamental class of H k g (m) (as a reduced substack), the papers [FP16,Sch18] consider a certain weighted fundamental class H k g,m ∈ A g (M g,n ) of H k g (m). To define this weighted class, let Z be an irreducible component of H k g (m). Denote by Γ the generic dual graph of a curve C in Z and let I be the 2 generic twist on Γ. Then it is shown in [FP16,Sch18] that Γ and I must be of a particular form. Indeed, the graph Γ is a so-called simple star graph, having a distinguished central vertex v 0 such that every edge has exactly one endpoint at the central vertex. The remaining vertices are called the outlying vertices. All markings i with m i negative or not divisible by k must be on the central vertex. Moreover, the twist I on Γ has the property that for all edges e = (h, h ′ ), with h incident to v 0 and h ′ incident to an outlying vertex, we have that I(h ′ ) is positive and divisible by k. By slight abuse of notation we write I(e) = I(h ′ ) in this case.
With this notation in place, we can define 3 the weighted fundamental class H k g,m of H k g (m) as where Z runs over the components of H k g (m) and as above Γ, I are the generic dual graph and twist on Z. Conjecture A above then relates these weighted fundamental classes to the explicit tautological cycles P g,k g (m) proposed by Aaron Pixton in [Pix14]. In our paper, we show how both the twisted differential space H k g (m) and its weighted fundamental class H k g,m naturally arise from a construction presented by the first-named author in [Hol19].

Extending the Abel-Jacobi map
Let J be the universal semi-abelian jacobian over M g,n , often written Pic 0 C/Mg,n . It has connected fibres, and parametrizes line bundles of multidegree zero on the fibres of the universal curve C → M g,n . Inside the open set M g,n ⊂ M g,n the strata H k g (m) of k-differentials can be obtained as the pullback of the zero-section e of J via the Abel-Jacobi section σ : M g,n → J , (C, p 1 , . . . , p n ) → ω k − n i=1 m i p i =: ω k (−mP ) .
While σ does not in general extend over M g,n , in [Hol19] the first author defines a "universal" stack M ♦ → M g,n , birational over M g,n , on which σ does extend to a morphism σ ♦ : M ♦ → J M ♦ , where J M ♦ is the pullback of J to M ♦ . Moreover, the scheme-theoretic pullback DRL ♦ of the unit section e of J M ♦ along σ ♦ is proper over M g,n . Denote DRC ♦ the cycle-theoretic pullback of the class [e] under σ ♦ , supported on DRL ♦ , and by DRC ∈ A g (M g,n ) its pushforward under the proper map DRL ♦ → M g,n .

Main result
Refining theorem 1.1, the main result of our paper is the following.
Theorem 1.2. The image of the double ramification locus DRL ♦ under the map M ♦ → M g,n is the moduli space H k g (m) ⊂ M g,n of twisted k-differentials. Moreover, for k 1 and at least one of the m i either negative or not divisible by k, we have that DRC = H k g,m ∈ A g (M g,n ). (1.3.1) In fact, this is true in the strong sense that these two cycles supported on H k g (m) have the same weight at each irreducible component (they are equal as cycles, not just cycle classes).
For the last point of the theorem, the equality of DRC and H k g,m on the cycle level, observe that the formula (1.1.2) allows us to define H k g,m as a cycle, not just a cycle class. And, under the assumptions of the theorem, the locus DRL ♦ has the expected codimension, and so DRC makes sense as a cycle, not just a cycle class. Then in fact the equality (1.3.1) holds as an equality of cycles, not only up to rational equivalence (in contrast to Conjecture A above, which only makes sense up to rational equivalence).
We give an outline of the proof in section 1.4, where we will also discuss in more detail how the multiplicities in the formula (1.1.2) come up for the cycle DRC. Our method of proof actually yields more precise information than required for the conjecture; we can compute not only the multiplicities of the cycle, but even give a presentation for the Artin local rings at generic points of the double ramification locus (see theorem 5.6).
The above theorem gives a concrete interpretation for the weights appearing in the definition of H k g,m . It is also a crucial component of the proof of Conjecture A.
Corollary 1.3. Conjecture A is true.
Proof. The equality DRC = 2 −g P g,k g (m) is proven in [BHP + 20], so this follows from theorem 1.2.
At the time this preprint was posted to the arXiv the equality DRC = 2 −g P g,k g (m) was known over the locus of compact-type curves by previous work [HPS19] with Pixton, showing Conjecture A to be true when restricted to the locus M ct g,n ⊂ M g,n of compact type curves.

Sketch of the proof
The main difficulty in the proof of theorem 1.2 is to compute the intersection multiplicity of the Abel-Jacobi map σ ♦ : M ♦ → J M ♦ with the unit section e of J M ♦ along the different components of DRL ♦ . For this, we use classical deformation theory to first compute the Zariski tangent space at a general point and then show how to extend this study to higher order deformations.
To set up the deformation theory, we first need to choose local coordinates on M ♦ . Here, it turns out that it is more convenient to work with a slight variant M m,1/k → M g,n of M ♦ , for which it is easier to write down local charts around the general points of DRL. The precise construction of M m,1/k is given in section 2 (where we also make more concrete the relationship with the construction of Marcus and Wise [MW20]), but for us the two key properties are (i) The map M m,1/k → M g,n is logétale and birational and the map σ : M → J sending (C, P ) to [ω k (−mP )] extends uniquely to a mapσ : M m,1/k → J .
(ii) We can compute the tangent space to M m,1/k explicitly.
The double ramification locus is then DRL =σ * e, where e is the unit section in J . The concrete local charts for M m,1/k can be used to show that the image of DRL in M g,n is exactly the twisted differential space H k g (m). With this setup established, the equality of weights in theorem 1.2 comes about in an interesting way: let Z be an irreducible component of H k g (m) with generic stable graph Γ and twist I. Then a general point p ∈ Z has exactly #{p ′ ∈ DRL over p} = k #E(Γ)−#V (Γ)+1 = k b 1 (Γ) preimages p ′ in DRL. This is something that can be easily checked in the local charts of M m,1/k .
In definition 2.11, we define a cycle DRC supported on DRL. At each preimage, its multiplicity is Hence, the pushforward DRC of DRC has multiplicity which is exactly the weight of [Z] in the class H k g,m . It is also easy to see that the cycle DRC on DRL equals the fundamental class of (the possibly nonreduced) DRL (see lemma 2.17), so we are left with studying the multiplicity of DRL at its generic points.
Section 3 is concerned with the computation of the tangent space to DRL. Suppose we are given a point p ∈ DRL ⊆ M m,1/k , which is a general point of some irreducible component of DRL. Let Γ be the generic stable graph and I be the generic twist on this component.
The maps e andσ induce maps on tangent spaces This induces an exact sequence it thus remains to analyse carefully the map b. For C p the stable curve corresponding to the point p, the domain and target of b are easily identified in terms of cohomology groups of sheaves on C p . Instead of studying the cokernel of b, it will be more convenient to use Serre duality and compute the kernel of the linear dual b ∨ , which is dual to coker(b). In theorem 4.2 we show that ker(b ∨ ) has a natural basis, with one element for each outlying vertex v of Γ connected to the central vertex only by edges with twists I > k. In theorem 4.3 we conclude that dim T p DRL = dim p DRL +#{e ∈ E(Γ) : I(e) > k}, so we have one "direction of nonreducedness" for each edge e with I(e) > k, corresponding to an infinitesimal deformation smoothing the corresponding node. While this description is quite simple, the deformation-theoretic computation that derives it is fairly long and involved. We decompose the tangent space T p M m,1/k into a direct sum of four pieces, corresponding to different types of deformations. Then the dual b ∨ decomposes in four summands accordingly and we compute the intersection of their kernels. In the course of these computations, we need to show that for the k-differential on the central component of C p , we have that sums of kth roots of its k-residues 4 at (subsets of the) nodes of C p are generically non-vanishing. We show a corresponding general result, which might be of independent interest, in appendix C. That the tangent space to the double ramification locus can be computed via first-order deformation theory is unsurprising, but in order to prove theorem 1.2, we need to compute the local rings of the double ramification locus, which is much more involved. It is not hard to show that an Artin local ring is determined by its functor of deformations, but reconstructing the Artin ring from the deformations is in practise often difficult.
Write E for the set of edges of the dual graph of the tautological stable curve C p over p. The universal deformation of C p comes with a natural projection map to Spec K[[x e : e ∈ E]], which we can see as the space of deformations which smooth the nodes. Here K is our base field, which we assume to be of characteristic zero. We slice DRL with a generic subvariety of codimension equal to the dimension of DRL, obtaining a space DRL ′ whose tangent space has dimension equal to the number of edges e with twist I(e) > k. We use our tangent space computation to show that the natural map DRL ′ → Spec K[[x e : e ∈ E]] is a closed immersion; it remains to identify the image. From the explanation above, one can reasonably guess that the image might be cut out by the ideal ] is exactly that all elements x I(e)/k e are sent to zero in A i .
Remark 1.4. If we had worked M ♦ instead of M m,1/k , a similar description would be possible, but both the multiplicities and the cardinalities of fibres of the double ramification locus over the twisted differential space would have to be expressed in terms of the gcd/lcm of the twists (though in the end everything would of course cancel to give the same answer). This would have made the deformation-theoretic calculation more complicated, and seemed to us better avoided.
Once again, the key input is our result in appendix C on the generic-non-vanishing of kresidues.

Relation to previous work and outlook
Compactification via log geometry In the paper [Gué16], Guéré uses logarithmic geometry to construct a moduli space of k-log canonical divisors sitting over H k g (m) and carrying a natural perfect obstruction theory and virtual fundamental class. For k = 1 and one of the m i negative, the pushforward of this virtual class equals the weighted fundamental class H k g,m . However, for general k the multiplicity of this pushforward at a component with stable graph Γ and twist I is equal to e∈E(Γ) I(e), and thus different from the multiplicities obtained here and conjectured in [Sch18]. This could indicate that for k > 1 the definition of the space in [Gué16] needs to be adapted. We hope that the computations in the present paper may shed some light on the necessary modifications.
The cases of excess dimension Until now, our paper has focused on the case k 1 and one of the m i negative or not divisible by k, in which case H k g (m) was of pure codimension g. In general, by [FP16,Theorem 21] all components of the space H k g (m) have at most codimension g. In these remaining cases, the behaviour is as follows: -for k = 0, the principal component H k g (m) is of codimension exactly g (unless all m i = 0), but there are components in the boundary of M g,n of various excess dimensions; -for k = 1 and all m i 0, the principal component H 1 g (m) is of pure codimension g − 1, with all other components supported in the boundary and of codimension g, -for k > 1 and all m i = km ′ i 0 divisible by k, the space H k g (m) decomposes as a disjoint union is the locus where the k-differential is a k-th power of a 1-differential, and is a union of components of codimension g − 1, with all other components (i.e. H k g (m) ′ and those supported in the boundary) having codimension g.
In all of these cases, the cycle DRC still makes sense and by theorem 1.2 it is indeed supported on the locus H k g (m) ⊂ M g,n . Similarly, the formula of Pixton's cycle P g,k g (m) makes sense in these cases, and in [Hol19], the first author shows that for k = 0 we have DRC = 2 −g P g,k g (m). We expect that in the cases k 1 and m = km ′ 0 the cycle DRC should behave as follows: as before (where Γ, I are the generic twist and dual graph), -on the components H 1 g (m ′ ) of codimension g − 1 it should be given by the first Chern class of an appropriate excess bundle (for the Abel-Jacobi section meeting the unit section) times the fundamental class of H 1 g (m ′ ).
It seems likely, that the deformation-theoretic tools in the present paper can be applied to prove these expectations, and explicitly identify the excess bundle.
The perspective above could also help shed further light on a second conjecture made in [Sch18]. There, for a nonnegative partition m ′ of 2g − 2, a class [H 1 g (m ′ )] vir was defined by the formula where Z runs through the boundary components of H 1 g (m ′ ) andm ′ = (m ′ 1 + 1, . . . , m ′ n + 1).
where Z runs through the boundary components of H k g (m).
From the perspective of defining the Double ramification cycle via an extension of the Abel-Jacobi map, this behaviour is expected: the space M ♦ for the partition km ′ of k(2g − 2) agrees with the space for the partition m ′ of 2g − 2, and the Abel-Jacobi section for km ′ is simply the composition of the section for m ′ with theétale morphism Thus, over the locus H 1 g (m ′ ), the intersection of the Abel-Jacobi section with the unit section should produce the same contribution to the cycle DRC.
Smoothing differentials The papers [BCG + 18, BCG + 19a] give criteria for a nodal curve (C, p 1 , . . . , p n ) to lie in the locus H k g (m). Being contained in this closure is equivalent to having some one-parameter deformation (C t , p 1,t , . . . , p n,t ) t∈∆ with the general curve being contained in H k g (m). The criteria of [BCG + 18, BCG + 19a] are phrased in terms of the existence of kdifferentials on the components of C satisfying some vanishing conditions for sums of k-th roots of their k-residues at nodes of C. On the other hand, in our deformation-theoretic computations in section 5 we see that for a point in a boundary component of the Double ramification locus, the obstruction to smoothing the nodes while remaining in the Double ramification locus is exactly related to a nonvanishing of such sums of k-th roots of k-residues. While these computations are not directly applicable to the problem of classifying H k g (m), it seems plausible that the methods of our paper can be applied in this direction. We thank Adrien Sauvaget for pointing out this connection and plan to pursue this in forthcoming work.
In a related direction, the recent paper [BCG + 19b] constructs a smooth compactification of the closure H k g (m) and gives a modular interpretation for this new compactification. Here, it is an interesting question how this relates to the compactification obtained by taking the closure of H k g (m) ⊂ DRL ♦ inside the Double ramification locus of M ♦ .

An overview of different definitions of Double ramification cycles
In this section, we want to summarize the existing definitions of Double ramification cycles in the literature and the known equivalences between them. Several authors gave elementary geometric constructions of the DR class on partial compactifications of M g,n inside M g,n (for example, the compact-type locus), and computed them it in the tautological ring. Examples include [Hai13], [GZ14b], [GZ14a], and [Dud18].
The following are the different constructions of a DR cycle on all of M g,n : -In the case k = 0, Li, Graber and Vakil gave a construction as the pushforward of a virtual fundamental class on spaces of rubber maps ( [Li02], [Li01],[GV05], see also [LR01]).
- Pixton ([Pix14]) proposed the formula 2 −g P g,k g (m) for the DR class as an explicit tautological class, defined via a graph sum. -Marcus and Wise used techniques from logarithmic and tropical geometry ([MW20]) to construct a space on which the Abel-Jacobi map extends.
-The first-named author gave a definition using a universal extension of the Abel-Jacobi map as described above ([Hol19]).
-Abreu and Pacini gave an explicit "tropical blowup" of M g,n (i.e. a blowup dictated by an explicit refinement of M trop g,n ) resolving the Abel-Jacobi map to the Esteves' compactified Jacobian over M g,n and use this to define a Double ramification cycle ([AP19]).
-Finally, for k 1 and one of the m i is negative or not divisible by k, there is the definition of the DR cycle as the weighted fundamental class H k g,m , proposed by Janda, Pandharipande, Pixton, and Zvonkine for k = 1 ( [FP16]) and the second-named author for k > 1 ( [Sch18]).
In fig. 2 we illustrate the known equivalences between these definitions. In particular, [BHP + 20] (which came out after the first appearance of this paper) completes the proof that they are all in fact equivalent.

Outline of the paper
The main purpose of this paper is to analyse very carefully the infinitesimal structure of the double ramification locus, eventually enabling us to compute the multiplicities of its components and thus compare it to the cycle of twisted differentials. In section 2 we describe the construction of the space M m,1/k , the variant of M ♦ on which we perform our computations (see section 1.4 above). We also make more concrete the relationship with the construction of Marcus and Wise [MW20]. Sections 3 and 4 are devoted to the computation to the tangent space to the double ramification locus. In the brief section 3 we compute the tangent space of the space M m,1/k , in which the double ramification locus naturally lives. Section 4 is much more substantial, and contains the computation of the tangent space of the double ramification locus itself. A key technical lemma on the non-vanishing of certain residues is postponed until appendix C, as it may be of independent interest and we wished to keep its exposition self-contained. Once we understand the tangent space to the double ramification locus, in section 5 we can compute explicitly its local ring, and in particular the length of the local ring. In section 5.1 we use this to deduce the desired formula of the Double ramification cycle as a weighted fundamental class.
Finally, in the appendices A and B we recall some standard results on Serre duality and deformation theory viaČech cocycles that are used in several places in the proof. This material is well-known, but we include it to fix notation, and because the very explicit forms of these results that we need are somewhat scattered about in the literature.
described in section 2.5 H 1 (Γ, K) the first cohomology group of the graph Γ with coefficients in K, cycle of length 2 given by the composition of the directed edge e with the inverse of the directed edge e ′ , assuming e, e ′ : v → v ′ have same source and same target

List of notations
Generalities We have fixed integers g 0, n > 0, k > 0 with 2g − 2 + n > 0, and integers m 1 , . . . , m n summing to k(2g − 2) with at least one m i < 0 or not divisible by k. We will write M for M g,n , M for M g,n etc. We write J for the universal semi-abelian jacobian over M, often written Pic We work throughout over a fixed field K, which we assume to have characteristic zero. Our proof is entirely algebraic, except for the crucial application of a result of Sauvaget [Sau19,Corollary 3.8] in appendix C, which we expect to admit an algebraic proof. When k > 1 we use very often the characteristic-zero assumption, but for k = 1 it can often be avoided; its main purpose is in allowing us to apply Sauvaget's result mentioned above, and in lemma 4.10 where we use that a function with vanishing differential is locally constant. As such it may well be possible with the methods here to determine what happens in small characteristic; it seems very likely that the multiplicities of the twisted differential space will be different in this case.
Remark 1.5. Our results do not require that the ground field K be algebraically closed. When we talk about the graph of a curve over a field, we are implicitly saying that the irreducible components are geometrically irreducible, and the preimages of the nodes in the normalisation are all rational points. At later points we will assert that various k-differentials locally have k-th roots; this should be interpreted over a suitable finite extension, (our characteristic-zero assumption ensures that adjoining k-th roots yields an etale extension, and thus does not affect the deformation theory. Alternatively, because the computations of the tangent spaces and lengths of local rings are invariant under etale extensions, the reader may assume without loss of generality that the ground field k is algebraically closed throughout sections 4 and 5.
We expect that most readers will be mainly interested in the case of algebraically closed fields, so to minimise clutter we do not explicitly discuss these field extensions, but allow the interested reader to insert them when necessary.
1.8.1 Graphs and twists A graph Γ 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, and a genus g : V → Z 0 . Graphs are connected, and the genus g(Γ) is the 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. Edges are sets {h, h ′ } (of cardinality 2) with i(h) = h ′ . Legs are fixed points of i, and L denotes the set of legs. 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.
The valence val(v) of a vertex is the number of non-leg half-edges incident to it, and we define the canonical degree can(v) = 2g(v) − 2 + val(v), so that 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.
A leg-weighted graph is a graph Γ together with a function m from the set L of legs to Z such that l∈L m(l) = k(2g(Γ) − 2). Definition 1.6. A twist of a leg-weighted graph is a function I from the half-edges to Z such that: (i) for all legs l ∈ L, we have m(l) = I(l).
We write Tw(Γ) for the (non-empty) set of twists of a leg-weighted graph Γ.
Remark 1.7. In [Hol19] these twists were called 'weightings', and were denoted w. The present notation is much closer to that used by [FP16]; we have made this change to facilitate comparison to [FP16], and because the letter w was already over-loaded.
Remark 1.8. Farkas and Pandharipande impose two additional conditions (which they call 'vanishing' and 'sign'), which together state that Γ cannot contain any directed cycle for which every directed edge h has I(h) 0, and at least one h has I(h) > 0.
We do not need to impose this condition as it will drop out automatically from our geometric setup; more precisely, the fibre of a chart M m I,U of M m over the origin in A E (see section 2.2 for this notation) is easily seen to be empty if either of these conditions is not satisfied. If one forgets the values of the integers I(h) and remembers only their signs and whether they vanish, the above condition is exactly equivalent to 'Suzumura consistency', a condition arising in decision theory [Bos08].
Definition 1.9. We say a leg-weighted graph Γ is a simple star graph if all legs with negative weight or weight not divisible by k are attached to the same vertex (which we call the central vertex ), and every edge has exactly one half-edge attached to the central vertex (in particular, there are no self-loops). We call the non-central vertices the outlying vertices, and the set of them is V out .

1.8.2
The weighted fundamental class of the space of twisted differentials In this section we recall the definition of the class H k g,m ∈ A g (M) given in [FP16,§A.4] (for k = 1) and [Sch18, Section 3.1] (for k > 1) and explain why it is equivalent to the definition as a weighted fundamental class of H k g (m) presented in the introduction. First, recall that given any integer k 1 and a partition m of k(2g − 2) of length n, we have the corresponding stratum of k-differentials. This closed, reduced substack has pure codimension g − 1 if k = 1 and all m i 0, and pure codimension g if there exists i such that m i is negative or not divisible by k. As before we denote by H k g (m) its closure in M g,n . Write S for the set of simple star graphs of genus g (see definition 1.9). We say a twist I of a simple star graph is positive (writing Tw + (Γ) for the set of positive twists) if I(h) > 0 and k divides I(h) for every half-edge h attached to an outlying vertex. In this case, by slight abuse of notation, we write I(e) = I(h) for the edge e = (h, h ′ ) to which h belongs.
with n(v 0 ) the number of halfedges attached to v 0 , and with weighting given by restricting the weighting m to those legs attached to v 0 , and given by −I(e) − k at the half-edge belonging to the edge e of Γ. The cycles on the outlying vertices v are defined analogously, where we use that all markings on them have weights m i divisible by k and all twists I are likewise divisible by k (again, see [Sch18] for details). Now we comment why this is a weighted fundamental class of the space H k g (m). Given a boundary component Z of this space, let Γ be the generic dual graph of a curve C in Z and let I be the twist on Γ such that the condition (1.1.1) is satisfied for this generic curve C. By [Sch18,Proposition A.1.] every node of C such that the corresponding edge has twist I = 0 can be smoothed while staying in H k g (m). Thus since Z is assumed a generic point of H k g (m), all edges of Γ must have nonzero twist. Then this condition tells us that the various components C v of C vary within appropriate strata of k-differentials. But the codimension of Z is at most g by [FP16,Theorem 21]. A short computation shows that this is only possible if at all but one of the vertices v, the curve C v varies in a stratum of k-th powers of holomorphic 1-differentials (which is the case of excess-dimension). This implies that all twists must be divisible by k and that there is exactly one vertex carrying all the negatively twisted half-edges as well as markings i with m i < 0 or not divisible by k. This easily implies that the generic dual graph Γ of Z is a simple star graph and that the twist I on Γ is positive.
Conversely, one checks that condition (1.1.1) is satisfied on all the loci on which the cycle H k g,m above is supported. This shows that it is indeed a weighted fundamental class of H k g (m). On the other hand, the weights agree with those given in the introduction: the closures of strata of differentials (which are pushed forward via ξ Γ ) are generically reduced and thus all have multiplicity 1. The factor 1/| Aut(Γ)| exactly accounts for the fact that the gluing morphism Γ has degree | Aut(Γ)|.
Thus the definition of H k g,m given in the introduction coincides with the definitions from [FP16,Sch18]. Given a leg-weighted graph Γ with edge set E, define 5 To any point a in A E we associate the graph Γ a obtained from Γ by contracting exactly those edges e such that a e is a unit at a. Denote by Definition 1.10. A combinatorial chart of M consists of a leg-weighted graph Γ and a diagram of stacks satisfying the following six conditions: Let p : Spec K → U be any geometric point, yielding natural maps Let C = f (p) and a = g(p), then we require an isomorphism 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 −m i to the leg-weighting on Γ g•p coming from that on Γ.
This definition is as in [Hol19] but with the logarithmic structures excised (since we do not need them). We see in [Hol19] that M can be covered by combinatorial charts.

Constructing suitable moduli spaces
2.1 Recalling the construction of DR We begin by recalling the basic construction of the cycle DR from [Hol19]. First one constructs a certain stack M ♦ /M such that the rational map σ : M → J extends to a morphism σ ♦ : M ♦ → J . Writing e for the unit section of J (viewed as a closed subscheme) and [e] for its Chow class, it is shown in [Hol19] that the scheme-theoretic pullback DRL ♦ of e along σ ♦ is proper over M. We would like to now take the cycle-theoretic pullback of the class of e along σ ♦ , but the latter is not (known to be) a regular closed immersion, so we do not know how to make sense of this pullback. Instead, we consider the induced section M ♦ → J M ♦ = J × M M ♦ , and pull back the class of the unit section along this section (using that the latter is a regular closed immersion as J is smooth over M) to obtain a cycle DRC ♦ on M ♦ . This cycle DRC ♦ is naturally supported on DRL ♦ , and so by properness can be pushed down to a cycle on M, which we denote DRC, the compactified double ramification cycle. Many more details and properties of the construction, and a comparison to other constructions in the literature, can be found in [Hol19], [HPS19] and [HKP18].
In this article we will work with a slight variant of the stack M ♦ of [Hol19]; this is only for convenience, but the intricacy of the calculations we have to carry out make every available bit of notational efficiency worth using. We also note that M ♦ depends not only on g and n, but also on the m i and k, hence the notation is not good -we will take the opportunity to correct this.
The stack M ♦ is built by glueing together normal toric varieties, in particular it is normal. We will begin by introducing a 'non-normal' analogue M m of M ♦ which is close to (but not yet quite) what we want. The resulting double ramification cycle will be unchanged, by compatibility of the refined gysin pullback with the proper pushforward, see section 2.6 for more details.

Construction of
(2.2.1) In the free abelian group on symbols a e : e ∈ E we consider the submonoid generated by the a e and by the expressions e∈E a Iγ(e) e (2.2.2) as γ runs over cycles in Γ, and we denote the spectrum of the associated monoid ring by A E I . Equivalently, A E I is the spectrum of the subring of K[a ±1 e : e ∈ E] generated by the a e and by the expressions in (2.2.2). Note that this is slightly different from the monoid rings constructed in [Hol19], where we worked with sub-polyhedral cones of Q E 0 , cut out by equations: monoids coming from cones are always saturated, and so yield normal varieties, whereas here we want to work with not-necessarily-saturated monoids. In section 2.3 we give explicit equations for (a slight variant on) the A E I . We write M m I,U for the pullback of A E I to U . We want to argue that these M m I,U naturally glue together to form a stack M m over M. The first part of the gluing can even be done over A E . Indeed, fixing a graph Γ, as I runs over twists of Γ the A E I naturally glue together as I, c.f. [Hol19, §3] 6 . We denote the glued object byÃ E → A E .
Example 2.1. In the case 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. Twists consist of a flow of a along edge e from u to v, and n − a along the other edge e ′ (again from u to v), for a ∈ Z: In this setting A E = Spec K[a e , a e ′ ]. There are two directed cycles, and the expression (2.2.2) yields a i e a i−n e ′ and a −i e a n−i e ′ . The form of A E I then depends on I: we have i < 0 : (2.2.4) A more detailed explanation of these equations can be found in (2.3.3) below. These patches are then all glued together along the torus Spec K[a ±1 e , a ±1 e ′ ] to formÃ E . Note that (in the case where n is not prime) this differs slightly from the example in [Hol19, remark 3.4] (where a toric interpretation is given) as the rings above are not normal for 0 < i < n whenever n and i have a common factor.
These patches can naturally be seen as charts of a (non-normal) toric blowup. In more involved examples (e.g. [Hol19, remark 3.5]) there is no canonical way to embed the patches in a blowup, though see also [AP19] for a general approach to compactifying.
While there are infinitely many charts glued together, only those for 0 i n are relevant, the others do not enlarge the space. This is how we glue infinitely many patches to obtain a quasi-compact space.
We now return to the general construction. For a fixed combinatorial chart U , pulling thesẽ A E back to U we obtain a stack covered by patches M The pushforward of DRC m to M makes sense by properness of DRL m → M, and the compatibility of the refined gysin pullback with the proper pushforward (see lemma 2.12) implies that the pushforward of DRC m to M coincides with pushforward of DRC ♦ to M. See section 2.6 for further details.

A partial normalisation of M m
As discussed in section 1.8.2, on the components of the Double ramification locus supported in the boundary, the twist I is generically divisible by k. since it arises by pulling back A E I ′ to U ). Let Υ be the set of cycles γ in Γ and recall that E is the set of edges in Γ. Then naturally we can see A E I ′ as a subscheme of A Υ × A E cut out by explicit equations. Let ((a γ ) γ∈Υ , (a e ) e∈E ) be coordinates on A Υ × A E , then the generators (2.3.1) translate into a system of equations in the a γ , a e . Indeed, given f ∈ Z Υ and e ∈ E define the integer . Then a set of equations cutting out A E I ′ ⊂ A Υ × A E is given by the vanishing of the as f runs through Z Υ . In particular, for any cycle γ we have for the inverted cycle i(γ), walking in opposite direction, that a γ a i(γ) = 1, which forces a γ = 0. Apart from that, the most simple equations in the system above are of the form (2.3.4) We will see later that these are the only equations that matter for computing the tangent space to M m,1/k .
To get the description of M m,1/k I ′ ,U over U one inserts for the variables a e the components of the function g : U → A E from our combinatorial chart, and obtains equations for M m,1/k Remark 2.4. A shorter but less explicit description of the polynomials Ψ f of (2.3.3) can be obtained by saturating an ideal obtained from the equations (2.3.4). Let R := K[a γ : γ ∈ Γ][a e : e ∈ E], and let A be the R-algebra obtained by formally adjoining inverses to the a e . Let I be the ideal of A generated by (2.3.5) and let I R be the intersection of I with R. Then I R is exactly the ideal generated by the Ψ f of (2.3.3). Note that this is not in general equal to the ideal generated by polynomials coming from expressions in the form (2.3.4).
The map M m,1/k → M m k|I is finite birational, but in general neither the source nor the target is normal, thus the map does not need to be an isomorphism. Indeed, we have Lemma 2.5. Let p ∈ M m k|I lie over a simple star graph Γ with outlying vertex set V out . Then the fibre over p of the map Lemma 2.7. Suppose that Γ is a simple star graph. Then M m,1/k I ′ ,U is a local complete intersection over K.
As in [Hol19], the stack M m,1/k I ′ ,U can be defined relative to Z, in which generality the same lemma holds, with the same proof. The requirement that Γ be a simple star seems necessary; the graph seems to give a counterexample in general, though we have not checked all details. → A E I ′ → Spec K; moreover the first morphism is smooth (and hence syntomic) as a base change of the smooth morphism U → A E . Thus it suffices to check that A E I ′ → Spec K is syntomic, i.e. that A E I ′ is a local complete intersection.
Step 1: Choosing a spanning tree in Γ induces a collection Υ ′ ⊆ Υ of cycles in Γ forming a basis of H 1 (Γ, Z). Given a cycle γ ∈ Υ, writing γ as an integral linear combination of elements of Υ ′ induces an element of Z Υ ′ (with all coefficients in {−1, 0, 1}), whose image in Z Υ under the natural inclusion Z Υ ′ → Z Υ we denote f γ . We denote by δ γ ∈ Z Υ the indicator function for γ. Then the corresponding expression Ψ fγ −δγ (as defined in (2.3.3)) contains no terms a e with non-zero exponents.
Step 2: Consider the collection of polynomials consisting of the Ψ fγ−δγ for γ ∈ Υ. We then claim that the subscheme Z of A Υ × A E cut out by these polynomials is smooth over K of dimension #Υ ′ + #E. First, for γ ∈ Υ ′ the equation Ψ fγ −δγ = 0 can be re-written as 1 = 1, so can be ignored. Then if i(γ) ∈ Υ ′ , the equation Ψ fγ −δγ = 0 yields a γ a i(γ) = 1, so a i(γ) is inverted. For all other γ ∈ Υ ′ we can move all the a γ ′ with γ ′ ∈ Υ ′ to the left side of the equation (perhaps inverting them), thus writing a γ as a product of a ±1 γ ′ . Thus in fact Z a graph of a suitable function ( obtained by solving for those coordinates a γ for γ ∈ Υ \ Υ ′ . In particular, Z is smooth of the claimed dimension. Step 3: Since A E I ′ has dimension #E, it suffices to show that it is cut out from Z by the Ψ δ γ ′ as γ ′ runs over Υ ′ . First, given γ ∈ Υ, we claim that Ψ δγ is contained in the ideal of Γ(Z, O Z ) generated by the Ψ δ γ ′ : γ ′ ∈ Υ ′ . We may assume neither γ nor i(γ) lies in Υ ′ . Then γ consists of two directed edges, say γ = e 1 • i(e 2 ), with the e i going from the central vertex to an outlying vertex. Suppose that the spanning tree contains the edge e 0 to that outlying vertex. Then γ can be written as a difference of two cycles in Υ ′ : γ = γ 1 − γ 2 with γ 1 = e 1 • i(e 0 ) and γ 2 = e 2 • i(e 0 ). Then which is evidently contained in the ideal generated by using that all a γ are invertible on Z.
Step 4: It remains to treat the case of a Ψ f coming from an arbitrary element f ∈ Z Υ . The element f induces an element of H 1 (Γ, Z), which we can view as a subset of Z E ; write F for the image of f in Z E . If F is zero then the equation Ψ f = 0 already holds on Z. If F is non-zero then there exist a cycle γ ∈ Υ satisfying the assumptions of lemma 2.8, and Ψ δγ is in our ideal by Step 3, so we may replace f by f − δ γ . Now the sum of the absolute values of the coefficients of F is a positive integer strictly greater than the corresponding term for f − δ γ , so this process must terminate.
Proof. A small calculation with the expressions (2.3.3).

Charts and coordinates on the universal curve
For the deformation theoretic computations later, it will be necessary to fix a system of coordinates on the universal curve in the neighbourhood of a given point. Suppose we have a combinatorial chart and a point u ∈ U (K) mapping to the origin in A E (if desired we make a finite separable extension of K so that this exists). Write U u = Spec O U,u . A smooth coordinate chart of the tautological stable curve C Uu /U u consists of an open subscheme V ֒→ C sm Uu of the smooth locus of C Uu over U u with connected fibre over u. A singular coordinate chart of C Uu /U u consists of an isomorphism from the strict henselisation 7 of C Uu at a non-smooth point (corresponding to an We will repeatedly make use of the following Situation 2.9. We fix lying over u; -an fpqc cover of the universal stable curve C Uu /U u consisting of a finite collection of smooth and non-smooth charts as described above. is that the multidegree of ω k (−mP ) can be 'corrected' to 0 by adding on vertical divisors supported over the boundary ('twistors'); details can be found in [Hol19].

The universal sectionσ
Later we will need an explicit description of the pullback of this 'corrected' bundle to the tautological curve over a (connected) scheme T → M m,1/k I,U such that the composite T → M m,1/k I,U → U factors via the strict henselisation at the point u (this is to ensure that the local coordinates z h , z h ′ below make sense on T ). We will describe this line bundle by giving its pullback to the fpqc cover chosen above, together with transition functions. We first define a 'correction' line bundle T . Begin by choosing a function such that λ(−e) = λ(e) −1 , and such that for every loop γ in Γ we have e∈γ λ(e) = a γ . This is possible: choose a spanning tree Γ ′ ⊂ Γ. Then the edges e in Γ but not Γ ′ correspond to a basis γ e of the space of cycles (where γ e first takes the edge e and then takes the unique path inside Γ ′ closing the loop). Given this, a possible choice of λ is to set λ = 1 on all edges of Γ ′ and λ(e) = a γe on the remaining edges.
Choose also an orientation on each non-loop edge. The bundle T will then be trivial on each chart of the cover, and we will choose a generating section 1 on the smooth charts, and τ e on the non-smooth chart corresponding to an edge e of Γ. The transition function on an intersection of smooth charts sends 1 to 1. When a non-smooth chart corresponding to an oriented edge e = {h, h ′ } meets a smooth chart, the connected components of the intersection will be contained in V (z h ) or V (z h ′ ). On a connected component contained in V (z h ′ ) the transition function on the overlap is given by τ = λ(e) k z I(h) h 1, and on a connected component contained in V (z h ) the transition function is given by τ = z When two non-smooth charts meet their intersection is necessarily contained in a smooth chart, and so the transition functions are uniquely determined by the previous cases. Then the sectionσ is defined by the line bundle ω k (−mP )⊗T ; the reader can check that it has multidegree 0, or can find the details in [Hol19,§5]. Moreover, one verifies that for a family with generically smooth fibre, the bundle T restricts to the trivial bundle on this smooth fibre. This means that on the smooth fibre, ω k (−mP ) ⊗ T is just the Abel-Jacobi section and from the separatedness of J it follows that ω k (−mP ) ⊗ T is indeed the unique extension to the whole family.
Later on we will want to make some of these choices in a 'natural' way on a simple star graph. Suppose thus that Γ is a simple star (see definition 1.9), and for each outlying vertex v choose one edge e v to v. We take the orientation to be the 'outgoing' one from the centre to the outlying vertices. And we uniquely determine λ by requiring it to take the value 1 on e v . A basis of cycles is given by going out along e v and back along a different edge. If a cycle γ is given by e v and e ′ then the glueing at the node corresponding to e ′ gets 'adjusted' by exactly a γ . Because of the choice of orientation, it is only the glueing on the outlying vertices that gets adjusted by the a γ .
Remark 2.10. Because we work on this particular normalisation M m,1/k , we can also define canonically a k-th root of T . The construction is similar to that of T ; we choose generating sections on the smooth and non-smooth charts (denoted 1 1/k and τ 1/k e respectively), then glue on overlaps by the formulae τ 1/k = λ(e)z Definition 2.11. We define the double ramification locus DRL 1/k to be the schematic pullback of the unit section of the universal jacobian along the mapσ. We define DRC = DRC 1/k to be the cycle-theoretic pullback of the unit section of the base change J × M M m,1/k along the section M m,1/k → J × M M m,1/k induced byσ, as a cycle class on DRL 1/k .
By [Sch18] we know that all the generic points of H k g (m) lie in the locus M m k|I of M m where the twists are divisible by k. We will use this to show in section 2.6 that to compute DRC m it suffices to compute the multiplicities of DRC 1/k . For most of the rest of this paper, we will be working to compute the multiplicities of DRC 1/k .

Comparing the various double ramification cycles and loci
Recall from section 2 that we have various moduli spaces and double ramification loci (with associated cycles), which we summarise in the following diagram: In lemma 2.13 below we will show that that DRL m → M factors set-theoretically through normalisation of an open subscheme of M m . In section 5 we compute the lengths of the local rings of the subscheme DRL 1/k M m,1/k . Ultimately we want to show an equality of cycles DRC = H k g,m on M (recalling that DRC is by definition the pushforward of DRC ♦ to M), so we need to compare the cycles on these various spaces, and to compare the length with the intersection multiplicity. We begin with a general lemma.
Lemma 2.12. Let X f → Y → M be birational representable morphisms of reduced stacks, with f proper (here 'birational' means inducing isomorphisms between some dense open substacks). Suppose that the morphism σ extends to σ X : X → J and σ Y : Y → J (necessarily unique, by reducedness). Define DRL X X and DRL Y Y by pulling back the unit section of J along σ X , resp. σ Y , and assume that they have the expected codimension g.
Define DRC X and DRC Y as cycles supported on DRL X resp. DRL Y as in definition 2.3. Then f * DRC X = DRC Y , an equality of cycles on DRL Y .
Proof. We proceed as in the proof of [Hol19, theorem 6.7]. Namely, we have a commutative diagram (here the upward-pointing arrows are closed immersions, by separatedness of J , so we can also see them as cycles). Since f is proper and birational, we see that f J * [e X ] = [e Y ]. By the com-mutativity of proper pushforward and the refined Gysin homomorphism 8 , we see that Recall that by the discussion of section 1.8.2, the underlying (reduced) substack of M corresponding to H k g,m is the twisted differential space H k g (m).
Lemma 2.13. The maps DRL m → M and DRL 1/k → M factor set-theoretically via H k g (m) M.
Proof. This is clear from the description of the universal bundle in section 2.5 (noting that the same construction of the latter works on M m as on M m,1/k ).
Lemma 2.14. Proof. We give the proof for DRL 1/k ; the other case is almost identical. Recall that DRL 1/k is constructed by intersecting two sections in the universal jacobian over M m,1/k , and the latter is smooth over M m,1/k of relative dimension g. As such, every generic point of DRL 1/k has codimension at most g, since DRL 1/k can be cut out locally by g equations. The substack H k g (m) has pure codimension g in M by construction, so we are done by combining lemmas 2.13 and 2.14.
Combining lemmas 2.13 to 2.15 also yields Lemma 2.16. Every generic point of DRL 1/k and of DRL m lies over a generic point of H k g (m).
Lemma 2.17. Let p be a generic point in H k g (m). Then the multiplicity of the cycle DRC at p is equal to the sum of the lengths of the Artin local rings of DRL 1/k at (necessarily generic) points p in DRL 1/k lying over p.
This lemma is almost obvious from the definition of the proper pushforward, but we must take a little care as the map M m,1/k → M m is not in general proper, and we must compare the cycle-theoretic multiplicity with the length. Proof. Combining lemmas 2.14 and 2.15 shows that any point in DRL 1/k mapping to p must be a generic point.
Fix a (minimal) combinatorial chart containing p. From section 2.2, the twist I on the dual graph of C p determines an affine patch of M m,1/k whose image in M contains p. Now in the fibre of M m,1/k over that point, the universal line bundle runs over all possible ways of glueing the bundle on the partial normalisation from [FP16, definition 1] to a bundle on the curve C p itself.
In particular, one of those 'glueings' yields the bundle ω k (−mP ) ⊗ T itself, so DRL 1/k meets that fibre.
Proposition 2.19. Let p be a generic point of H k g (m) and let Γ be the dual graph of C p , such that Γ is a simple star graph. Then there are exactly k #E−#V out points of DRL 1/k mapping to p.
Proof. Recall that we have the following diagram of maps and inclusions For every point p ′ ∈ DRL m over a generic point p ∈ H k g (m), by lemma 2.5 there are k #E−#V out points of DRL 1/k mapping to p ′ . Thus it suffices to show that there is a unique point p ′ ∈ DRL m mapping to p.
By lemma 2.18 there is at least one such p ′ . On the other hand, fixing a combinatorial chart M ← U → A E , the spaces M So it is enough to check that for a general point p of a component Z of H k g (m) supported in the boundary, there is at most one positive twist I satisfying the twisted-differential condition. Let Γ be the generic dual graph of Z, which is a simple star graph.
Let (C, P ) be a general point of Z, so we have an identification of the components/nodes of C with the vertices/edges of Γ. By assumption there is some m i with m i < 0 or m i not divisible by k. Then the vertex v 0 carrying the marking p i must be the central vertex of the star graph. Thus from the abstract dual graph Γ we know uniquely which was the central vertex and which the outlying vertices v ∈ V out .
For each outlying vertex v we take the corresponding component C v ⊂ C together with its inherited markings and preimages of nodes. As explained in section 1.8.2 this is a generic point of some H 1 g(v) (µ ′ ) for a nonnegative partition µ ′ of 2g(v) − 2. In particular g(v) 1. If we knew all such µ ′ , this would allow us to reconstruct the positive twist I on Γ. But by lemma 2.20, knowing a general point C v of the space H 1 g(v) (µ ′ ) uniquely determines the partition µ ′ . Thus indeed the twist I is uniquely determined.
Together, the results of this section reduce the problem of computing the double ramification cycle to that of computing the length of its local ring at a generic point. This computation will occupy the remainder of the paper.

Relation to the construction of Marcus and Wise
The construction of Marcus and Wise [MW20] produces a stack Div g,m over M; we will explain how it is related to our M m . We will not need this in what follows, but we feel it may be useful to sketch the connection. We begin by outlining the construction of Div g,m . First, we define the tropical multiplicative group to be the functor on log schemes sending X to G trop m (X) = Γ(X, M gp X ), and a tropical line over a log scheme S to be a G trop m -torsor over S. Then Div is the stack in the strictétale topology on logarithmic schemes whose S-points are triples (C, P, α) where C is a logarithmic curve over S, P is a tropical line over S, and α : C → P is an S-morphism. We write Div g,m for the open substack where the underlying curve has genus g, the marked points are labelled 1, . . . , n and the outgoing slope at the marked point labelled i is given by m i . Now Div g,m comes with a natural forgetful map to the stack of all log curves, and we write Div st g,m for the pullback of the locus of stable log curves. It also comes with an 'abel-jacobi' map aj : Div g,a → Pic g,n , described in [MW20, §4]. Write Π for the fibrewise connected component in Pic g,n of the section ω k (− i m i p i ). We write Div st,Π g,m for the pullback of Π along aj to Div st g,m . Then unravelling the definitions yields a natural isomorphism of log stacks In particular, combining with lemma 2.12 yields Lemma 2.21. The double ramification cycle DRC ∈ A * (M) coincides with that constructed in [MW20].
3. The tangent space to M m,1/k at a simple star In this section we will give an explicit description of the tangent space T p M m,1/k . Sinceétale maps are isomorphisms on tangent spaces we will not distinguish between tangent spaces to a stack and tangent spaces to its charts. We will write T u M and T u A E in place of T f •u M and T g•u A E . The natural map M m,1/k → M induces a map T p M m,1/k → T u M, and we will describe its kernel and image.
An element of T u M m,1/k is given by a pointed map from the spectrum of K[t]/t 2 to M m,1/k .
Suppose we are given a pointed map from Spec K[t]/t 2 to M. For each edge e of Γ we denote by a e ∈ tK[t]/t 2 the image of ℓ e under the given map O M → K[t]/t 2 (see definition 1.10 for the ℓ e ; in particular, the choice of these coordinates means ℓ e is really well-defined, not only up to units). By the description given in section 2.2, the point p corresponds to giving an element (a γ ) p ∈ K × for every cycle γ, subject to some compatibility conditions. Then, to specify a vector in T p M m,1/k is to give an element a γ ∈ (K[t]/t 2 ) × for every cycle γ in Γ, lifting the element (a γ ) p ∈ K × , and subject to the relations (2.3.3).
This description is valid for graphs of any shape, but we now specialise to the case where Γ is a simple star graph (c.f. section 1.8.2), for which things become simpler. For each outlying vertex v write E v for the set of directed edges from the central vertex v 0 to v, and choose one edge e v ∈ E v . For e, e ′ ∈ E v distinct edges, let γ(e, e ′ ) be the directed cycle going out along e and back along e ′ . Then the cycles γ(e v , e ′ ) for v ∈ V out , e ′ ∈ E v \ {e v } yield a basis of the homology of Γ. In cases (1) and (2), the lifts of a γ from K × to (K[t]/t 2 ) × can be chosen completely freely for γ in a basis of the cycles between v 0 and v, and these determine all the other a γ . To see this, we can split the equations (2.3.3) into two types. We consider first those where the element f ∈ Z Υ corresponds to a trivial class in the homology of Γ. Then the resulting equation does not contain any instances of the a e , and simply imposes that product of the a γ is 1 (i.e. the group homomorphism from the free abelian group on cycles to (K[t]/t 2 ) × factors via the homology). On the other hand, if f ∈ Z Υ does not correspond to a trivial class in homology, then the resulting equation will have terms a e appearing on both sides (since twists are positive on outgoing edges). Since the a e lie in tK[t]/t 2 , any lift of (a γ ) p from K × to (K[t]/t 2 ) × will automatically satisfy these equations (as t 2 = 0). Summarising, we can choose the lifts a γ freely for γ in a basis of the homology, and the rest are uniquely determined.
In case (3)  Using this analysis, we easily determine the kernel and image of the map T p M m,1/k → T u M.
The kernel is given by the set of lifts of the zero tangent vector, i.e. all a e = 0 in the above discussion. Then it is clear that we are always in case (1), and the lift of a γ from K × to (K[t]/t 2 ) × can be freely chosen. The set of these choices yields a copy of K, and we see that the kernel of T p M m,1/k → T u M is given by H 1 (Γ, K).
To understand the image of T p M m,1/k → T u M we must distinguish carefully between the cases (1), (2) and (3). Since each a e ∈ tK[t]/t 2 , we know that a I ′ (e) e = 0 whenever I ′ (e) > 1. Thus, at a vertex v with at least one incident edge e satisfying I ′ (e) > 1, we can never be in case (2). Then we are in case (1) if and only if a e ′ = 0 for all e ′ at v with I ′ (e ′ ) = 1, and we can choose a e ∈ tK[t]/t 2 arbitrarily for all e with I ′ (e) > 1.
It remains to understand the contribution of the vertices v with all edges e satisfying I ′ (e) = 1. We define L v ⊆ e∈Ev K to be {0} if there exists an e ∈ E v with I ′ (e) > 1, and otherwise to be the set of tuples (l e ) e satisfying the linear equations (a γ(e,e ′ ) ) p l e ′ = l e for all e, e ′ ∈ E v .
Note that in this latter case, L v has dimension 1. Then the image of T p M m,1/k → T u M is given by Here the first factor corresponds to the locally trivial deformations, not smoothing the nodes. These correspond to setting a e = 0 for all e and thus can always be lifted by the analysis above.
The second factor to the free choices of a e ∈ tK[t]/t 2 ∼ = K for I ′ (e) > 0, and the last factor to the contribution from vertices with all incident edges e satisfying I ′ (e) = 1. This yields an exact sequence but we can write down a 'natural' splitting by setting a γ = (a γ ) p for all γ, so we get

The tangent space to the double ramification locus
In this section we will compute the tangent space to the double ramification cycle at the generic point of an irreducible component. Having in section 3 analysed the tangent space to M m,1/k and decomposed it into direct summands, we will begin by describing the Abel-Jacobi map on the tangent space. We will decompose the dual of the tangent map of the Abel-Jacobi map into four factors (section 4.1), and then in sections 4.2 to 4.5 compute the intersection of the kernels of these four maps. Along the way we will need a result on the non-vanishing of sums of kth roots of k-residues of k-differentials, which we state and prove in appendix C as it is somewhat disjoint from the rest of our story, and may be of some independent interest.

The Abel-Jacobi map on the tangent space
Assume we are in situation 2.9 with p lying in DRL 1/k , soσ(p) = e(p) ∈ J (K). Taking the difference of the induced maps T pσ and T p e on tangent spaces yields another map T p M m,1/k → The kernel of b consists of exactly those vectors on which T pσ and T p e agree, hence ker b = T p DRL 1/k . Dualising the natural exact sequence we obtain an exact sequence In the following we compute , where the latter equality holds by Serre duality.
To describe the proof strategy, note that from (3.0.3) we have a chosen isomorphism We write the restrictions of b to each subspace appearing as a direct summand as Elementary linear algebra yields In sections 4.2 to 4.5 we compute the intersection on the right. To describe the final result, note that for a stable curve C = C p with dual graph Γ there exists a natural inclusion v∈V (Γ) (4.1.1) taking differentials on the normalizations C v of the components of C and descending them to C. The image is exactly the space of differentials on C with vanishing residues at all nodes. As a second ingredient, recall from section 1.8.2 that for p a general point of a boundary component of DRL 1/k , the stable graph of C p is a simple star and on the components of C p corresponding to the outlying vertices, the twisted k-differential on C p is the kth power of a holomorphic abelian differential. To be more precise, recall that p lying in DRL 1/k means the line bundle ω k (−mP ) ⊗ T is trivial on C p . Let be a generating section (unique up to scaling). Recall from section 2.5 that the bundle T has a generating section 1 on the smooth part of C p . Let 1 v be the restriction of 1 to the components C v of C = C p , then ϕ 0 /1 v is a meromorphic section of ω k (−mP ) on C v . As described in section 1.8.2, for v an outlying vertex, this section is actually a holomorphic k-differential, which is moreover a kth power of a holomorphic differential. Denote one such choice of a kth root by ( ϕ 0 1v ) 1/k .
As a final piece of notation, let V >1 ⊂ V (Γ) denote the set of outlying vertices, such that at least one edge e incident to v has I ′ (e) > 1, and denote by V 1 the remaining outlying vertices. Then we can state the main result of this section (whose proof follows lemma 4.13).
Theorem 4.2. Let p be the generic point of a boundary component of DRL 1/k . Then the kernel of b ∨ inside H 0 (C p , ω) is given by the injection sending (c v ) v to the section given by 0 on the smooth locus of the central vertex, and c v ( ϕ 0 1v ) 1/k on the smooth locus of the outlying vertex v.
Theorem 4.3. In the situation of theorem 4.2, for the exact sequence the cokernel of b has dimension equal to #V >1 . The dimension of T p DRL 1/k is given by dim DRL 1/k plus the number of edges e in the star graph having I ′ (e) > 1.
Proof. The assertion about the cokernel of b comes from theorem 4.2, and the equality of dim ker(b ∨ ) with dim coker b. To compute the dimension of T p DRL 1/k we must first compute the dimension of T p M m,1/k ; following (3.0.3) it is given by where we use that L v has dimension 1 for v ∈ V 1 . Then H 0 (C v , ω Cv ) → H 0 (C, ω), as described in (4.1.1).
Proof. We start by recalling some generalities about line bundles on nodal curves. Let C be a nodal curve with dual graph Γ = (V, E). Choose some orientation for the edges e ∈ E such that we can uniquely identify source and target s(e), t(e) ∈ V of each edge. Moreover let (C v ) v∈V be the set of components of the normalization of C and for an edge e let n ′ (e) ∈ C s(e) , n ′′ (e) ∈ C t(e) be the preimages of the nodes corresponding to the edge.
Then a line bundle L on C is given by a collection of line bundles (L v ) v∈V on all components of its normalization together with identifications of the fibres (σ e : L s(e) | n ′ (e) ∼ − → L t(e) | n ′′ (e) ) e∈E of these line bundles at the pairs of points mapping to the same node. These identifications (σ e ) e∈E have a natural action by the group (G m ) E by componentwise multiplication. The set of such identifications is a torsor under this action. Moreover, different identifications can give the same line bundle: multiplying all the fibres on a given vertex v ∈ V by the same constant µ, i.e. going from (σ e ) e∈E to (σ e · µ δ v,t(e) −δ v,s(e) ) e∈E does not change the line bundle. Moreover, multiplying all fibres by the same constant does not even change the set of identifications (σ e ) e∈E . This means we have an effective action of the group G V m /G m on G E m which does not change the line bundle on C. Making suitable choices we can identify the quotient of G E m by G V m /G m with the torus T = Hom (H 1 (Γ, Z), G m ). Now let p ∈ M m,1/k with underlying stable curve C = C p ; this determines a kth root T 1/k of the correction bundle T on C as described in remark 2.10. In fact, in the local charts for M m,1/k , the additional coordinates a γ exactly parametrize the gluing data for T 1/k . Allowing T to act on the bundle T 1/k as described above yields a faithful action of T on the fibre of M m,1/k over C, whose orbit is open (it is exactly the fibre of M m,1/k I,U for the relevant weighting I at p and any neighbourhood U of p). Hence the summand H 1 (Γ, K) in T p ′ M m,1/k is canonically identified with the tangent space to T at 1. On the other hand, there is also an action of T on the jacobian of C, where elements of T act on line bundles on C in the way described above. However, we want to take the action obtained from this usual action by composing with the group morphism T → T, t → t k . With respect to this new action, the Abel-Jacobi map is equivariant. This follows since, on the fibre in M m,1/k over C ∈ M, the Abel-Jacobi map just sends T 1/k → ω k C (−mP ) ⊗ (T 1/k ) k . Hence the tangent map b Γ : H 1 (Γ, K) → H 1 (C, O C ) to the Abel-Jacobi map on this fibre is given by the tangent map for the action of T on the jacobian of C. One then verifies that for the map coming from the long exact sequence of 0 → O C → π * OC → K nodes → 0, whereC → C is the normalisation, we have b Γ = kψ. Thus, for computing kernels and cokernels we may as well work with the map ψ above. The kernel of b ∨ Γ is then equal to the left kernel of the Serre Duality pairing This left kernel is equal to the inclusion v H 0 (C v , ω Cv ) ⊂ H 0 (C, ω) from the statement of the lemma. To see that v H 0 (C v , ω Cv ) is contained in the left kernel, note that the cocycles in the image of ψ are represented by constant functions on the overlaps of theČech cover, so multiplying them with holomorphic differentials from v H 0 (C v , ω Cv ) does not produce poles. Hence the residue pairing indeed vanishes. However, note that the map ψ is injective from the long exact sequence we used to define it. Thus, the dimension of the left kernel is

Though we will not need it in what follows, we mention
Lemma 4.5. The projection map T p DRL 1/k → T p M is injective.
Proof. We have T p DRL 1/k ⊂ T p M m,1/k → T p M and the kernel of T p M m,1/k → T p M is given by H 1 (Γ, K). On the other hand, T p DRL 1/k is the kernel of b, the differential of the Abel-Jacobi map, and the restriction of b to H 1 (Γ, K) is b Γ = kψ. So an element of the kernel of T p DRL 1/k → T p M is an element of the kernel of ψ and ψ is injective.

Ω
We put ourselves in situation 2.9 with p a K-point of DRL 1/k . As described in section 4.1, we can choose a generating section We will use the description in appendix A to compute explicitly the map Recall that for covers of C p as in situation 2.9, all non-trivial intersections of charts map to the smooth locus of C p /K, and the line bundle T comes with a trivialisation on that smooth locus, described by a generating section 1.
Lemma 4.6. Working inČech cohomology for a cover of C p as in situation 2.9, the map is induced by the map For an interpretation of this formula, note that on the smooth locus of C p we can interpret the g ij as tangent fields on the overlaps U ij of the cover. Thus the pairing g k ij ϕ 0 1 of g k ij with the meromorphic differential ϕ 0 1 makes sense as a meromorphic function. Applying the external derivative d gives a meromorphic differential, which we again pair with g ij to obtain a meromorphic function. Given the simple nature of the formula, it feels as if there should be a simple conceptual proof of this lemma. But our description of the line bundle T was somewhat ad-hoc, making it necessary to keep careful track of all the glueing data. Since this bookkeeping is quite subtle, we have written the proof out in a painful amount of detail.
Proof. To prove the statement, we make (more) explicit the calculations from appendix A. Our short exact sequence 0 and we represent an element g ∈Ȟ 1 (C p , Ω ∨ (−P )) as a cocycle (g ij ) ij on the cover {V i } i of C p .
Then ϕ 0 is a generating section of the line bundle L K = ω k (−mP ) ⊗ T .
Recall that g encodes a locally trivial deformation C A ′ /A ′ of C p . To obtain it, define U i = V i × K K[t]/t 2 , then we can choose isomorphisms U ij → V ij × K K[t]/t 2 such that the inclusion f i : U ij → U i is just the base-change of V ij → V i to K[t]/t 2 , and the inclusion f j : U ij → U j is given by applying Spec to the ring homomorphism r → r + tg ij (dr). Glueing the U i together along the U ij yields the desired locally trivial deformation C A ′ /A ′ .
Recall from appendix A that, to compute the image of g inȞ 1 (C p , ω k (−mP ) ⊗ T ), we should consider the line bundle L A ′ = ω k C A ′ (−mP ) ⊗ T , choose generating sections ϕ i on U i , ϕ j on U j , then pull them both back to U ij and compare. It will be very important to ensure that the pullback is performed in a functorial way, so we can effectively compare these pullbacks.
For the first tensor factor of L A ′ , we identify f * i (ω C A ′ | U i ) with ω U ij via the differential df i (and similarly with df j ). The fact that the differential is naturally functorial later ensures that this gives compatible identifications.
The second factor T is, in a sense, more tricky, because we defined T in terms of a cover with gluing maps and thus we need to be extra careful how to identify pullbacks of T under various compositions of maps. First, to fix terminology we recall the following very standard description of the pullback of a line bundle given by glueing data: Digression (Pullbacks of line bundles) Let f : X → Y be a morphism of schemes and Y = i U i an open cover. Assume a line bundle L on Y is given by fixing Giving a section s of L on Y means giving local sections Then the pullback of L under f is given by the cover X = i f −1 (U i ) with new generating sections 1 f −1 (U i ) and gluing functions η ij = f * ρ ij = ρ ij • f . The pullback of the section s is specified by the local sections (s i • f ) · 1 f −1 (U i ) .

End digression.
Now for the line bundle T on C A ′ we use the cover by the U i above. For smooth U i we have a trivializing section 1, which we here call 1 U i to be more precise. For the singular chart U j associated to an edge e = {h, h ′ } we have the trivializing section τ e . The transition functions were 1 between two smooth charts, and for a smooth chart U i and a singular chart U j , the transition function is τ e = λ(e) k z is a chart on an outlying vertex and τ e = (z h ′ ) I(h ′ ) 1 U i if U i is a chart on the central vertex. Since the formula uses the coordinates z h , z h ′ on U j , we implicitly identify U ij as a subset of U j here (since on U i the expression z h has no meaning). Note that overlaps between singular charts are already contained within the overlaps of smooth and singular charts, so the values of the cocycles on these patches are uniquely determined by those we have already listed. Now we proceed to choose generating sections ϕ i on U i , ϕ j on U j , pull them both back to U ij and compare. We consider the case when U i is a smooth chart and U j is a singular chart on an outlying vertex. Denote by π i : U i → V i and π j : U j → V j the natural projections. Also, denote by 1 V i and τ e,V j the trivializing sections of T | Cp on V i , V j . Then the section ϕ 0 /1 V i is a generating section of ω k V i (−mP ) on V i and soφ i := π * i (ϕ 0 /1 V i ) is a generating section of ω k U i (−mP ) on U i . Denoting by 1 U i the trivializing section of T | U i , we choose ϕ i :=φ i ⊗ 1 U i as the generating section of ω k (−mP ) ⊗ T on U i .
For U j something different happens: on V j we have that ϕ 0 /τ e,V j is a generating section of ω k V j (−mP ). Denote byφ j := π * j (ϕ 0 /τ e,V j ) the section of ω k U j (−mP ) and by τ e,U j the section of T on U j , then ϕ j :=φ j ⊗ τ e,U j is our chosen generating section of ω k (−mP ) ⊗ T on U j .
On the other hand, f j is the composition of the automorphism Ψ g ij : U ij → U ij (obtained as Spec of the ring map r → r + tg ij (dr)) with the inclusion U ij ⊂ U j . But now note that when restricting ϕ j to U ij we obtain However, by the original gluing data of T we have The crucial thing to observe is that the transition function λ(e) k z is actually a pullback from V ij (it 'does not contain the variable t'). Denote by π ij : U ij → V ij the projection. Using again thatφ j = π * j (ϕ 0 /τ e,V j ), we make a quick sanity check: To conclude, by lemma A.4 the class inȞ 1 (C p , ω k (−mP ) ⊗ T ) that we seek is given by Using lemma 4.7 below, we see . This allows us to conclude pulling back along the 'multiplication by t' isomorphism from A to J yields the element To translate this to an element of Lemma 4.7. Let B be an A ′ -algebra, and g : Ω B K /K → B K a B K -linear map. Define f : B → B; r → r + tg(d(r| B K )), where we use that the map B K → B, u → t · u is well-defined.
Then the map f Ω : Ω k B/A ′ → Ω k B/A ′ induced by the differential of f is given by f Ω (w) = w + tg 1−k d(g k (w| B K )), where we use that Ω k Proof. We prove the result for local generators w = r 0 dr 1 · · · dr k of Ω k B . In the computation below we implicitly use that tdr j = t(dr j | B K ) and that the restriction to the fibre B K commutes with taking differentials. We then obtain Note that in the computation we used a natural extension of the differential d to tensor powers of the tangent sheaf of B, when we apply it to g and g k .

Recall that
, ω)) denotes the map induced by applying Serre Duality to b Ω . Our next goal is to give an explicit formula for this map; or rather, for its restriction to ker , since this is all we need later.
We define a map of coherent sheaves on the smooth locus C sm p : Note in particular that this map makes sense at markings of C p : while the differential d log((ϕ 0 /1) −1 s k ) can have a pole of order 1 at a marking, we have that f is a local section of Ω ∨ (−P ), so a vector field vanishing at the marking. This cancels the possible pole.
Inside H 0 (C p , ω) there is the subspace ⊕ v H 0 (C v , ω Cv ) of global sections of ω with vanishing residues at all nodes. The following result shows that it makes sense to apply β sm to elements in this subspace.
, ω) on C sm p extends uniquely to all of C p and we thus obtain a map Proof. What we need to check is that for every local section f of Ω ∨ (−P ) around a node of C p , the section of ω on C sm p extends over the node. In other words, for each branch of the node we need to show that this differential has at most a simple pole at the node and that the residues at both sides of the node add to zero.
For this, we make the following observation about f : workingétale-locally we may assume a neighbourhood of the node is given by the spectrum of R = K[x, y]/(xy), so that Ω R/K = R dx,dy xdy+ydx . Since f is O Cp -linear we have xf (dy) + yf (dx) = 0, and hence f (dx) (resp. f (dy)) is divisible by x (resp. by y). Thus, on both branches of the node, we can regard f as a tangent field vanishing to order at least 1 at the node.
Then, as before, the term d log((ϕ 0 /1) −1 s k ) has at most a pole of order 1, cancelling with the zero of f . So in fact the differential (4.3.1) is regular at the nodes (it does not even have simple poles) and in particular the residues vanish on both sides and thus add to zero.
where −, − denotes the respective Serre-Duality pairings on both sides. Inserting the formulas above, we obtain Thus difference of the two sides of the equality has the form dη for the element η = (−ksg ij ) ij ∈ H 1 (C p , O Cp ) and thus is zero. Here we use again that on the overlaps of our cover, we can pair the differential s and the sections g ij of Ω ∨ (−P ) to obtain a local section of O Cp . The computation above is taken from Mondello's unpublished note [Mon].
Recall from section 4.1, that on the components of C p corresponding to outlying vertices, we could choose kth roots ( ϕ 0 1v ) 1/k of the twisted differential, all unique up to scaling. Lemma 4.10. The kernel of b ∨ Ω ⊕ b ∨ Γ is given by the map sending (c v ) v to the section given by 0 on the smooth locus of the component C v 0 of the central vertex v 0 , and c v ( ϕ 0 1v ) 1/k on the smooth locus of the component C v for the outlying vertices v.
Note that while the sections ( ϕ 0 1v ) 1/k are only unique up to scaling, the image of the map (4.3.2) is independent of these choices.

Proof. Since every element of the kernel of
Γ , it suffices to know the description of b ∨ Ω on this subspace. On the smooth locus we have Hom(Ω ∨ (−P ), ω) = ω(P ), and on this locus b ∨ Ω (s) = sd log(s k (ϕ 0 /1) −1 ). This vanishes iff s k (ϕ 0 /1) −1 is a locally constant function. In other words, up to scaling, s should be a kth root of ϕ 0 /1.
On the central vertex, this implies s| Cv 0 = 0. Indeed, by assumption there is a marking i with m i < 0 or m i not divisible by k. By the discussion in section 1.8.2 this marking must be on the central vertex v 0 , hence there cannot be a holomorphic differential s with kth power ϕ 0 /1. On the other hand, on the outlying vertices there do exist the sections ( ϕ 0 1v ) 1/k , so the kernel of b ∨ Ω is exactly given by the map (4.3.2).
4.4 Computing the residue pairing with an element of the kernel of b ∨ Ω ⊕ b ∨ Γ In section 4.5 and in lemma 5.5 it will be important to compute the value of the residue pairing between an element of the kernel of b ∨ Ω ⊕ b ∨ Γ and a particular element δ of H 1 (C, ω k (−mP ) ⊗ T ). To avoid duplication we will carry out this computation here, in sufficient generality for both our applications. Let be a short exact sequence of K-modules, where A and A ′ have the structure of (Artin local) K-algebras, the map A ′ → A is a K-algebra homomorphism, and Suppose we are given a stable marked curve C A ′ over A ′ , whose fibre over m A ′ is our curve C p .
We consider a formallyétale fpqc cover of C A ′ consisting of charts and gluing morphisms as follows: Since U v is smooth and affine 9 , it is (non-canonically) a trivial deformation of its central fibre Note that in particular, U v is the complement of finitely many A ′ points of the smooth projective curve (ii) for each node (corresponding to an edge e of Γ), a formallyétale neighbourhood of the form Note the following important assumption about the smoothing parameters ℓ e : We assume throughout this section that ℓ This will be the case in our applications and in general for first order deformations, i.e. those over A ′ = K[t]/(t 2 ). Indeed, it is always true that ℓ e ∈ J = tK[t]/(t 2 ) since otherwise the chart U e is not nodal. Now assume C A lies in the double ramification locus, so we can choose a generating section ϕ 0 for the line bundle ω k C A (−mP ) ⊗ T . For each of the charts U e we choose a lift ϕ e of ϕ 0 , and for each U v a lift ϕ v . The differences ϕ * − ϕ * ′ on the overlaps restrict to 0 over A, hence give an element δ ′ of H 1 (C p , ω k Cp (−mP ) ⊗ T ) ⊗ K J (see appendix A). Using the isomorphism ω k Cp (−mP ) ⊗ T ∼ = O Cp via ϕ 0 | Cp , we can convert this into an element δ ∈ H 1 (C p , O Cp ) Now let c = (c v ( ϕ 0 1v ) 1/k ) v be an element of the kernel of b ∨ Ω ⊕ b ∨ Γ (recall that we fixed the k-th root of ϕ 0 /1 v on the outlying vertices v). Then the product δc lies in H 1 (C p , ω Cp ) ⊗ K J, which is isomorphic via the residue pairing to J. In this section we make the image of δc in J explicit.
Recall from remark 2.10 that we have a canonically-defined root T 1/k of T , and on the smooth charts U v a generating section 1 1/k v of T 1/k which is a kth root of 1 v . Multiplying with the section (ϕ 0 /1 v ) 1/k , we obtain a well-defined k-th root of ϕ 0 on the outlying components, which extends to the nodes. Now, we should not expect this to extend to a root over the whole of the central component. However, since ϕ 0 is a generating section of the line bundle ω k (−mP ) ⊗ T , this root will extend uniquely to a k-th root ϕ where q e is the point on C v 0 corresponding to the edge e. Thus when taking the kth root defined locally around the nodes we see that ϕ where γ(e) is the cycle in Γ going out along the distinguished edge e v : v 0 → v and back along e (in particular a γ(ev ) = 1).
Some remarks on the statement, before giving the proof: -This root ϕ 1/k 0 (defined above) only makes sense locally around nodes, but this is all we need for (4.4.1) to make sense.
-We emphasise that in (4.4.1), the residue is taken on the central component C v 0 of C.
-Recall that we assume throughout this section that ℓ Proof. We compute the residue one-point-at-a-time, as in appendix B.
Case 1: smooth points. On the smooth points of C v 0 for the central vertex v 0 , the section coming from c vanishes. On the outlying vertices v, the sections ϕ v extending ϕ 0 do not have a pole. Hence in both cases the residue vanishes.
Case 2: nodes. Here there is only one possible choice of patch. But to compute the residue we have to sum the residues coming from the two preimages of our point under the normalisation map π :C p → C p . 10 Here we use that the marked points P are disjoint from the smallétale neighbourhood of the nodes Case 2.1: lift to the central vertex. Here again the section coming from c is zero, hence the residue vanishes.
Case 2.2: lift to an outlying vertex. Let q be the chosen point on the outlying component C v mapping to a node. Our strategy will be as follows: as described above, the overlap of the singular chart and the smooth chart on the outlying vertex is given by sitting inside the singular chart U e = Spec R ′ via the ring map where the branch z h = 0 corresponds to the central vertex. On the other hand, since C v is smooth at q, we can take the inclusion D → U v to be the product of a small punctured formallyétale We will compute the difference ϕ e − ϕ v on D and, by the deformation theory in the appendix, this gives an element Recall that the collection of such differences exactly describes a cocycle representing the element Then we can take the residue of this at t = 0 and obtain the contribution to the pairing from the node q.
Now instead of abstractly using that ϕ e −ϕ v ∈ H 0 (D K , ω k D K ⊗T | D K )⊗ K J and multiplying with the section c v (ϕ 0 ) −1+1/k 1 −1/k v defined over K, we can instead use any section ρ ∈ H 0 (D, ω −k+1 ⊗ T −1+1/k ) such that ρ restricts to c v (ϕ 0 ) −1+1/k over K and compute the residues of ρ(ϕ e −ϕ v )/1 We now propose a particular choice of ρ: by lemma 4.12 below there is a unique section ϕ 1/k e of ω ⊗ T 1/k on U e such that (ϕ 1/k e ) k = ϕ e and such that ϕ 1/k e | A = ϕ 1/k 0 . We choose the section ρ = c v (ϕ 1/k e ) −k+1 , which obviously has the properties mentioned above. Thus the value of the pairing at q is given by (lying in J ⊂ A ′ ) and we will see that our choice of ρ allows us to compute the residues from the terms involving ϕ e and ϕ v separately.
For the first term we observe that (ϕ 1/k e ) −k+1 ϕ e = ϕ 1/k e is (the restriction to D of) a section of ω ⊗ T 1/k on U e . Thus on U e it has a representation Pulling this back to D (and using the gluing maps of the sections of T 1/k ) gives us Dividing by 1 1/k v and Taylor expanding yields Here we use that λ(e) = a γ(e) is a valid choice according to the construction presented in section 2.5 (for the spanning tree Γ ′ ⊂ Γ we choose the tree formed by the distinguished edges e v mentioned in lemma 4.11).
To compute the residue at t = 0 we look at the terms whose order in t is exactly −1. Sinceφ e is a power series in z h , z ′ h , this forces i I ′ (e). On the other hand, by our assumption l I ′ (e) e ∈ J and also l e ∈ m A ′ , so for i > I ′ (e) we have l i e = l i−I ′ (e) e l I ′ (e) e = 0 since m A ′ · J = 0. Thus all terms for i > I ′ (e) vanish and hence the only term of the above sum making a possibly-non-zero contribution to the residue occurs when i = I ′ (e), and the residue is given by This finishes the computation of the residue for the term of (4.4.2) involving ϕ e . Now we want to argue that the sum (over the nodes q connecting C v to C v 0 ) of the residues of the terms (ϕ For this, we now choose some splitting of the short exact sequence 0 → J → A ′ → A → 0 of K-vector spaces, allowing us to write A ′ = A ⊕ J in some non-canonical way. Then on D we can write where as before ϕ −1+1/k 0 and ϕ 0 are sections of the base-changes of ω −k+1 ⊗ T −1+1/k and ω k ⊗ T on D to A (i.e. ϕ .), and R 1 , R 2 are corresponding sections which vanish modulo J (i.e. there exist representatives with coefficients in J). We note here that since Using that J 2 = 0 we can write out the product above and obtain three terms By assumption, the first term ϕ 1/k 0 /1 1/k v has order I ′ (e)− 1 0 at t = 0, so it does not contribute a residue. On the other hand, for the third term we observe that R 1 is the restriction to D of a section R 1 on U e killed by J (the 'J-valued' part of ϕ −1+1/k e ). If we write ] then restricting to D we have but ψ(t, le t ) = ψ(t, 0) since ψ has coefficients in J and since l e ∈ m A ′ so that l e · J = 0. Looking at ϕ 0 we know . Combining the two terms (and using I(h) = I ′ (h) · k) we obtain so the order of this term at t = 0 is nonnegative and hence the residue vanishes.
Finally we look at the remaining term ϕ −1+1/k 0 R 2 /1 1/k v . Now using lemma 4.12 we can find a section ϕ Since R 2 is killed by J, this means that ϕ −1+1/k 0 Using that R 2 is the restriction of the k-differential R 2 , we see that ϕ −1+1/k 0 is the complement of finitely many A ′ -points, we can see this as a meromorphic differential of the smooth projective curve is just the residue of this differential at the A ′ -point corresponding to the node q. Then, by the Residue theorem for curves over Artin rings ([Con00,Appendix B.1.,page 272]), the sum of all residues at nodes q connecting C v to C v 0 is zero.
Summarising the above, we see that the image of δc in J under the residue pairing is given by where z h ′ is a local coordinate on the central vertex for the node.
Since l I ′ (e) e ∈ J and m A ′ J = 0 we see that in the formula we can replaceφ e by its restrictioñ ϕ 0 to K. Looking back at the definition ofφ e , its restrictionφ 0 over K satisfies that on U e × A ′ K we have

From this it is clear that the expression
is the residue of the differential ϕ In this lemma our assumption of characteristic zero (or more precisely that k is invertible on A) is essential.
Proof. Let us first discuss uniqueness. The condition l ′ + JM ′ = l means l ′ is unique up to an element of JM ′ . For a different l ′′ = l ′ + jm we have (l ′′ ) ⊗k = l ′⊗k + kj(l ′ ) ⊗k−1m (here we use M ′ being locally free so we can commute tensor products). The fact that m ′ is generating implies that l ′ is generating. Thus (l ′′ ) ⊗k = m ′ = l ′⊗k is only possible for kjm = 0. Since k is invertible in A ′ , this implies jm = 0 hence l ′′ = l ′ .
By the uniqueness part, we can work locally and so assume that M ′ is free, so take M ′ = R ′ , M = R (and identify M ⊗k = R etc also). Choose any liftl of l to R ′ , and define ǫ =l k −m ′ ∈ JR ′ . Now l generates R since m ′ generates R ′ , and k is invertible, so there is a unique j ∈ J such that kl k−1 j = ǫ (recall Jm A ′ = 0). Then

Intersecting with the kernels of b ∨
Lv and b ∨

>1
Recall from section 4.1 the maps b >1 : The goal of this section is to prove the following lemma, which describes exactly the intersection of the kernels of b ∨ Ω , b ∨ Γ , b ∨ >1 and b ∨ Lv (c.f. lemma 4.1). It immediately allows us to conclude the proof of theorem 4.2.
But recall that b >1 and b Lv come from tangent maps of the Abel-Jacobi section for first-order deformations of C p locally smoothing various nodes of C p . This is exactly the situation analysed in lemma 4.11. To be more precise, we apply lemma 4.11 for A = K, A ′ = K[t]/t 2 , and J = tK.
Then from the definition of b ∨ >1 and b ∨ Lv we see that the local smoothing parameters ℓ e are exactly ℓ e = a e t ∈ J and we have We check that the pairing of c with δ vanishes for all choices of a iff c v = 0 for all v ∈ V 1 .
First, we claim that b ∨ >1 vanishes on the kernel of b ∨ Ω ⊕ b ∨ Γ . Indeed, in the formula (4.4.1) from lemma 4.11, all terms ℓ Hence to prove that the element (4.5.1) is nonzero for c v , a ev = 0, we need to show that e:v 0 →v But the summands above are just some choice of kth roots of k-residues 11 for the meromorphic k-differential ϕ 0 | Cv 0 /1 v 0 on C v 0 . Since p was generically chosen in its component, we have that C v 0 is generic in a suitable stratum of meromorphic k-differentials. Then the non-vanishing of (4.5.2) follows from theorem C.1.

The length of the double ramification cycle
In this section we will compute the length of the Artin local ring obtained by localising DRL For a generic linear subspace H in A N through our chosen point p ∈ DRL 1/k , where H has codimension 2g − 3 + n, we denote DRL ′ = (DRL 1/k ∩H) p the intersection of DRL 1/k with H, localized at our point p. Since dim DRL 1/k = 2g − 3 + n, this is an Artin-local K scheme with residue field p sufficiently generically chosen in that component. induces a map DRL ′ → A E . Our goal for this section is to prove the following Theorem 5.1. The map DRL ′ → A E is a closed immersion, with image cut out by the ideal (ℓ I ′ (e) e : e ∈ E).
An immediate corollary of this theorem is that the length of DRL 1/k at p is given by e I ′ (e). We will deduce the theorem from the next lemma, for which we need a little notation. Set : e ∈ E) has a section, so in particular it is surjective on tangent spaces. Since the tangent spaces have the same dimension by theorem 4.3, the map is necessarily bijective on tangent spaces. Now, since DRL ′ and A E are evidently (locally) of finite presentation over K, by [Sta13, Tag 00UV (8)] injectivity of the tangent map implies that DRL ′ → A E is unramified. Since being a closed immersion isétale-local on the target, we conclude that DRL ′ → A E is a closed immersion by applying [Sta13, Tag 00UY] together with the fact that the source is Artin local.
It is then clear from another application of lemma 5.2 that the image is cut out exactly by (ℓ I ′ (e) E : e ∈ E).
We want to apply deformation theory to prove lemma 5.2, but the kernel of B → K; ℓ e → 0 is not necessarily killed by m. So we decompose it into steps. For every integer r 2 we have a short exact sequence Fixing some r 2, denote the non-zero terms in the above sequence by J, A ′ and A respectively (remembering the R-algebra structures of the latter two), then we have a surjection A ′ → A of Artin local K-algebras with residue field K, and the kernel J is killed by the maximal ideal of A ′ . Suppose we are given an A-point of DRL ′ . We want to understand when this lifts to an A ′point of DRL ′ (again, as a map over Spec R). We say an R-algebra B is I-constrained if for every edge e the element ℓ I ′ (e) e maps to zero in B. Note that K = R/m is automatically I-constrained. Lemma 5.4. If A ′ is I-constrained then the pseudotorsor M (A, A ′ ) is a torsor (i.e. is nonempty).
Proof. Firstly, since by assumption the map U → A E from our combinatorial chart is smooth, we can always lift the ) we must substitute for a e the image of l e in A ′ . Since A ′ is I-constrained and since all exponents M e are divisible by I ′ (e) it turns out that all of the defining equations, except for a ′ γ a ′ i(γ) = 1, become trivial. So indeed we can choose any lift (a ′ γ ) γ∈Υ of (a γ ) γ∈Υ satisfying these equations. such that µ + δ : Spec A ′ → M m,1/k lands in DRL 1/k .
Proof of lemma 5.3 assuming lemma 5.5. If M (A, A ′ ) is empty then by lemma 5.4 we see that A ′ is not I-constrained, and clearly no lift to an A ′ -point of DRL ′ exists. Hence we may as well assume M (A, A ′ ) to be non-empty, hence a torsor under H 1 (C p , Ω ∨ (−P ) ⊗ J) ⊕ H 1 (Γ, J). If A ′ is not I-constrained then lemma 5.5 shows that no element of M (A, A ′ ) lands in DRL 1/k . If A ′ is I-constrained then by lemma 5.5 there exists an element µ ′ ∈ M (A, A ′ ) which lands in DRL 1/k . To finish the proof we now need to show that given an element in M (A, A ′ ) contained in DRL 1/k we can construct another element of M (A, A ′ ) which is also contained in DRL ′ .
Recall that we have anétale coordinate chart with a generic linear subspace H through the origin of codimension 2g − 3 + n (and localizing at p). Denote W = T 0 DRL 1/k red ⊂ T 0 A N the tangent space to the reduced double ramification cycle, which we consider as a linear subspace of A N . Recall that since DRL 1/k has dimension 2g − 3 + n, the space W also has dimension 2g − 3 + n. As H was assumed generic and the two linear subspaces W, H are of complementary dimensions, there exists a linear projection h : A N → W with h| W = id W and h −1 (0) = H. For this to lie in DRL ′ = (DRL 1/k ∩H) p we want that the composition ϕ : Spec(A ′ ) → M m,1/k I,U h − → W is zero. We know this is true on Spec(A), since Spec(A) factored through DRL ′ . Thus the difference between ϕ and the zero map is an element ǫ ∈ (T 0 W ) ⊗ J. But note we can shift our map Spec(A ′ ) → DRL 1/k ⊂ M m,1/k I,U around by elements of (T 0 W ) ⊗ J ⊂ (T 0 M m,1/k I,U ) ⊗ J. Indeed, tangent vectors to the reduced DRL 1/k -component are locally trivial deformations (the reduced DRL 1/k -component is contained in the preimage of the boundary of M), so the shift by W ⊗ J does not change the map to A E we want to lift. Also, clearly it does not change the composition with the Abel-Jacobi map, so we stay in DRL 1/k . But note that the tangent map h is the identity, since h was assumed to restrict to the identity of W . So indeed, we can shift our map Spec(A ′ ) → DRL 1/k by −ǫ ∈ (T 0 W ) ⊗ J, to obtain an A ′ -point of DRL ′ .
Proof of lemma 5.5. The sections e andσ of the universal jacobian J induce a map Φ : M (A, A ′ ) → T e J p ⊗ K J, which is a pseudotorsor under the map Choose an element µ ∈ M (A, A ′ ). We need to decide when there exists δ ∈ H 1 (C p , Ω ∨ (−p) ⊗ J) ⊕ H 1 (Γ, J) such that Φ(µ + δ) = 0, i.e. such that µ + δ : Spec A ′ → M m,1/k I,U lands in DRL 1/k . Now the existence of such a δ is equivalent to Φ(µ) mapping to zero in the cokernel of α, which is in turn equivalent to Φ(µ) pairing to zero with the kernel of α ∨ . Hence we are reduced to showing (*) A ′ is I-constrained if and only if Φ(µ) pairs to zero with the kernel of α ∨ .
By lemma 4.10 the kernel of α ∨ is given by the injection sending c = (c v ) v to the section given by 0 on the smooth locus of C v 0 for the central vertex v 0 , and c v ( ϕ 0 1v ) 1/k on the smooth locus of C v for the outlying vertices v. By lemma 4.11 (where the notation is also defined), the image of Φ(µ)c in J under the residue pairing is given by with a γ(e) invertible in A ′ , it follows ℓ I ′ (e) e = 0 ∈ A ′ for all e, finishing the proof.

Concluding the proof of the formula
Finally, we conclude the proof of the equality of the double ramification cycle DRC and the cycle H k g,m of Janda, Pandharipande, Pixton, Zvonkine and the second-named author (section 1.8.2) in the Chow ring A g (M).
Finally we can easily deduce the main result of this paper, that DRC = H k g,m .
Proof of theorem 1.2. Let p be the general point of some component of H k g (m). If p lies in the interior of the moduli space (i.e. C p is smooth), then DRC has multiplicity 1 at p, agreeing with its multiplicity in H k g,m . This multiplicity follows from the computation in [Sch18, Proposition 1.2].
Thus we can assume that C p lies in the boundary with associated simple star graph Γ and positive twist I. Combining lemma 2.16 and lemma 2.17, it suffices to show that the sum of the lengths of the Artin local rings of DRL 1/k at points lying over p is given by the formula e∈E(Γ) I(e) k #V (Γ)+1 (c.f. (1.1.2)). By proposition 2.19 there are exactly k #E−#V −1 of these points and by theorem 5.1 the multiplicity of DRL 1/k at each of them is e∈E I ′ (e). Using I ′ (e) = I(e)/k the result is then immediate.

Presentation of the local rings of the double ramification locus
Our strategy for computing the multiplicities of the double ramification cycle was somewhat indirect, as we began section 5 by slicing with a generic hyperplane. In this section we do a little gentle bootstrapping to extract a presentation for the local rings of DRL 1/k itself at generic points. We begin by resuming the notation from the start of section 5, and write η for the generic point of DRL 1/k containing the point p in its closure. The local ring O DRL 1/k ,η admits (by the Cohen Structure Theorem [Sta13, Tag 0323]) a non-canonical structure as an algebra over κ(η), the residue field of η (compatible with the K-algebra structure). In this section we show Theorem 5.6. The κ(η)-algebra structure on O DRL 1/k ,η can be chosen so that : e ∈ E) . (5.2.1) We are very close to proving this in theorem 5.1, except that latter concerns DRL ′ , which is obtained from DRL 1/k by cutting with a generic linear subspace (see the discussion at the beginning of section 5). Noting that we can work over any field of characteristic zero, we can in particular base-change the whole setup to the residue field κ(η). The gap is then filled by the following lemma.
Lemma 5.7. Let Z A N K be an irreducible closed subscheme of dimension d, with generic point η. Write ∆ : η → η × K η for the diagonal (see the diagram below), and let H ⊆ A N η be a generic linear subspace of codimension d through ∆ (the latter viewed as a point of A N η ). Then there exists an isomorphism of complete K-algebras (5.2.2) be a short exact sequence of K-modules, where A and A ′ have the structure of (Artin local) K-algebras, the map A ′ → A is a K-algebra homomorphism, and Jm A ′ = 0 where m A ′ is the maximal ideal of A ′ . This generality will only be needed in section 5; for computations of tangent spaces, it is enough to look at the special case Before we start with deformation theory, we need to introduce some technical results, which we use later.
Remark A.1. Now J is an A ′ -module and A ′ a K-algebra, hence J is also a K-module. Writinḡ a ∈ K for the reduction of a ∈ A ′ we see thatāj = aj for all j ∈ J, since the difference between a and the image ofā in A ′ lies in m A ′ , and m A ′ J = 0.
Lemma A.2. Let M be an A ′ -module, then the K-bilinear map is in fact A ′ -bilinear, and the induced map Proof. This is a special case of Bourbaki, Algebra I, Chapter 2, paragraph 3.6 (p.254).
We will apply the following well-known lemma to the jacobian of the universal curve.
Lemma A.3. Let X be a K-scheme and η : Spec A → X a map over K with image point q ∈ X. Denote Then M (η, A ′ ) is naturally a pseudotorsor under T X,q ⊗ K J. Moreover, for g : X → Y a morphism of K-schemes, the natural map M (η, A ′ ) → M (g • η, A ′ ) is a pseudotorsor under the natural map Let C A ′ → Spec A ′ be a family of stable curves (i.e. a map Spec A ′ → M) and L A ′ a line bundle on C A ′ . Assume that the restriction L A of L A ′ to the fibre C A = C A ′ × A ′ A is trivial, with a trivialising section ϕ 0 ∈ H 0 (C A , L A ).
In particular, this implies that L A ′ has multidegree zero, so the line bundles L A ′ and O C A ′ induce maps σ 1 , σ 2 : Spec A ′ → J into the universal Jacobian J → M. By the assumption L A ∼ = O C A , the restrictions η : Spec A → Spec A ′ → J of these maps to Spec A agree. Thus both give elements 12 in M (η, A ′ ), in the notation of lemma A.3, with X = J . Furthermore, their compositions with the projection J → M to the moduli space of curves agree (on all of Spec A ′ ), since for both σ 1 , σ 2 the underlying family of curves is C A ′ .
Let C = C A ′ × A ′ K, then by lemma A.3 the set M (η, A ′ ) is a pseudotorsor under the group T (C,O) J ⊗ K J, so the difference of σ 1 , σ 2 gives a unique element δ ∈ T (C,O) J ⊗ K J. Furthermore, it must lie in the kernel of the map 12 J is onlyétale-locally a scheme, but this is enough for these infinitesimal considerations.
which is exactly T e J C ⊗ J = H 1 (C, O C K ) ⊗ J = H 1 (C, L K ) ⊗ J, where J C is the Jacobian of C, and the last isomorphism is via the restriction ϕ 0 | C of ϕ 0 to the fibre over K.
Our goal here is to describe how to obtain this element δ ∈ H 1 (C, L K ) ⊗ J usingČech cohomology for a suitable cover U = (U i ) i of C A ′ .
Suppose there exists U = {U i } i∈I an fpqc cover of C A ′ by affines such that for every i there exists a section We fix a cover U and sections ϕ i as above. For the overlaps U i,j = U i × C A ′ U j we see from the definition of the ϕ i that lies in the kernel of the 'reduction mod J' map We now want to identify this kernel with H 0 (U ij , L K ) ⊗ K J. To see this note that we can interpret the sequence 0 → J → A ′ → A → 0 as an exact sequence of sheaves on Spec A ′ . Since L A ′ is flat over A ′ , we obtain an exact sequence 2) on global sections, so the kernel of that map is given by Lemma A.4. For the cover (U i ) i of C K = C A ′ × A ′ K, the element (ψ i,j ) i,j defines a 1-cocycle in H 1 (C, L K ) ⊗ K J, which represents the class δ ∈ T e J C ⊗ J we want to compute.

Appendix B. Explicit Serre duality
For the convenience of the reader, and to fix notation, we recall here the standard description of Serre duality on a curve in terms ofČech cocycles.
We consider first the case of a smooth proper (possibly non-connected) curve C/K. We choose aČech cover U = {U i } i of C, where i runs over some indexing set I = {0, . . . , n}. For a sheaf of abelian groups F on C we write C i U (F) for the group ofČech i-cochains, Z i U (F) for the group of i-cocycles, andȞ i (C, F) for the ithČech cohomology group. The point of Serre duality is that the 'residue map'Ȟ 1 (C, ω) → K is an isomorphism of K-vector spaces; our goal here is to make this residue map explicit. We approximately follow [Forster, section 17.2]. Write K for the sheaf of fractions of O C on C. Fix an element w = (w ij ) i<j ∈ Z 1 U (ω). Choose an elementw ∈ C 0 U (K ⊗ O C ω) such that for all i < j we havẽ w i −w j = w ij ∈ ω(U ij ) ⊆ K ⊗ ω(U ij ).
For example, if all U 0 is dense in C we could setw 0 = 0 and for i = 0 letw i be any meromorphic differential extending w i0 . For a point p ∈ C, choose i such that p ∈ U i and define res pw = res pwi . To see that this is independent of the choice of i, note that if p ∈ U ij thenw i −w j is by assumption holomorphic around p, and so has zero residue. Finally, res pw is not independent of the choice ofw, but the global residue p∈C res pw is independent of all choices, and we define this to be res w. This gives a well-defined residue map H 1 (C, ω) → K. Now for the case of nodal curves. We resume the notation from above, but we allow C to have nodal singularities. We write π :C → C for the normalisation of C. Suppose again we are given w = (w ij ) i<j ∈ Z 1 U (ω). WritingŨ for the cover ofC obtained by pulling back U , we have a natural pullback map Z 0 U (ω ⊗ K) → Z 0 U (ω ⊗ K). We choosew ∈ Z 0 U (ω ⊗ K) such that w i −w j ∈ ω(U ij ) ⊆ K ⊗ ω(U ij ), andw i −w j = w ij . Given a point p ∈ C we choose i with p ∈ U i , and define res p w = π(q)=p res q π * w i . As before we should check that this is independent of the choice of i. For p in the smooth locus of C this proceeds exactly as before. If p is a node, write π −1 p = {q, q ′ }. Assume 13 p ∈ U ij , then we need to show that res qwi + res q ′w i = res qwj + res q ′w j .
(B.0.1) But sincew i −w j ∈ ω(U ij ) we have that the residues ofw i −w j at q and q ′ sum to zero, giving exactly the above equality. Thus for a given choice ofw we have a well-defined residue map at all points of C, and the global residue p∈C res pw is independent of all choices, giving a well-defined residue map H 1 (C, ω) → K.

Appendix C. Generic non-vanishing of k-residues
In this section we are concerned with the vanishing of sums of kth roots of k-residues (defined just below) of k-differentials on smooth curves. We fix integers g 0, n 2 with 2g − 2 + n > 0. Let k > 0, and m a vector of n integers m i summing to k(2g − 2), and assume that some m i is either negative or not divisible by k. Write DRL g M g,n for the locus of (smooth, marked) curves admitting a k-differential with divisor i m i p i , and let Y be an irreducible component of DRL g (by [Sch18], such a Y is necessarily smooth and of pure codimension g). Let η denote a general K-point of Y (we assume in this section that K is algebraically closed, so that this exists; otherwise one simply works with the generic point, but the notation becomes slightly less convenient), and let ξ η be a differential on the curve C η with the prescribed divisor. We recall from [BCG + 19a, prop 3.1] the notion of k-residue of a k-differential ξ on a smooth curve C. Assume that ξ has multiplicity m < 0 at a point P ∈ C (i.e. a pole of order |m| at P ). Assume furthermore that k | m. Then after suitable choice of local coordinate z on C (with z = 0 at P ) we can write for m = −k and ξ = z m/k + s z k (dz) k , for m < −k, respectively. Here s is an element of K whose k-th power is well defined, denoted Res k P (ξ) = s k , the k-residue.
Theorem C.1. Let 0 < n ′ < n, and assume that m i < 0 and k | m i for all 1 i n ′ . Also assume that m n ′ +1 is either negative or not divisible by k. For each 1 i n ′ , let r i be any kth root of the k-residue of ξ η at p i . Then r 1 + · · · + r n ′ = 0, (C.0.1) independent of the choices of kth roots r i .
For this result to hold, it is essential that the point η be general in Y . In the case k = 1, we know that the sum of all the residues vanishes, and this result tells us that the sum of any proper subset of the residues is generically non-vanishing. Our result is closely related to those of Gendron and Tahar [GT17]. The key difference is that on one hand we need to treat the connected components Y of DRL g separately but on the other hand, we are only interested at the behaviour at the generic point. Note that for k = 1 our result follows from [GT17, Proposition 1.3].
The proof will occupy the remainder of this section, and we break it into a number of steps.
Step 1: the case dim Y = 0. In general we will argue by showing that the sum in (C.0.1) varies non-trivially in Y , and thus cannot vanish at a general point; this argument fails if dim Y = 0, so we treat this case separately. Now dim Y = 2g−3+n and n 2, so we must have g = 0, n = 3. Note that in this case necessarily n ′ = 1, since for n ′ = 2 we have m 1 , m 2 −k (since they are negative and divisible by k). But m 1 + m 2 + m 3 = −2k forcing that k | m 3 and m 3 = −2k − m 1 − m 2 0, a contradiction to the assumptions of the theorem.
So we are in the case g = 0, n ′ = 1, n = 3. Then we have DRL g = M 0,3 is a single point and we can assume that C = P 1 with (p 1 , p 2 , p 3 ) = (0, 1, ∞). Then w is uniquely determined (up to scaling) as w = z m 1 (1 − z) m 2 (dz) k .
Let g(z) be a kth root of (1 − z) m 2 around z = 0 with g(0) = 1. Then form 1 = −m 1 /k we have w = (z −m 1 g(z)dz) k and Res k 0 (w) = Res 0 (z −m 1 g(z)dz)) k = 1 (m 1 − 1)! d dz One verifies that to compute the derivative of g we can just apply the usual rules for derivatives for the formula g(z) = (1 − z) m 2 /k and obtain Since m 2 is either negative or not divisible by k, this is a nonzero number for b = −m 1 /k − 1 and thus the k-residue of w at 0 does not vanish, as claimed.
Step 2: Canonical covers of curves with k-differentials We now move on to the general case, where dim Y > 0. Given a curve C η with the k-differential ξ η , we are going to use its canonical cover π : C η → C η (see [BCG + 19a]). This is a cyclic cover π : C η → C η of degree k obtained by extracting a kth root of the section ξ η of the line bundle ω ⊗k (−mP ). This means that there exists a 1-differential ξ η on C η with ( ξ η ) k = π * ξ η . Moreover, for τ : C η → C η an automorphism over C η generating the Galois group, it satisfies τ * ξ η = ρ k ξ η where ρ k is a primitive kth root of unity. Note that the map π isétale outside of the preimages of the points p i (since over points where the k-differential ξ η is not zero, there are exactly k choices of a root).
There is a unique maximal b 1 such that ξ η is a bth power of a k ′ = (k/b)-differential. The number b is also the number of connected components of the cover C η , and each such component is the canonical cover for the suitable k ′ -differential on C η . The component Y of DRL g is then just a component of a space of k ′ -differentials and the k-th roots r i of the k-residues are exactly k ′ -th roots of the corresponding k ′ -residues. Since the canonical cover of the k ′ -differential is connected, it suffices to show the statement of the theorem for connected canonical covers if we show it for all k 1. So from now on we assume that C η is connected.
Let g be the genus of C η . There are gcd(m i , k) preimages of each p i (and g is determined by Riemann-Hurwitz). Then we write H for the stack of 'cyclic covers with the same degree and ramification data as C η → C η '; more precisely, the objects of H consist of -A (smooth, connected, proper) curve C of genus g with n marked points p 1 , . . . , p n ; -A (smooth, connected, proper) curve C of genus g with i gcd(m i , k) marked points q i,j : 1 i n, 1 j gcd(m i , k); -A cyclic cover π : C → C of degree k mapping the q i,j to p i .
For a full definition and the properties of the stacks H that we will use, we refer the reader to [Sv18] and the references therein.
The stack H comes with a map δ : H → M g,n remembering the target curve (C, (p i ) i ) and a map φ : H → M g,r remembering the domain curve ( C, (q i,j ) i,j ); here r = n i=1 gcd(m i , k). The map δ isétale and φ is unramified.
Recall that we write DRL g ⊂ M g,n for the 'double ramification' locus where there exists a k-differential ξ with divisor mP . If q i,j is a marked point on C mapping to a marking p i on C, then the canonical cover C → C has multiplicity f i = k/ gcd(m i , k) at q i,j and the 1-differential ξ has multiplicity m ′ i := (m i + k)/ gcd(m i , k) − 1. We write DRL g ⊂ M g,r for the locus where there exists a 1-differential ξ with divisor m ′ Q = i,j m ′ i q i,j .
Lemma C.2. Let π : ( C, (q i,j ) i,j ) → (C, (p i ) i ) be a point of H given by the canonical cover of a curve (C, (p i ) i ) ∈ DRL g . Then ( C, (q i,j ) i,j ) ∈ DRL g , and inside H, in a neighbourhood of the point π : ( C, (q i,j ) i,j ) → (C, (p i ) i ) of H we have Proof. For the inclusion φ −1 (DRL g ) ⊆ δ −1 (DRL g ) let π : ( C, (q i,j ) i,j ) → (C, (p i ) i ) be a point of H such that ( C, (q i,j ) i,j ) ∈ DRL g , i.e. such that there exists a 1-differential w on C with multiplicity m ′ i at the points q i,j . Then for the cyclic automorphism τ of the cover π we have τ * w = λw for some λ ∈ K, since τ * w has the same pattern of zeros and poles as w. Since τ has order k, it follows that λ is a kth root of unity. But then the kth power w ⊗k of w is invariant under τ , and hence descends to a k-differential on C with suitable zeros and poles. This shows φ −1 (DRL g ) ⊆ δ −1 (DRL g ).
The other inclusion is not true globally, but we only need it on a neighbourhood of our point π which already lies in φ −1 (DRL g ). If we can show that every infinitesimal deformation of π which lies in δ −1 (DRL g ) also lies in φ −1 (DRL g ) then we are done, since all these moduli stacks are of finite presentation. A deformation ( C t , (q i,j;t ) i,j ) of C lying in δ −1 (DRL g ) implies that the line bundle ω ⊗k Ct (− i,j km ′ i q i,j;t ) is trivial, i.e. ω Ct (− i,j m ′ i q i,j;t ) is k-torsion. Since the k-torsion points are discrete in the relative Picard of the family C t and since at C = C 0 this bundle is trivial (since C ∈ DRL g ), it stays trivial in the deformation C t , so C t ∈ DRL g .
Step 3: Tangent space computations We know that T ( C,(q i,j ) i,j ) M g,r = H 1 ( C, Ω ∨ C (− q i,j )) and we have an action of Z/kZ on C induced by the automorphism τ of C. This in turn induces an action of Z/kZ on T ( C,(q i,j ) i,j ) M g,r , and where we see the tangent space to H as a subspace of the tangent space to M g,r via the unramified map φ.
One also checks that the tangent space to DRL g ( contained in T ( C,(q i,j ) i,j ) M g,r ) is stable under the Z/kZ-action (for fixed i, all markings q i,j , which form a Z/kZ-orbit, have the same weight m ′ i in the definition of the double ramification locus DRL g ).
Step 4: Residue maps on the tangent space From now on we focus on a curve C coming from a general K-point of the component Y in the statement of the theorem, and take the cover π : C → C by extracting a kth root of the given differential ξ, as in the previous step. Write ξ for canonical kth root of ξ on C. Suppose that k | m i . Then there are exactly k markings q i,j of C lying over p i (the cover π is unramified there), and the residues of ξ at the markings q i,j lying over p i are exactly the kth roots of the k-residue of ξ at p i . The chosen roots r 1 , . . . , r n ′ in the statement of the theorem thus correspond uniquely to certain markings q i,0 lying over the p i ; they are given exactly by the residue of ξ at the q i,0 .
Write f : C → DRL g for the universal curve over DRL g . After perhaps restricting to an open subset of Y , on DRL g the coherent sheaf f * ω C (− i,j m ′ i q i,j ) is invertible. We write n for the number of markings q i,j with negative weight, and we let H ⊆ O ⊕ n DRL g be the subspace defined by the vanishing of the sum of all the coordinates. Then the (usual) residue gives a map Let C → C in H lie over a general point in Y , and choose a non-zero section ξ of ω k (−mP ) over C, leading to a differential ξ over C. This gives a point in the total space of f * ω C (− i,j m ′ i q i,j ), and we can consider the tangent map T R at such a point.
Step 5: Concluding the proof with a lemma of Sauvaget Recall that we have reduced to the case where the curve C is connected. We can then apply [Sau19,Corollary 3.8] to see that T R is a surjection.
The morphism τ induces an automorphism of the pair ( C, ξ), where it acts on the differential by pulling back and dividing by ρ k . This induces an action of the group Z/kZ on the deformations of the pair, in other words on the source of T R. This group also acts on the target (by permutation of markings and multiplication by suitable roots of unity), and the map T R is then equivariant for the action. Since Z/kZ is linearly reductive, the induced map on the invariant subspaces is also surjective.