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Infinitesimal structure of the pluricanonical double ramification locus

Published online by Cambridge University Press:  14 September 2021

David Holmes
Affiliation:
Mathematisch Instituut, Universiteit Leiden, Postbus 9512, 2300 RALeiden, Netherlandsholmesdst@math.leidenuniv.nl
Johannes Schmitt
Affiliation:
Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115Bonn, Germanyschmitt@math.uni-bonn.de
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Abstract

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.

Type
Research Article
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original article is properly cited. Compositio Mathematica is © Foundation Compositio Mathematica.
Copyright
© 2021 The Author(s)

1. Introduction

Inside the moduli space $\mathcal {M}_{g,n}$ of smooth pointed curves $(C,p_1, \ldots , p_n)$ there are natural closed subsets

(1.0.1)\begin{equation} \mathcal{H}_g^k(\mathbf{m}) = \bigg\{(C,p_1, \ldots, p_n) : \omega_C^{{\otimes} k} \cong \mathcal{O}_C\bigg(\sum_{i=1}^n m_i p_i\bigg)\bigg\} \subset \mathcal{M}_{g,n}, \end{equation}

where $\mathbf {m}=(m_1, \ldots , m_n) \in \mathbb {Z}^n$ 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 [Reference ZorichZor06, Reference WrightWri15, Reference ChenChe17] 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 $\overline{\mathcal {M}}_{g,n}$ 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 $\mathbb {P}^1$ [Reference LiLi01, Reference LiLi02, Reference Li and RuanLR01, Reference Graber and VakilGV05]. This is based on the observation that for a smooth curve $C$, a meromorphic $0$-differential on $C$ as above corresponds to a morphism $C \to \mathbb {P}^1$ with given ramification profiles over $0,\infty$. The second series of approaches, viable for any $k \geq 0$, uses that $\mathcal {H}_g^k(\mathbf {m})$ can be obtained by pulling back the zero section $e$ of the universal Jacobian $\mathcal {J} \to \mathcal {M}_{g,n}$ via the Abel–Jacobi section

\[ \sigma : \mathcal{M}_{g,n} \to \mathcal{J}, (C,p_1, \ldots, p_n) \mapsto \omega^k \bigg(-\sum_{i=1}^n m_i p_i\bigg). \]

The map $\sigma$ does not extend naturally to $\overline{\mathcal {M}}_{g,n}$, but various geometric extensions of its domain and target have been proposed that yield cycles on $\overline{\mathcal {M}}_{g,n}$ [Reference Holmes, Kass and PaganiHKP18, Reference Abreu and PaciniAP21, Reference HolmesHol21, Reference Kass and PaganiKP19, Reference Marcus and WiseMW20]. These constructions all produce the same cycle class on $\overline{\mathcal {M}}_{g,n}$, which we denote $\overline{\operatorname {DRC}}$; an overview of one construction is given in § 1.2.

Pixton [Reference PixtonPix14] defined a class $P_g^{g,k}(\tilde{\mathbf{m}})$ in the tautological ring of $\overline{\mathcal {M}}_{g,n}$. The equality

(1.0.2)\begin{equation} \overline{\operatorname{DRC}} = 2^{{-}g} P_g^{g,k}(\tilde{\mathbf{m}}) \end{equation}

was conjectured by Pixton for $k=0$, and in [Reference HolmesHol21] for all $k$. An introduction to Pixton's formula in the case $k=0$ can be found in [Reference PandharipandePan18, § 6.4], and in the general case in [Reference Janda, Pandharipande, Pixton and ZvonkineJPPZ17]. The conjectured equality $2^{-g}P_g^{g,k}(\tilde{\mathbf{m}}) = \overline{\operatorname {DRC}}$ for $k=0$ was proven in [Reference Janda, Pandharipande, Pixton and ZvonkineJPPZ17]. Since the preprint of the present paper was posted, the equality $2^{-g}P_g^{g,k}(\tilde{\mathbf{m}}) = \overline{\operatorname {DRC}}$ for all $k$ has been established in [Reference Bae, Holmes, Pandharipande, Schmitt and SchwarzBHP+20].

A new geometric approach to extending the cycle appears for $k \geq 1$: assuming that one of the integers $m_i$ is negative or not divisible by $k$, the papers [Reference Farkas and PandharipandeFP16, Reference SchmittSch18] define a cycle $H_{g,\mathbf {m}}^k$ obtained as a weightedFootnote 1 fundamental class of an explicit closed subset $\widetilde {\mathcal {H}}_g^k(\mathbf {m}) \subset \overline{\mathcal {M}}_{g,n}$ extending $\mathcal {H}_g^k(\mathbf {m})$, and propose the following conjecture.

Conjecture A [Reference Farkas and PandharipandeFP16, Reference SchmittSch18]

Let $k\geq 1$ and $\mathbf {m}=(m_1, \ldots , m_n) \in \mathbb {Z}^n$ with $m_1 + \cdots + m_n = k(2g-2)$. Assume that one of the $m_i$ is negative or not divisible by $k$ and let $\tilde{\mathbf{m}} = (m_1 +k, \ldots , m_n +k)$. Then

\[ H_{g,\mathbf{m}}^k = 2^{{-}g} P_{g}^{g,k}(\tilde{\mathbf{m}}) \in A^g(\overline{\mathcal{M}}_{g,n}). \]

At the time these papers were written the geometric class $\overline{\operatorname {DRC}}$ had not been defined for $k>0$; from our current perspective it seems most natural simply to conjecture that all three classes ($\overline{\operatorname {DRC}}$, $2^{-g}P_g^{g,k}(\tilde{\mathbf{m}})$ and $H_{g,\mathbf {m}}^k$) are equal whenever they are defined.

The main result of our paper is the following theorem.

Theorem 1.1 For $k \geq 1$ and at least one of the $m_i$ either negative or not divisible by $k$, the equality

(1.0.3)\begin{equation} \overline{\operatorname{DRC}} = H_{g,\mathbf{m}}^k \end{equation}

holds in the Chow ring of $\overline{\mathcal {M}}_{g,n}$.

Combined with the recent proof of $2^{-g}P_g^{g,k}(\tilde{\mathbf{m}}) = \overline{\operatorname {DRC}}$ in [Reference Bae, Holmes, Pandharipande, Schmitt and SchwarzBHP+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 $H_{g,\mathbf {m}}^k$ arise from intersection multiplicities of the Abel–Jacobi section with the zero section in the construction of [Reference HolmesHol21]. Further, with a little extra work our method allows us not only to compute the multiplicities of the cycle, but even to 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 $H_{g,\mathbf {m}}^k$ and the construction of $\overline{\operatorname {DRC}}$ from [Reference HolmesHol21] before stating a more refined version of our main result in § 1.3. We give a sketch of the proof in § 1.4 and discuss some future research directions in § 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.

1.1 The moduli space of twisted $k$-differentials

A first idea for extending the stratum $\mathcal {H}_g^k(\mathbf {m})$ of $k$-differentials is to consider its closure $\overline{\mathcal {H}}_g^k(\mathbf {m}) \subset \overline{\mathcal {M}}_{g,n}$. Stable curves $(C,p_1, \ldots , p_n)$ in this closure have been characterized in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+18, Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+19a] in terms of existence of $k$-differentials on the components of $C$ satisfying certain residue conditions. For $k=1$ and all $m_i\geq 0$, the closure $\overline{\mathcal {H}}_g^1(\mathbf {m})$ is of pure codimension $g-1$, and [Reference Pixton, Pandharipande and ZvonkinePPZ19] gives a conjectural relation of the fundamental class of this closure to Witten's $r$-spin classes.

A larger compactification containing the closure $\overline{\mathcal {H}}_g^1(\mathbf {m})$, the moduli space of twisted $k$-differentials $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$, has been proposed by Farkas and Pandharipande in [Reference Farkas and PandharipandeFP16]. The idea here is that as the curve $C$ becomes reducible, it is no longer reasonable to ask for an isomorphism of line bundles $\omega _C^{\otimes k} \cong \mathcal {O}_C\big (\sum _{i=1}^n m_i p_i\big )$, 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 $\Gamma$ of $C$. This is a map $I$ from the set of half-edges of $\Gamma$ to the integers, satisfying $I(h)=-I(h')$ if $(h,h')$ forms an edge, together with a further combinatorial condition (see Definition 1.6 for details). Given a twist $I$ on the dual graph $\Gamma$ of a stable curve $C$, let $\nu _I: C_I \to C$ be the map normalizing the nodes $q \in C$ belonging to edges $(h,h')$ with $I(h) \neq 0$. Let $q_h, q_{h'} \in C_I$ be the corresponding preimages of $q$ under $\nu _I$. Then the curve $(C,p_1, \ldots , p_n)$ is contained in $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ if and only if there exists a twist $I$ on its stable graph, such that we have an isomorphism of line bundles

(1.1.1)\begin{equation} \omega_{C_I}^{{\otimes} k} \cong \mathcal{O}_{C_I}\Bigg(\sum_{i=1}^n m_i p_i + \sum_{\substack{(h,h')\in E(\Gamma)\\I(h)\neq 0}} (I(h)-k) q_h + (I(h')-k) q_{h'}\Bigg) \end{equation}

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 $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ is a closed subset of $\overline{\mathcal {M}}_{g,n}$ containing $\overline{\mathcal {H}}_g^k(\mathbf {m})$ but possibly having additional components supported in the boundary of $\overline{\mathcal {M}}_{g,n}$. It turns out that these extra components are essential when trying to associate a natural cycle class to the extension of the strata of $k$-differentials. Assume we are in the case where $k \geq 1$ and that at least one of the $m_i$ is either negative or not divisible by $k$. Then it is shown in [Reference Farkas and PandharipandeFP16] (for $k=1$) and [Reference SchmittSch18] (for $k>1$) that $\widetilde {\mathcal {H}}_g^k(\mathbf {m}) \subset \overline{\mathcal {M}}_{g,n}$ has pure codimension $g$. In this situation, instead of studying the fundamental class of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ (as a reduced substack), the papers [Reference Farkas and PandharipandeFP16, Reference SchmittSch18] consider a certain weighted fundamental class $H_{g,\mathbf {m}}^k \in A^g(\overline{\mathcal {M}}_{g,n})$ of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$.

To define this weighted class, let $Z$ be an irreducible component of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$. Denote by $\Gamma$ the generic dual graph of a curve $C$ in $Z$ and let $I$ be theFootnote 2 generic twist on $\Gamma$. Then it is shown in [Reference Farkas and PandharipandeFP16, Reference SchmittSch18] that $\Gamma$ and $I$ must be of a particular form. Indeed, the graph $\Gamma$ 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 $\Gamma$ 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 a slight abuse of notation we write $I(e)=I(h')$ in this case; see Figure 1 for an example.

Figure 1. Example of a simple star graph for $g=4$, $k=3$ and $\mathbf {m}=(-2,5,3,12)$ with the twists $I$ of the half-edges and the weights $m_i$ of the marked points indicated in grey.

With this notation in place, we can defineFootnote 3 the weighted fundamental class $H_{g,\mathbf {m}}^k$ of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ as

(1.1.2)\begin{equation} H_{g,\mathbf{m}}^k = \sum_{(Z,\Gamma,I)} \frac{\prod_{e \in E(\Gamma)} I(e)}{k^{\# V(\Gamma)-1}} [Z] \in A^g(\overline{\mathcal{M}}_{g,n}), \end{equation}

where $Z$ runs over the components of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ and, as above, $\Gamma , I$ are the generic dual graph and twist on $Z$.

Conjecture A above then relates these weighted fundamental classes to the explicit tautological cycles $P_g^{g,k}(\tilde{\mathbf{m}})$ proposed by Aaron Pixton in [Reference PixtonPix14]. In our paper, we show how both the twisted differential space $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ and its weighted fundamental class $H_{g,\mathbf {m}}^k$ naturally arise from a construction presented by the first-named author in [Reference HolmesHol21].

1.2 Extending the Abel–Jacobi map

Let $\mathcal {J}$ be the universal semi-abelian Jacobian over $\overline{\mathcal {M}}_{g,n}$, often written $\operatorname {Pic}^{\underline {0}}_{\mathcal {C}/\overline{\mathcal {M}}_{g,n}}$. It has connected fibres, and parametrizes line bundles of multidegree zero on the fibres of the universal curve $\mathcal {C} \to \overline{\mathcal {M}}_{g,n}$. Inside the open set $\mathcal {M}_{g,n} \subset \overline{\mathcal {M}}_{g,n}$ the strata $\mathcal {H}_g^k(\mathbf {m})$ of $k$-differentials can be obtained as the pullback of the zero section $e$ of $\mathcal {J}$ via the Abel–Jacobi section

\[ \sigma : \mathcal{M}_{g,n} \to \mathcal{J}, (C,p_1, \ldots, p_n) \mapsto \omega^k \bigg(-\sum_{i=1}^n m_i p_i\bigg)=:\omega^k (-\mathbf{m} P ). \]

While $\sigma$ does not in general extend over $\overline{\mathcal {M}}_{g,n}$, in [Reference HolmesHol21] the first author defines a ‘universal’ stack $\mathcal {M}^\lozenge \to \overline{\mathcal {M}}_{g,n}$, birational over $\overline{\mathcal {M}}_{g,n}$, on which $\sigma$ does extend to a morphism $\sigma ^\lozenge : \mathcal {M}^\lozenge \to \mathcal {J}_{\mathcal {M}^\lozenge }$, where $\mathcal {J}_{\mathcal {M}^\lozenge }$ is the pullback of $\mathcal {J}$ to $\mathcal {M}^\lozenge$. Moreover, the scheme-theoretic pullback $\operatorname {DRL}^\lozenge$ of the unit section $e$ of $\mathcal {J}_{\mathcal {M}^\lozenge }$ along $\sigma ^\lozenge$ is proper over $\overline{\mathcal {M}}_{g,n}$. Denote by $\operatorname {DRC}^\lozenge$ the cycle-theoretic pullback of the class $[e]$ under $\sigma ^\lozenge$, supported on $\operatorname {DRL}^\lozenge$, and by $\overline{\operatorname {DRC}} \in A^g(\overline{\mathcal {M}}_{g,n})$ its pushforward under the proper map $\operatorname {DRL}^\lozenge \to \overline{\mathcal {M}}_{g,n}$.

1.3 Main result

Refining Theorem 1.1, the next theorem is the main result of our paper.

Theorem 1.2 The image of the double ramification locus $\operatorname {DRL}^\lozenge$ under the map $\mathcal {M}^\lozenge \to \overline{\mathcal {M}}_{g,n}$ is the moduli space $\widetilde {\mathcal {H}}_g^k(\mathbf {m}) \subset \overline{\mathcal {M}}_{g,n}$ of twisted $k$-differentials. Moreover, for $k \geq 1$ and at least one of the $m_i$ either negative or not divisible by $k$, we have that

(1.3.1)\begin{equation} \overline{\operatorname{DRC}} = H_{g,\mathbf{m}}^k\in A^g(\overline{\mathcal{M}}_{g,n}). \end{equation}

In fact, this is true in the strong sense that these two cycles supported on $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ 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 $\overline{\operatorname {DRC}}$ and $H_{g,\mathbf {m}}^k$ on the cycle level, observe that formula (1.1.2) allows us to define $H^k_{g, \mathbf {m}}$ as a cycle, not just a cycle class. And, under the assumptions of the theorem, the locus $\operatorname {DRL}^\lozenge$ has the expected codimension, and so $\overline{\operatorname {DRC}}$ 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 § 1.4, where we will also discuss in more detail how the multiplicities in formula (1.1.2) come up for the cycle $\overline{\operatorname {DRC}}$. Our method of proof actually yields more precise information than required for the conjecture; we can not only compute 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 $H_{g,\mathbf {m}}^k$. It is also a crucial component of the proof of Conjecture A.

Corollary 1.3 Conjecture A is true.

Proof. The equality $\overline{\operatorname {DRC}} = 2^{-g}P_g^{g,k}(\tilde{\mathbf{m}})$ is proven in [Reference Bae, Holmes, Pandharipande, Schmitt and SchwarzBHP+20], so this follows from Theorem 1.2.

At the time the preprint of the present paper was posted to the arXiv the equality $\overline{\operatorname {DRC}} = 2^{-g}P_g^{g,k}(\tilde{\mathbf{m}})$ was known over the locus of compact-type curves by previous work [Reference Holmes, Pixton and SchmittHPS19] with Pixton, showing Conjecture A to be true when restricted to the locus $\overline{\mathcal {M}}_{g,n}^{ct} \subset \overline{\mathcal {M}}_{g,n}$ of compact type curves.

1.4 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 $\sigma ^\lozenge : \mathcal {M}^\lozenge \to \mathcal {J}_{\mathcal {M}^\lozenge }$ with the unit section $e$ of $\mathcal {J}_{\mathcal {M}^\lozenge }$ along the different components of $\operatorname {DRL}^\lozenge$. 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 $\mathcal {M}^\lozenge$. Here, it turns out that it is more convenient to work with a slight variant $\overline{\mathcal {M}}^{\mathbf {m}, 1/k} \to \overline{\mathcal {M}}_{g,n}$ of $\mathcal {M}^\lozenge$, for which it is easier to write down local charts around the general points of $\operatorname {DRL}$. The precise construction of $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$ is given in § 2 (where we also make more concrete the relationship with the construction of Marcus and Wise [Reference Marcus and WiseMW20]), but for us the two key properties are as follows.

  1. (i) The map $\overline{\mathcal {M}}^{\mathbf {m}, 1/k} \to \overline{\mathcal {M}}_{g,n}$ is log étale and birational and the map $\sigma : \mathcal {M} \to \mathcal {J}$ sending $(C, P)$ to $[\omega ^k(-\mathbf {m} P)]$ extends uniquely to a map $\bar \sigma :\overline{\mathcal {M}}^{\mathbf {m}, 1/k} \to \mathcal {J}$.

  2. (ii) We can compute the tangent space to $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$ explicitly.

The double ramification locus is then $\operatorname {DRL} = \bar \sigma ^* e$, where $e$ is the unit section in $\mathcal {J}$. The concrete local charts for $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$ can be used to show that the image of $\operatorname {DRL}$ in $\overline{\mathcal {M}}_{g,n}$ is exactly the twisted differential space $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$.

With this set-up established, the equality of weights in Theorem 1.2 comes about in an interesting way. Let $Z$ be an irreducible component of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ with generic stable graph $\Gamma$ and twist $I$. Then a general point $p \in Z$ has exactly

\[ \# \{p' \in \operatorname{DRL} \text{ over } p\} = k^{\#E(\Gamma) - \#V(\Gamma)+1} = k^{b_1(\Gamma)} \]

preimages $p'$ in $\operatorname {DRL}$. This is something that can be easily checked in the local charts of $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$. In Definition 2.11, we define a cycle $\operatorname {DRC}$ supported on $\operatorname {DRL}$. At each preimage, its multiplicity is

\[ \operatorname{mult}_{p'} \operatorname{DRC} = \prod_{e \in E(\Gamma)} \frac{I(e)}{k} = \frac{\prod_{e \in E(\Gamma)} I(e)}{k^{\#E(\Gamma)}}. \]

Hence, the pushforward $\overline{\operatorname {DRC}}$ of $\operatorname {DRC}$ has multiplicity

\[ \operatorname{mult}_p \overline{\operatorname{DRC}} = k^{\#E(\Gamma) - \#V(\Gamma)+1} \frac{\prod_{e \in E(\Gamma)} I(e)}{k^{\#E(\Gamma)}} = \frac{\prod_{e \in E(\Gamma)} I(e)}{k^{\# V(\Gamma)-1}}, \]

which is exactly the weight of $[Z]$ in the class $H_{g,\mathbf {m}}^k$. It is also easy to see that the cycle $\operatorname {DRC}$ on $\operatorname {DRL}$ equals the fundamental class of (the possibly non-reduced) $\operatorname {DRL}$ (see Lemma 2.17), so we are left with studying the multiplicity of $\operatorname {DRL}$ at its generic points.

Section 3 is concerned with the computation of the tangent space to $\operatorname {DRL}$. Suppose we are given a point $p \in \operatorname {DRL} \subseteq \overline{\mathcal {M}}^{\mathbf {m}, 1/k}$, which is a general point of some irreducible component of $\operatorname {DRL}$. Let $\Gamma$ be the generic stable graph and $I$ be the generic twist on this component.

The maps $e$ and $\bar \sigma$ induce maps on tangent spaces

and the difference $b=T_p\bar \sigma - T_pe$ factors via the tangent space $T_e \mathcal {J}_p$ to the fibre $\mathcal {J}_p$ of $\mathcal {J}$ over $p$. This induces an exact sequence

\[ 0 \to T_p\operatorname{DRL} \to T_p\overline{\mathcal{M}}^{\mathbf{m}, 1/k} \stackrel{b}{\to} T_e\mathcal{J}_p; \]

it thus remains to analyse carefully the map $b$. For $\mathcal {C}_p$ 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 $\mathcal {C}_p$. Instead of studying the cokernel of $b$, it will be more convenient to use Serre duality and compute the kernel of the linear dual $b^\vee$, which is dual to $\operatorname {coker}(b)$. In Theorem 4.2 we show that $\ker (b^\vee )$ has a natural basis, with one element for each outlying vertex $v$ of $\Gamma$ connected to the central vertex only by edges with twists $I>k$. In Theorem 4.3 we conclude that

\[ \dim T_p \operatorname{DRL} = \dim_p \operatorname{DRL} + \# \{e \in E(\Gamma) : I(e)>k\}, \]

so we have one ‘direction of non-reducedness’ 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 $T_p\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$ into a direct sum of four pieces, corresponding to different types of deformations. Then the dual $b^\vee$ decomposes into 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 $\mathcal {C}_p$, we have that sums of $k$th roots of its $k$-residuesFootnote 4 at (subsets of the) nodes of $\mathcal {C}_p$ 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 practice often difficult.

Write $E$ for the set of edges of the dual graph of the tautological stable curve $\mathcal {C}_p$ over $p$. The universal deformation of $\mathcal {C}_p$ comes with a natural projection map to $\operatorname {Spec} K[[x_e:e \in E]]$, 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 $\operatorname {DRL}$ with a generic subvariety of codimension equal to the dimension of $\operatorname {DRL}$, obtaining a space $\operatorname {DRL}'$ 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 $\operatorname {DRL}'\to \operatorname {Spec} K[[x_e:e \in E]]$ 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

\[ (x_e^{I(e)/k} : e \in E) \subseteq K[[x_e:e \in E]]. \]

We conclude the proof by showing that for an Artin ring $A'$, a map $\operatorname {Spec} A' \to \operatorname {Spec} K[[x_e:e \in E]]$ lifts along $\operatorname {DRL}'\to \operatorname {Spec} K[[x_e:e \in E]]$ if and only if the elements $x_e^{I(e)/k}$ are sent to zero under the corresponding ring map $K[[x_e:e \in E]] \to A'$. The proof works by writing $A'$ as an iterated extension of Artin rings and lifting the map one step at a time. That is, we have Artin rings $A_0=K, A_1, \ldots , A_M=A'$ and short exact sequences

\[ 0 \to J_i \to A_{i} \to A_{i-1} \to 0 \]

of $K$-vector spaces, such that $A_i \to A_{i-1}$ is a morphism of $K$-algebras with kernel $J_i \subset A_i$ satisfying $J_i \frak m_{A_i}=0$ for the maximal ideal $\frak m_{A_i}$ of $A_i$. Then we show that for each $i$, the obstruction of lifting an $A_i$-point of $\operatorname {DRL}'$ to an $A_{i+1}$-point of $\operatorname {DRL}'$ over $\operatorname {Spec} K[[x_e:e \in E]]$ is exactly that all elements $x_e^{I(e)/k}$ are sent to zero in $A_i$.

Remark 1.4 If we had worked with $\mathcal {M}^\lozenge$ instead of $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$, 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 greatest common divisor/least common multiple 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 $k$-residues.

1.5 Relation to previous work and outlook

Compactification via log geometry

In the paper [Reference GuéréGué16], Guéré uses logarithmic geometry to construct a moduli space of $k$-log canonical divisors sitting over $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ 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 $H_{g,\mathbf {m}}^k$. However, for general $k$ the multiplicity of this pushforward at a component with stable graph $\Gamma$ and twist $I$ is equal to $\prod _{e \in E(\Gamma )} I(e)$, and thus different from the multiplicities obtained here and conjectured in [Reference SchmittSch18]. This could indicate that for $k>1$ the definition of the space in [Reference GuéréGué16] 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 $k \geq 1$ and one of the $m_i$ negative or not divisible by $k$, in which case $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ was of pure codimension $g$. In general, by [Reference Farkas and PandharipandeFP16, Theorem 21] all components of the space $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ have at most codimension $g$. In these remaining cases, the behaviour is as follows:

  1. for $k=0$, the principal component $\overline{\mathcal {H}}_g^k(\mathbf {m})$ is of codimension exactly $g$ (unless all $m_i = 0$), but there are components in the boundary of $\overline{\mathcal {M}}_{g,n}$ of various excess dimensions;

  2. for $k=1$ and all $m_i \geq 0$, the principal component $\overline{\mathcal {H}}_g^1(\mathbf {m})$ is of pure codimension $g-1$, with all other components supported in the boundary and of codimension $g$;

  3. for $k>1$ and all $m_i = k m_i' \geq 0$ divisible by $k$, the space $\mathcal {H}_{g}^k(\mathbf {m})$ decomposes as a disjoint union

    \[ \mathcal{H}_{g}^k(\mathbf{m}) = {\mathcal{H}}_g^1(\mathbf{m}') \cup \mathcal{H}_{g}^k(\mathbf{m})', \]
    where ${\mathcal {H}}_g^1(\mathbf {m}')$ is the locus where the $k$-differential is a $k$th power of a $1$-differential, and $\mathcal {H}_{g}^k(\mathbf {m})'$ is the complement. Then $\overline{\mathcal {H}}_g^1(\mathbf {m}') \subset \widetilde {\mathcal {H}}_g^k(\mathbf {m})$ is a union of components of codimension $g-1$, with all other components (i.e. $\overline{\mathcal {H}}_{g}^k(\mathbf {m})'$ and those supported in the boundary) having codimension $g$.

In all of these cases, the cycle $\overline{\operatorname {DRC}}$ still makes sense and by Theorem 1.2 it is indeed supported on the locus $\widetilde {\mathcal {H}}_g^k(\mathbf {m}) \subset \overline{\mathcal {M}}_{g,n}$. Similarly, the formula of Pixton's cycle $P_g^{g,k}(\tilde{\mathbf{m}})$ makes sense in these cases, and in [Reference HolmesHol21] the first author shows that for $k=0$ we have $\overline{\operatorname {DRC}}=2^{-g} P_g^{g,k}(\tilde{\mathbf{m}})$.

We expect that in the cases $k \geq 1$ and $\mathbf {m}=k \mathbf {m}' \geq 0$ the cycle $\overline{\operatorname {DRC}}$ should behave as follows:

  1. on a component $Z$ of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ of codimension equal to $g$, it should be

    \[ \frac{\prod_{e \in E(\Gamma))} I(e)}{k^{\#V(\Gamma)-1}} [Z] \]
    as before (where $\Gamma ,I$ are the generic twist and dual graph);
  2. on the components $\overline{\mathcal {H}}_g^1(\mathbf {m}')$ 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 $\overline{\mathcal {H}}_g^1(\mathbf {m}')$.

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 [Reference SchmittSch18]. There, for a non-negative partition $\mathbf {m}'$ of $2g-2$, a class $[\overline{\mathcal {H}}_g^1(\mathbf {m}')]^{\mathrm {vir}}$ was defined by the formula

\[ [\overline{\mathcal{H}}_g^1(\mathbf{m}')]^{\mathrm{vir}} + \sum_{(Z,\Gamma,I)} \bigg({\prod_{e \in E(\Gamma))} I(e)} \bigg) [Z] = 2^{{-}g} P_g^{g,1}(\tilde{\mathbf{m}}'), \]

where $Z$ runs through the boundary components of $\widetilde {\mathcal {H}}_g^1(\mathbf {m}')$ and $\tilde{\mathbf{m}}'=(m_1'+1, \ldots , m_n'+1)$. The idea was that $[\overline{\mathcal {H}}_g^1(\mathbf {m}')]^{\mathrm {vir}}$ should be a contribution to the double ramification cycle of the partition $\mathbf {m}'$, supported on $\overline{\mathcal {H}}_g^1(\mathbf {m}')$. From our perspective, this should just be the contribution of $\overline{\operatorname {DRC}}$ supported there. Then, since the locus $\overline{\mathcal {H}}_g^1(\mathbf {m}')$ appears as a component of $\overline{\mathcal {H}}_g^k(k \mathbf {m}')$ for any $k>1$, the following conjecture was made.

Conjecture A′ [Reference SchmittSch18]

Let $k \geq 1$ and $\mathbf {m}=k \mathbf {m}'$ for a non-negative partition $\mathbf {m}'$ of $2g-2$. Then we have

\[ [\overline{\mathcal{H}}_g^1(\mathbf{m}')]^{\mathrm{vir}} + [\overline{\mathcal{H}}_{g}^k(\mathbf{m})'] + \sum_{(Z,\Gamma,I)} \frac{\prod_{e \in E(\Gamma))} I(e)}{k^{\#V(\Gamma)-1}} [Z] = 2^{{-}g} P_g^{g,k}(\tilde{\mathbf{m}}), \]

where $Z$ runs through the boundary components of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$.

From the perspective of defining the double ramification cycle via an extension of the Abel–Jacobi map, this behaviour is expected: the space $\mathcal {M}^\lozenge$ for the partition $k \mathbf {m}'$ of $k(2g-2)$ agrees with the space for the partition $\mathbf {m}'$ of $2g-2$, and the Abel–Jacobi section for $k \mathbf {m}'$ is simply the composition of the section for $\mathbf {m}'$ with the étale morphism

\[ \mathcal{J} \to \mathcal{J}, (C,\mathcal{L}) \mapsto (C,\mathcal{L}^{{\otimes} k}). \]

Thus, over the locus $\overline{\mathcal {H}}_g^1(\mathbf {m}')$, the intersection of the Abel–Jacobi section with the unit section should produce the same contribution to the cycle $\overline{\operatorname {DRC}}$.

Smoothing differentials

The papers [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+18, Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+19a] give criteria for a nodal curve $(C,p_1, \ldots , p_n)$ to lie in the locus $\overline{\mathcal {H}}_g^k(\mathbf {m})$. Being contained in this closure is equivalent to having some one-parameter deformation $(C_t, p_{1,t}, \ldots , p_{n,t})_{t \in \Delta }$ with the general curve being contained in ${\mathcal {H}}_g^k(\mathbf {m})$. The criteria of [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+18, Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+19a] are phrased in terms of the existence of $k$-differentials 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 § 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 non-vanishing of such sums of $k$th roots of $k$-residues. While these computations are not directly applicable to the problem of classifying $\overline{\mathcal {H}}_g^k(\mathbf {m})$, 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 [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG+19b] constructs a smooth compactification of the closure $\overline{\mathcal {H}}_g^k(\mathbf {m})$ 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 ${\mathcal {H}}_g^k(\mathbf {m}) \subset \operatorname {DRL}^\lozenge$ inside the double ramification locus of $\mathcal {M}^\lozenge$.

1.6 An overview of different definitions of double ramification cycles

In this subsection 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 $\mathcal {M}_{g,n}$ inside $\overline{\mathcal {M}}_{g,n}$ (e.g. the compact-type locus), and computed them in the tautological ring. Examples include [Reference Hain, Farkas and MorrisonHai13], [Reference Grushevsky and ZakharovGZ14b], [Reference Grushevsky and ZakharovGZ14a], and [Reference DudinDud18].

The following are the different constructions of a DR cycle on all of $\overline{\mathcal {M}}_{g,n}$:

  1. 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 ([Reference LiLi01, Reference LiLi02, Reference Graber and VakilGV05]; see also [Reference Li and RuanLR01]).

  2. Pixton [Reference PixtonPix14] proposed the formula $2^{-g} P_g^{g,k}(\tilde{\mathbf{m}})$ for the DR class as an explicit tautological class, defined via a graph sum.

  3. Kass and Pagani proposed extending the cycle as the pullback of a universal Brill–Noether class on a compactified Jacobian via the Abel–Jacobi section ([Reference Kass and PaganiKP19] and [Reference Holmes, Kass and PaganiHKP18, § 2.4]).

  4. Marcus and Wise used techniques from logarithmic and tropical geometry [Reference Marcus and WiseMW20] to construct a space on which the Abel–Jacobi map extends.

  5. The first-named author gave a definition using a universal extension of the Abel–Jacobi map as described above [Reference HolmesHol21].

  6. Abreu and Pacini gave an explicit tropical blowup of $\overline{\mathcal {M}}_{g,n}$ (i.e. a blowup dictated by an explicit refinement of $\overline{\mathcal {M}}_{g,n}^{{\rm trop}}$) resolving the Abel–Jacobi map to the Esteves’ compactified Jacobian over $\overline{\mathcal {M}}_{g,n}$ and use this to define a double ramification cycle [Reference Abreu and PaciniAP21].

  7. Finally, for $k \geq 1$ and one of the $m_i$ negative or not divisible by $k$, there is the definition of the DR cycle as the weighted fundamental class $H_{g,\mathbf {m}}^k$, proposed by Janda, Pandharipande, Pixton, and Zvonkine for $k=1$ [Reference Farkas and PandharipandeFP16] and the second-named author for $k > 1$ [Reference SchmittSch18].

In Figure 2 we illustrate the known equivalences between these definitions. In particular, [Reference Bae, Holmes, Pandharipande, Schmitt and SchwarzBHP+20] (which came out after the preprint of the present paper) completes the proof that they are all in fact equivalent.

Figure 2. Equivalences between different definitions of double ramification cycles.

1.7 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 § 2 we describe the construction of the space $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$, the variant of $\mathcal {M}^\lozenge$ on which we perform our computations (see § 1.4 above). We also make more concrete the relationship with the construction of Marcus and Wise [Reference Marcus and WiseMW20].

Sections 3 and 4 are devoted to the computation to the tangent space to the double ramification locus. In the brief § 3 we compute the tangent space of the space $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$, 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 § 5 we can explicitly compute its local ring, and in particular the length of the local ring. In § 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 via Čech 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.

1.8 Notation and conventions

Notation

Generalities

We have fixed integers $g \ge 0$, $n >0$, $k >0$ with $2g-2 + n>0$, and integers $m_1, \ldots , m_n$ summing to $k(2g-2)$ with at least one $m_i$ negative or not divisible by $k$. We will write $\mathcal {M}$ for $\mathcal {M}_{g,n}$, $\overline{\mathcal {M}}$ for $\overline{\mathcal {M}}_{g,n}$ etc. We write $\mathcal {J}$ for the universal semi-abelian Jacobian over $\overline{\mathcal {M}}$, often written $\operatorname {Pic}^{\underline {0}}_{\mathcal {C}/\overline{\mathcal {M}}}$. Then the section

\[ \sigma : \mathcal{M} \to \mathcal{J}, (C,p_1, \ldots, p_n) \mapsto \omega^k \bigg(-\sum_{i=1}^n m_i p_i\bigg)=:\omega^k \big(-\mathbf{m} P \big) \]

lives naturally in $\mathcal {J}(\mathcal {M})$, but in general does not extend to the whole of $\overline{\mathcal {M}}$.

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 [Reference SauvagetSau19, Corollary 3.8] in Appendix C, which we expect to admit an algebraic proof. When $k >1$ we very often use 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 the ground field $K$ to 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 normalization 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 étale 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 étale extensions, the reader may assume without loss of generality that the ground field $k$ is algebraically closed throughout §§ 4 and 5.

We expect that most readers will be mainly interested in the case of algebraically closed fields, so to minimize 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 $\Gamma$ consists of a finite set $V$ of vertices, a finite set $H$ of half-edges, a map ‘$\operatorname {end}$’ from the half-edges to the vertices, an involution $i$ on the half-edges, and a genus $g: V \to \mathbb {Z}_{\ge 0}$. Graphs are connected, and the genus $g(\Gamma$) 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 $\operatorname {end}(h)$ its source and $\operatorname {end}(i(h))$ its target, and sometimes write it as $h: \operatorname {end}(h) \to \operatorname {end}(i(h))$. We write $E = E(\Gamma )$ for the set of edges.

The valence $\operatorname {val}(v)$ of a vertex is the number of non-leg half-edges incident to it, and we define the canonical degree $\mathrm {can}(v) = 2g(v) - 2 + \operatorname {val}(v)$, so that

\[ 2g(\Gamma) - 2 = \sum_v \mathrm{can}(v). \]

A closed walk in $\Gamma$ 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 $\Gamma$ together with a function $\mathbf {m}$ from the set $L$ of legs to $\mathbb {Z}$ such that $\sum _{l \in L}\mathbf {m}(l) = k(2g(\Gamma )-2)$.

Definition 1.6 A twist of a leg-weighted graph is a function $I$ from the half-edges to $\mathbb {Z}$ such that:

  1. (i) for all legs $l \in L$, we have $\mathbf {m}(l) = I(l)$;

  2. (ii) if $i(h) = h'$ and $h \neq h'$ then $I(h) +I(h') =0$;

  3. (iii) for all vertices $v$, $\sum _{\operatorname {end}(h) = v} I(h) - k \cdot \mathrm {can}(v)$ = 0.

We write $\operatorname {Tw}(\Gamma )$ for the (non-empty) set of twists of a leg-weighted graph $\Gamma$.

Remark 1.7 In [Reference HolmesHol21] these twists were called ‘weightings’, and were denoted $w$. The present notation is much closer to that used by [Reference Farkas and PandharipandeFP16]; we have made this change to facilitate comparison to [Reference Farkas and PandharipandeFP16], and because the letter $w$ was already overloaded.

Remark 1.8 Farkas and Pandharipande impose two additional conditions (which they call ‘vanishing’ and ‘sign’), which together state that $\Gamma$ cannot contain any directed cycle for which every directed edge $h$ has $I(h) \ge 0$, 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 set-up; more precisely, the fibre of a chart $\overline{\mathcal {M}}^\mathbf {m}_{I,U}$ of $\overline{\mathcal {M}}^\mathbf {m}$ over the origin in $\mathbb {A}^E$ (see § 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 [Reference Bossert, Pattanaik, Tadenuma, Xu and YoshiharaBos08].

Definition 1.9 We say that a leg-weighted graph $\Gamma$ 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 $V^{{\rm out}}$.

1.8.2 The weighted fundamental class of the space of twisted differentials

Here we recall the definition of the class $H_{g,\mathbf {m}}^k \in A^{g}(\overline{\mathcal {M}})$ given in [Reference Farkas and PandharipandeFP16, § A.4] (for $k=1$) and [Reference SchmittSch18, § 3.1] (for $k>1$) and explain why it is equivalent to the definition as a weighted fundamental class of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$ presented earlier in this introduction.

First, recall that given any integer $k \geq 1$ and a partition $\mathbf {m}$ of $k(2g-2)$ of length $n$, we have

\[ \mathcal{H}_{g}^{k}(\mathbf{m}) = \bigg\{(C,p_1, \ldots, p_{n}) : \omega_{C}^{k}\bigg(-\sum_{i} m_i p_i\bigg) \cong \mathcal{O}_{C}\bigg\} \subset \mathcal{M}_{g,n}, \]

the corresponding stratum of $k$-differentials. This closed, reduced substack has pure codimension $g-1$ if $k=1$ and all $m_i \geq 0$, and pure codimension $g$ if there exists $i$ such that $m_i$ is negative or not divisible by $k$. As before we denote by $\overline{\mathcal {H}}_{g}^{k}(\mathbf {m})$ its closure in $\overline{\mathcal {M}}_{g,n}$.

Write $S$ for the set of simple star graphs of genus $g$ (see Definition 1.9). We say that a twist $I$ of a simple star graph is positive (writing $\operatorname {Tw}^+(\Gamma )$ 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 a slight abuse of notation, we write $I(e)=I(h)$ for the edge $e=(h,h')$ to which $h$ belongs.

With this notation, Janda, Pandharipande, Pixton, Zvonkine (for $k =1$) and the second author (for $k > 1$) define

\begin{align*} H_{g,\mathbf{m}}^k &= \sum_{\Gamma \in S}\sum_{I \in \operatorname{Tw}^+(\Gamma)}\frac{\prod_{e \in E(\Gamma)}I(e)}{|\operatorname{Aut}(\Gamma)|k^{\# V^{{\rm out}}}}\xi_{\Gamma *}\bigg[ \big[\overline{\mathcal{H}}_{g(v_0)}^{k}(\mathbf{m}|_{v_0}, -I|_{v_0} - k)\big]\\ &\quad \cdot\prod_{v \in V^{{\rm out}}(\Gamma)}\bigg[ \bar{\mathcal{H}}_{g(v)}^1\bigg(\frac{\mathbf{m}}{k}|_v, \frac{I}{k}|_v - 1\bigg)\bigg]\bigg]. \end{align*}

Here $\mathcal {H}_{g(v_0)}^{k}(\mathbf {m}|_{v_0}, -I|_{v_0} - k)$ denotes the cycle in $\mathcal {M}_{g(v_0), n(v_0)}$ with $n(v_0)$ the number of half-edges attached to $v_0$, and with weighting given by restricting the weighting $\mathbf {m}$ to those legs attached to $v_0$, and given by $-I(e)-k$ at the half-edge belonging to the edge $e$ of $\Gamma$. The cycles $\mathcal {H}_{g(v)}^1({\mathbf {m}}/{k}|_v, {I}/{k}|_v - 1)$ 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 [Reference SchmittSch18] for details).

Now we comment on why this is a weighted fundamental class of the space $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$. Given a boundary component $Z$ of this space, let $\Gamma$ be the generic dual graph of a curve $C$ in $Z$ and let $I$ be the twist on $\Gamma$ such that condition (1.1.1) is satisfied for this generic curve $C$. By [Reference SchmittSch18, Proposition A.1] every node of $C$ such that the corresponding edge has twist $I=0$ can be smoothed while staying in $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$. Thus since $Z$ is assumed a generic point of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$, all edges of $\Gamma$ must have non-zero 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 [Reference Farkas and PandharipandeFP16, 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 $m_i$ negative or not divisible by $k$. This easily implies that the generic dual graph $\Gamma$ of $Z$ is a simple star graph and that the twist $I$ on $\Gamma$ is positive.

Conversely, one checks that condition (1.1.1) is satisfied on all the loci on which the cycle $H_{g,\mathbf {m}}^k$ above is supported. This shows that it is indeed a weighted fundamental class of $\widetilde {\mathcal {H}}_g^k(\mathbf {m})$. On the other hand, the weights agree with those given earlier in this introduction: the closures of strata of differentials (which are pushed forward via $\xi _\Gamma$) are generically reduced and thus all have multiplicity $1$. The factor $1/|\operatorname {Aut}(\Gamma )|$ exactly accounts for the fact that the gluing morphism $\Gamma$ has degree $|{\operatorname {Aut}(\Gamma)}|$.

Thus the definition of $H_{g,\mathbf {m}}^k$ given earlier coincides with the definitions from [Reference Farkas and PandharipandeFP16, Reference SchmittSch18].

1.8.3 Combinatorial charts

If $p : \operatorname {Spec} K \to \overline{\mathcal {M}}$ is a geometric point corresponding to a curve $C$, the associated graph $\Gamma _C$ comes with a leg weighting from the integers $m_i$. If a node of the curve over $p$ has local equation $xy - r$ for some $r \in \mathcal {O}^{et}_{\overline{\mathcal {M}}, p}$, then the image of $r$ in the monoid $\mathcal {O}^{et}_{\overline{\mathcal {M}}, p}/(\mathcal {O}^{et}_{\overline{\mathcal {M}}, p})^\times$ is independent of the choice of local equation. In this way, for each edge $e \in E(\Gamma _p)$ we obtain an element $\ell _e \in \mathcal {O}^{et}_{\overline{\mathcal {M}}, p}/(\mathcal {O}^{et}_{\overline{\mathcal {M}}, p})^\times$, recalling that edges of the graph correspond to nodes of the curve.

Given a leg-weighted graph $\Gamma$ with edge set $E$, defineFootnote 5

\[ \mathbb{A}^E = \operatorname{Spec}K[a_e:e \in E]. \]

To any point $a$ in $\mathbb {A}^E$ we associate the graph $\Gamma _a$ obtained from $\Gamma$ by contracting exactly those edges $e$ such that $a_e$ is a unit at $a$. Denote by $[a_e] \in \mathcal {O}^{et}_{\mathbb {A}^E, p}$ the image of the function $a_e$ on $\mathbb {A}^E$ in $\mathcal {O}^{et}_{\mathbb {A}^E, p}$.

Definition 1.10 A combinatorial chart of $\overline{\mathcal {M}}$ consists of a leg-weighted graph $\Gamma$ and a diagram of stacks

\[ \overline{\mathcal{M}} \stackrel{f}{\longleftarrow} U \stackrel{g}{\longrightarrow} \mathbb{A}^E \]

satisfying the following six conditions.

  1. (i) $U$ is a connected scheme.

  2. (ii) $g: U \to \mathbb {A}^E$ is smooth.

  3. (iii) $f: U \to \overline{\mathcal {M}}$ is étale.

  4. (iv) the pullbacks of the boundary divisors in $\overline{\mathcal {M}}$ and $\mathbb {A}^E$ to $U$ coincide.

  5. (v) $0 \in \mathbb {A}^E$ is in the image of $g$.

Let $p: \operatorname {Spec}K \to U$ be any geometric point, yielding natural maps

\[ \mathcal{O}^{et}_{\overline{\mathcal{M}}, f \circ p} \stackrel{f^\flat}{\to} \mathcal{O}^{et}_{U, p} \stackrel{g^\flat}{\leftarrow} \mathcal{O}^{et}_{\mathbb{A}^E, g \circ p}. \]
  1. (vi) Let $C=f(p)$ and $a=g(p)$. Then we require an isomorphism

    \[ \varphi_p: \Gamma_C \to \Gamma_a \]
    such that $f^\flat (\ell _e) = g^\flat ([a_{\varphi _p(e)}])$ up to units in $\mathcal {O}^{et}_{U, p}$ for every edge $e$ (which necessarily makes this $\varphi _p$ unique if it exists). Moreover, the map $\varphi _p$ sends the leg weighting on $\Gamma _{f \circ p}$ coming from the $-m_i$ to the leg weighting on $\Gamma _{g \circ p}$ coming from that on $\Gamma$.

This definition is as in [Reference HolmesHol21] but with the logarithmic structures excised (since we do not need them). We see in [Reference HolmesHol21] that $\overline{\mathcal {M}}$ can be covered by combinatorial charts.

2. Constructing suitable moduli spaces

2.1 Recalling the construction of $\operatorname {DR}$

We begin by recalling the basic construction of the cycle $\operatorname {DR}$ from [Reference HolmesHol21]. First one constructs a certain stack $\mathcal {M}^\lozenge /\overline{\mathcal {M}}$ such that the rational map $\sigma : \mathcal {M} \to \mathcal {J}$ extends to a morphism $\sigma ^\lozenge : \mathcal {M}^\lozenge \to \mathcal {J}$. Writing $e$ for the unit section of $\mathcal {J}$ (viewed as a closed subscheme) and $[e]$ for its Chow class, it is shown in [Reference HolmesHol21] that the scheme-theoretic pullback $\operatorname {DRL}^\lozenge$ of $e$ along $\sigma ^\lozenge$ is proper over $\overline{\mathcal {M}}$. We would now like to take the cycle-theoretic pullback of the class of $e$ along $\sigma ^\lozenge$, 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 $\mathcal {M}^\lozenge \to \mathcal {J}_{\mathcal {M}^\lozenge } = \mathcal {J} \times _{\overline{\mathcal {M}}} \mathcal {M}^\lozenge$, and pull back the class of the unit section along this section (using that the latter is a regular closed immersion as $\mathcal {J}$ is smooth over $\overline{\mathcal {M}}$) to obtain a cycle $\operatorname {DRC}^\lozenge$ on $\mathcal {M}^\lozenge$. This cycle $\operatorname {DRC}^\lozenge$ is naturally supported on $\operatorname {DRL}^\lozenge$, and so by properness can be pushed down to a cycle on $\overline{\mathcal {M}}$, which we denote $\overline{\operatorname {DRC}}$, 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 [Reference HolmesHol21], [Reference Holmes, Pixton and SchmittHPS19] and [Reference Holmes, Kass and PaganiHKP18].

In this paper we will work with a slight variant of the stack $\mathcal {M}^\lozenge$ of [Reference HolmesHol21]; this is only for convenience, but the intricacy of the calculations we have to carry out makes every available bit of notational efficiency worth using. We also note that $\mathcal {M}^\lozenge$ 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 $\mathcal {M}^\lozenge$ is built by gluing together normal toric varieties; in particular, it is normal. We will begin by introducing a ‘non-normal’ analogue $\overline{\mathcal {M}}^\mathbf {m}$ of $\mathcal {M}^\lozenge$ 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 § 2.6 for more details.

2.2 Construction of $\overline{\mathcal {M}}^\mathbf {m}$

Fix a combinatorial chart

\[ \overline{\mathcal{M}} \stackrel{f}{\longleftarrow} U \stackrel{g}{\longrightarrow} \mathbb{A}^E \]

and a twist $I$ on $\Gamma$. If $e = \{ h, h'\}$ is an edge of $\Gamma$, and $\gamma$ is a cycle in $\Gamma$, we define

(2.2.1)\begin{equation} I_\gamma(e) = \left\{\begin{array}{@{}ll} 0 & \text{if } h \notin \gamma \text{ and } h' \notin \gamma \\ I(h) & \text{if } h \in \gamma\\ I(h') & \text{if } h' \in \gamma.\end{array}\right. \end{equation}

In the free abelian group on symbols $a_e: e \in E$ we consider the submonoid generated by the $a_e$ and by the expressions

(2.2.2)\begin{equation} \prod_{e \in E} a_e^{I_\gamma(e)} \end{equation}

as $\gamma$ runs over cycles in $\Gamma$, and we denote the spectrum of the associated monoid ring by $\mathbb {A}^E_I$. Equivalently, $\mathbb {A}^E_I$ is the spectrum of the subring of $K[a_e^{\pm 1}:e \in E]$ 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 [Reference HolmesHol21], where we worked with sub-polyhedral cones of $\mathbb {Q}_{\geq 0}^E$, 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 § 2.3 we give explicit equations for (a slight variant on) the $\mathbb {A}^E_I$.

We write $\overline{\mathcal {M}}^\mathbf {m}_{I,U}$ for the pullback of $\mathbb {A}^E_I$ to $U$. We want to argue that these $\overline{\mathcal {M}}^\mathbf {m}_{I,U}$ naturally glue together to form a stack $\overline{\mathcal {M}}^\mathbf {m}$ over $\overline{\mathcal {M}}$. The first part of the gluing can even be done over $\mathbb {A}^E$. Indeed, fixing a graph $\Gamma$, as $I$ runs over twists of $\Gamma$ the $\mathbb {A}^E_I$ naturally glue together as $I$; cf. [Reference HolmesHol21, § 3].Footnote 6 We denote the glued object by $\tilde {\mathbb {A}}^E \to \mathbb {A}^E$.

Example 2.1 In the case $k=0$, suppose the graph $\Gamma$ 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 $a \in \mathbb {Z}$:

(2.2.3)

In this setting $\mathbb {A}^E = \operatorname {Spec} K[a_e, a_{e'}]$. There are two directed cycles, and expression (2.2.2) yields $a_e^i a_{e'}^{i-n}$ and $a_e^{-i} a_{e'}^{n-i}$. The form of $\mathbb {A}^E_I$ then depends on $I$: we have

(2.2.4)\begin{equation} \begin{aligned} i<0: & \quad \mathbb{A}^E_I = \operatorname{Spec} K[a_e^{{\pm} 1}, a_{e'}^{{\pm} 1}],\\ i=0: & \quad \mathbb{A}^E_I = \operatorname{Spec} K[a_e, a_{e'}^{{\pm} 1}],\\ 0 < i < n: & \quad \mathbb{A}^E_I = \operatorname{Spec} \frac{K[a_e, a_{e'}, s^{{\pm} 1}]}{(a_{e'}^{n-i}s - a_e^i)} = \operatorname{Spec} \frac{K[a_e, a_{e'}, t^{{\pm} 1}]}{(a_{e}^{i}t - a_{e'}^{n-i})},\\ i=n: & \quad \mathbb{A}^E_I = \operatorname{Spec} K[a_e^{{\pm} 1}, a_{e'}] ,\\ i>n: & \quad \mathbb{A}^E_I = \operatorname{Spec} K[a_e^{{\pm} 1}, a_{e'}^{{\pm} 1}]. \end{aligned} \end{equation}

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 $\operatorname {Spec} K[a_e^{\pm 1}, a_{e'}^{\pm 1}]$ to form $\tilde {\mathbb {A}}^E$. Note that (in the case where $n$ is not prime) this differs slightly from the example in [Reference HolmesHol21, 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. [Reference HolmesHol21, Remark 3.5]) there is no canonical way to embed the patches in a blowup, though see also [Reference Abreu and PaciniAP21] for a general approach to compactifying.

While there are infinitely many charts glued together, only those for $0 \le i \le n$ 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 these $\tilde {\mathbb {A}}^E$ back to $U$ we obtain a stack covered by patches $\overline{\mathcal {M}}^\mathbf {m}_{I,U}$. Then running over a cover of $\overline{\mathcal {M}}$ by combinatorial charts yields a collection of stacks over $\overline{\mathcal {M}}$ which are easily upgraded to a descent datum. We denote the resulting ‘descended’ object by $\overline{\mathcal {M}}^\mathbf {m}$. Comparing with the construction of $\mathcal {M}^\lozenge$ in [Reference HolmesHol21], one sees that the normalization of $\overline{\mathcal {M}}^\mathbf {m}$ is $\mathcal {M}^\lozenge$. Imitating the proof of [Reference HolmesHol21, Theorem 3.5] shows that the map $\overline{\mathcal {M}}^\mathbf {m} \to \overline{\mathcal {M}}$ is separated, of finite presentation, relatively representable by algebraic spaces, and an isomorphism over $\mathcal {M}$. If we equip the above objects with their natural log structures, it is also log étale. From separatedness and the implication $(1) \implies (2)$ of [Reference HolmesHol21, Lemma 4.3], we see that the map $\sigma : \mathcal {M} \to \mathcal {J}$ extends (uniquely) to a morphism $\sigma ^\mathbf {m}: \overline{\mathcal {M}}^\mathbf {m} \to \mathcal {J}$.

Definition 2.2 We define the double ramification locus $\operatorname {DRL}^\mathbf {m} \rightarrowtail \overline{\mathcal {M}}^\mathbf {m}$ to be the schematic pullback of the unit section of $\mathcal {J}$ along $\sigma ^\mathbf {m}$.

Now $\mathcal {M}^\lozenge \to \overline{\mathcal {M}}^\mathbf {m}$ is proper, and by [Reference HolmesHol21, Proposition 5.2] the map $\operatorname {DRL}^\lozenge \to \overline{\mathcal {M}}$ is proper. But since $\operatorname {DRL}^\lozenge \to \operatorname {DRL}^\mathbf {m}$ is surjective, by [Sta13, Tag 03GN] $\operatorname {DRL}^\mathbf {m} \to \overline{\mathcal {M}}$ is also proper.

Definition 2.3 We define the double ramification cycle $\operatorname {DRC}^\mathbf {m}$ to be the cycle-theoretic pullback of the unit section of $\mathcal {J}\times _{\overline{\mathcal {M}}} \overline{\mathcal {M}}^\mathbf {m}$ along the section induced by $\sigma ^\mathbf {m}$, yielding a cycle on $\operatorname {DRL}^\mathbf {m}$.

The pushforward of $\operatorname {DRC}^\mathbf {m}$ to $\overline{\mathcal {M}}$ makes sense by properness of $\operatorname {DRL}^\mathbf {m} \to \overline{\mathcal {M}}$, and the compatibility of the refined Gysin pullback with the proper pushforward (see Lemma 2.12) implies that the pushforward of $\operatorname {DRC}^\mathbf {m}$ to $\overline{\mathcal {M}}$ coincides with pushforward of $\operatorname {DRC}^\lozenge$ to $\overline{\mathcal {M}}$. See § 2.6 for further details.

2.3 A partial normalization of $\overline{\mathcal {M}}^\mathbf {m}$

As discussed in § 1.8.4, on the components of the double ramification locus supported in the boundary, the twist $I$ is generically divisible by $k$. If we restrict the construction in § 2.2 to twists $I$ taking values in $k \mathbb {Z}$, we obtain an open substack $\overline{\mathcal {M}}^\mathbf {m}_{k | I}$ of $\overline{\mathcal {M}}^\mathbf {m}$. We can define a finite surjective map $\overline{\mathcal {M}}^{\mathbf {m}, 1/k} \to \overline{\mathcal {M}}^\mathbf {m}_{k | I}$ by replacing the generators in (2.2.2) by

(2.3.1)\begin{equation} \prod_{e \in \gamma} a_e^{{\pm} I(e)/k}. \end{equation}

More concretely, we obtain $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$ by gluing together patches $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I,U}$ similarly to the procedure in § 2.2. But now, the patch $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I,U}$ is the pullback of the space $\mathbb {A}^E_{I'} \to \mathbb {A}^E$ to $U$, where $I'(e) = I(e)/k$. For clarity and later use, we now give explicit equations for the $\mathbb {A}^E_{I'}$ (and hence implicitly for $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I,U}$ since it arises by pulling back $\mathbb {A}^E_{I'}$ to $U$).

Let $\Upsilon$ be the set of cycles $\gamma$ in $\Gamma$ and recall that $E$ is the set of edges in $\Gamma$. Then naturally we can see $\mathbb {A}^E_{I'}$ as a subscheme of $\mathbb {A}^{\Upsilon } \times \mathbb {A}^{E}$ cut out by explicit equations. Let $((a_\gamma )_{\gamma \in \Upsilon }, (a_e)_{e \in E})$ be coordinates on $\mathbb {A}^{\Upsilon } \times \mathbb {A}^{E}$. Then the generators (2.3.1) translate into a system of equations in the $a_\gamma , a_e$. Indeed, given $f \in \mathbb {Z}^\Upsilon$ and $e \in E$, define the integer

(2.3.2)\begin{equation} M_{e, f} = \sum_{\gamma} f_\gamma I'_\gamma(e), \end{equation}

where $I'_\gamma = I_\gamma /k$ (cf. (2.2.1)). Then a set of equations cutting out $\mathbb {A}^E_{I'} \subset \mathbb {A}^{\Upsilon } \times \mathbb {A}^{E}$ is given by the vanishing of the

(2.3.3)\begin{equation} \Psi_f = \bigg(\prod_{\gamma \in \Upsilon : f_\gamma >0} a_\gamma^{f_\gamma} \bigg) \cdot \prod_{e \in E: M_{e, f}<0} a_e^{{-}M_{e, f}} - \bigg(\prod_{\gamma \in \Upsilon : f_\gamma <0} a_\gamma^{{-}f_\gamma} \bigg) \prod_{e \in E: M_{e, f}>0} a_e^{M_{e, f}} \end{equation}

as $f$ runs through $\mathbb {Z}^{\Upsilon }$. In particular, for any cycle $\gamma$ we have for the inverted cycle $i(\gamma )$, walking in the opposite direction, that $a_\gamma a_{i(\gamma )}=1$, which forces $a_\gamma \neq 0$. Apart from that, the simplest equations in the system above are of the form

(2.3.4)\begin{equation} a_\gamma \prod_{e \in \gamma:I(e)<0}a_e^{{-}I'(e)} = \prod_{e \in \gamma:I(e)>0}a_e^{I'(e)}. \end{equation}

We will see later that these are the only equations that matter for computing the tangent space to $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$.

To get the description of $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I',U}$ over $U$ one inserts for the variables $a_e$ the components of the function $g : U \to \mathbb {A}^E$ from our combinatorial chart, and obtains equations for $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I',U} \subset U \times \mathbb {A}^{\Upsilon }$.

Remark 2.4 A shorter but less explicit description of the polynomials $\Psi _f$ of (2.3.3) can be obtained by saturating an ideal obtained from (2.3.4). Let $R := K[a_\gamma : \gamma \in \Gamma ][a_e: e \in E]$, 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)\begin{equation} a_\gamma - \prod_{e \in \gamma} a_e^{I'(e)}, \end{equation}

and let $I_R$ be the intersection of $I$ with $R$. Then $I_R$ is exactly the ideal generated by the $\Psi _f$ 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 $\overline{\mathcal {M}}^{\mathbf {m}, 1/k} \to \overline{\mathcal {M}}^\mathbf {m}_{k | I}$ is finite birational, but in general neither the source nor the target is normal, thus the map need not be an isomorphism. Indeed, we have the following lemma.

Lemma 2.5 Let $p \in \overline{\mathcal {M}}^\mathbf {m}_{k | I}$ lie over a simple star graph $\Gamma$ with outlying vertex set $V^{{\rm out}}$. Then the fibre over $p$ of the map $\overline{\mathcal {M}}^{\mathbf {m}, 1/k} \to \overline{\mathcal {M}}^\mathbf {m}_{k | I}$ contains exactly $k^{\#E(\Gamma )-\#V^{{\rm out}}} = k^{b_1(\Gamma )}$ points.

Proof. Let $\overline{\mathcal {M}}^\mathbf {m}_{I,\Gamma }$ be a chart containing $p$ with $I=k \cdot I'$ (here we use $p \in \overline{\mathcal {M}}^\mathbf {m}_{k | I}$). Then the preimage of $\overline{\mathcal {M}}^\mathbf {m}_{I,\Gamma }$ in $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}$ is $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I',\Gamma }$ and the map $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I',\Gamma } \to \overline{\mathcal {M}}^\mathbf {m}_{I,\Gamma }$ is a base change of the map

(2.3.6)\begin{equation} \mathbb{A}^E_{I'} \to \mathbb{A}^E_I, ((a_\gamma)_\gamma, (a_e)_e) \mapsto (((a_\gamma)^k)_\gamma, (a_e)_e). \end{equation}

Now $p$ corresponds to a point where all the $a_e = 0$, and the values of $a_\gamma$ for $\gamma$ in a basis of $H^1(\Gamma ,\mathbb {Z})$ can be chosen freely. Once these values of $a_\gamma$ are fixed, all other $a_{\gamma '}$ are determined by (2.3.3). Thus the number of preimage points under the map (2.3.6) is exactly $k^{b_1(\Gamma )}$, and so the same is true for the pullback $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I',\Gamma } \to \overline{\mathcal {M}}^\mathbf {m}_{I,\Gamma }$.

Remark 2.6 The reader only interested in the case $k = 1$ will note that in this case the maps

\[ \overline{\mathcal{M}}^{\mathbf{m}, 1/k} \to \overline{\mathcal{M}}^\mathbf{m}_{k | I} \to \overline{\mathcal{M}}^\mathbf{m} \]

are all isomorphisms, and $I'(e) = I(e)$.

Lemma 2.7 Suppose that $\Gamma$ is a simple star graph. Then $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I', U}$ is a local complete intersection over $K$.

As in [Reference HolmesHol21], the stack $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I', U}$ can be defined relative to $\mathbb {Z}$, in which generality the same lemma holds, with the same proof. The requirement that $\Gamma$ be a simple star seems necessary; the graph

(2.3.7)

seems to give a counterexample in general, though we have not checked all details.

Proof. Recall the notion of a syntomic morphism [Sta13, Tag 01UB] generalizing the definition of being a local complete intersection over a field. In particular, the stack $\overline{\mathcal {M}}^{\mathbf {m}, 1/k}_{I', U}$ is a local complete intersection over $K$ if and only if