1.1 Log (twisted) curves

We will be interested in enriching families of curves with some combinatorial data; for this reason, we adopt the language of logarithmic and tropical geometry. Let $(S,M_S)$ be a logarithmic scheme. F. Kato defined log curves over S as integral, saturated and logarithmically smooth morphisms $\pi \colon (C,M_C)\to (S,M_S)$ with one-dimensional geometric fibres. He also proved the following more explicit characterisation [Reference KatoKat96, Theorem 1.3], which we may take as a definition.

Definition-Proposition 1.1. A family of log curves is a morphism of logarithmic schemes

$$\begin{align*}\pi\colon (C,M_C) \to (S,M_S)\end{align*}$$

such that

1. 1. the underlying morphism of schemes is a flat family of nodal curves;

2. 2. the log structure admits the following description: for each geometric point p of C, there exists an étale local neighbourhood with a strict étale morphism over S to one of the following:

   • smooth point $\mathbb {A}^1_S$ with the log structure pulled back from the base $\pi ^*M_S$ ;

   • marking $\mathbb {A}^1_S$ with the log structure generated by the zero section and $\pi ^*M_S$ ;

   • node $\mathcal {O}_S[\![x,y]\!]/(xy=t)$ for some $t\in \mathcal {O}_S$ , with semistable log structure induced by the multiplication map $\mathbb {A}^2_S\to \mathbb {A}^1_S$ and $t\colon S\to \mathbb {A}^1$ .

In the last case, $\operatorname {log}(t)$ (a local section of $M_S$ ) and t are called smoothing parameters of the node.

An n-pointed family of log curves consists of a family of log curves $\pi : (C, M_C) \to (S, M_S)$ and a tuple of disjoint sections $\sigma _i : \underline S \to \underline C$ , $i = 1, \ldots , n$ (not logarithmic), such that for each $p \in C$ with the log structure of a marking, the zero section of $\mathbb {A}^1_S$ agrees with exactly one of the sections $\sigma _i$ on the étale local neighborhood of p in the definition above. A family of n-pointed log curves is stable if the same is true of its underlying morphism of schemes.

The stacks $\mathfrak {M}_{g,n}$ and $\overline {\mathcal {M}}_{g,n}$ of prestable (resp. stable) n-pointed curves of genus g admit log structures such that they represent the moduli of prestable (resp. stable) families of log curves over the category of logarithmic schemes [Reference GillamGil12]. The log structure in each case is easily described: it is the log structure associated to the divisor of singular curves. In particular, the stalk of the characteristic sheaf $\overline {M}_{\mathfrak {M}_{g,n}}$ at a strict geometric point $s \to \mathfrak {M}_{g,n}$ is isomorphic to $\mathbb {N}^{\#E(s)}$ , where $E(s)$ is the set of nodes of the nodal curve $C_s$ associated to $s \to \mathfrak {M}_{g,n}$ . Given a family of prestable log curves $\pi : (C, M_C) \to (S,M_S)$ , the logarithmic structure on $\underline {S}$ obtained by pulling back the log structure of $\mathfrak {M}_{g,n}$ along the classifying morphism $\overline {\pi } : S \to \mathfrak {M}_{g,n}$ is called the minimal logarithmic structure and denoted by $M_S^{\text {can}}$ . It has the following universal property: there exists a unique log curve $\pi ^{\text {can}}\colon (\underline {C},M_C^{\text {can}})\to (\underline {S},M_S^{\text {can}})$ and a unique morphism of log schemes $\mu \colon (\underline {S},M_S)\to (\underline {S},M_S^{\text {can}})$ covering $\operatorname {id}_{\underline S}$ , such that $\pi $ is the pullback of $\pi ^{\text {can}}$ along $\mu $ .

Generalising Kato’s result, M. Olsson [Reference OlssonOls07] proved that there is an equivalence of categories between (balanced) twisted curves in the sense of [Reference Abramovich and VistoliAV02] and log smooth curves $\pi \colon (C,M_C)\to (S,M_S)$ with a Kummer extension $M_S\hookrightarrow M_S^\prime $ and a choice of an integer $a_i$ for every marking $p_i$ of C. We recall that a map of fs monoids $h\colon Q\to P$ is called Kummer if it is injective and for every $p\in P$ , there exists a $b\in \mathbb {N}$ such that $bp=h(q)$ for some $q\in Q$ (see, for instance, [Reference IllusieIll02]). The Kummer extension $M_S\hookrightarrow M_S^\prime $ induces a Kummer log ètale morphism $S^\prime \to S$ of Deligne–Mumford stacks, roughly speaking, when S is log smooth over a point with trivial log structure, an iterated root stack introducing roots of the smoothing parameters $\operatorname {log}(t)=b\operatorname {log}(s)$ . By pulling back C to $S^\prime $ and taking a further root stack of $C\times _S S^\prime $ along components over the nodal divisor, we arrive at a twisted curve $C^\prime $ with local model:

$$\begin{align*}\left[\operatorname{Spec}\left(\mathcal{O}_{S^\prime}[\![u,v]\!]/(uv-s)\right)/\mu_{b}\right],\end{align*}$$

where $x=u^b,y=v^b$ and $\mu _b$ acts with weights $(1,-1,0)$ on $(x,y,s)$ . Similarly, $C^\prime $ entails an $a_i$ -root stack of $C\times _S S^\prime $ along the marking $p_i$ .

The stack of twisted curves $\mathfrak {M}_{g,n}^{\text {tw}}$ admits a canonical locally free log structure and a Kummer log ètale map to $\mathfrak {M}_{g,n}$ .

1.2 Tropicalisation and conewise-linear functions

To a family of log smooth curves $\pi \colon C\to S$ one can associate a family of tropical curves $\operatorname {trop}(\pi )\colon \Gamma \to S$ [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, Section 7.2]. Over a geometric point s, the tropical curve $\Gamma _s$ is the dual graph of $C_s$ metrised in $\overline {M}_{S,s}$ , where the length of an edge $e_q$ , corresponding to the node $q\in C_s$ , is the class of the smoothing parameter $\operatorname {log}(t_q)$ in the stalk of the characteristic sheaf $\overline {M}_{S,s}$ . For each specialisation $s_1\rightsquigarrow s_0$ , there is an associated map $\Gamma _{s_0} \to \Gamma _{s_1}$ which applies the induced map $\overline {M}_{S,s_0} \to \overline {M}_{S,s_1}$ to each edge length and contracts the edges whose length goes to $0$ . Thus, $\Gamma _s$ can be thought of as a family of (standard, $\mathbb {R}_{\geq 0}$ metrised) tropical curves over the dual cone $\sigma _s=\operatorname {\mathrm {Hom}}(\overline {M}_{S_s},\mathbb R_{\geq 0})$ . These cones $\sigma _s$ can be glued together into the tropicalisation of S, as we proceed to explain next, and $\Gamma $ can be equivalently thought of as a family of tropical curves over $\operatorname {trop}(S)$ .

Indeed, tropical geometry can be embedded in algebraic geometry by means of the Artin fan machinery [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, Section 6.3]: an Artin fan is a (relative) $0$ -dimensional Artin stack with log structure, admitting a strict ètale cover by Artin cones. These are quotients (in the sense of stacks) of affine toric varieties by their dense tori: $\mathcal {A}_\sigma =[\operatorname {Spec}(R[\sigma ^\vee \cap \overline {M}^{\text {gp}}])/\operatorname {Spec}(R[\overline {M}^{\text {gp}}])]$ . In fact, the $2$ -category of Artin fans is equivalent to the $2$ -category of stacks over rational polyhedral cones. Artin fans provide a cover of the Olsson stack of logarithmic structures [Reference OlssonOls03], with the property that every logarithmic scheme admits a universal strict morphism to an Artin fan $X\to \mathcal {A}_X$ encoding all the combinatorial (and none of the geometric) information about X. Thanks to the above-mentioned equivalence, we may think of $\mathcal {A}_X$ as a cone stack, which we call the tropicalisation of X, and sometimes denote by $\operatorname {trop}(X)$ .

Let C be a log smooth curve over a geometric point, and let $q\in C$ be a node. The groupification of the characteristic monoid at q can be identified with

$$\begin{align*}\overline M_{C,q}^{\text{gp}}\simeq\{(\overline{\gamma}_1,\overline{\gamma}_2)\in\overline M_{S,\pi(q)}^{\text{gp},\oplus2}|\ \overline{\gamma}_2-\overline{\gamma}_1\in\mathbb{Z}\overline{\operatorname{log}(t_q)}\}.\end{align*}$$

This allows for an identification of global sections of the characteristic group on C with conewise linear (CL) functionsFootnote ² on the tropicalisation $\Gamma $ with values in $\overline M_S^{\text {gp}}$ and integral slopes along the edges:

$$\begin{align*}H^0(C,\overline M_C^{\text{gp}})=\{\mathbb{Z}-\text{CL functions on }\Gamma\text{ with values in }\overline M_S^{\text{gp}}\};\end{align*}$$

see [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, Remark 7.3]. Moreover, the fundamental exact sequence

$$\begin{align*}0\to \mathcal{O}_C^\times\to M_C^{\text{gp}}\to\overline M_C^{\text{gp}}\to 0,\end{align*}$$

and its long exact sequence in cohomology show that to every section $\overline {\gamma }\in H^0(C,\overline M_C^{\text {gp}})$ , there is an associated $\mathcal {O}_C^\times $ -torsor of lifts of $\overline {\gamma }$ to $M_C^{\text {gp}}$ ; this can be filled into a line bundle in two ways, and we choose the convention such that for $\overline {\gamma }\in H^0(C,\overline M_C)$ (i.e., $\overline {\gamma }\geq 0$ in the partial order on $\overline M_C^{\text {gp}}$ induced by $\overline M_C$ ), the log structure induces a section $\mathcal {O}_C\to \mathcal {O}_C(\overline {\gamma })$ . The restriction of $\mathcal {O}_C(\overline {\gamma })$ to the component $C_v$ of C corresponding to a vertex v of $\Gamma $ is made explicit in [Reference Ranganathan, Santos-Parker and WiseRSPW19a, Proposition 2.4.1]:

(1.1) $$ \begin{align} \mathcal{O}_{C_v}(\overline{\gamma})\simeq \mathcal{O}_{C_v}\left(\sum s(\overline{\gamma},e_q)q\right)\otimes\pi^*\mathcal{O}_S(\overline{\gamma}(v)), \end{align} $$

where $s(\overline {\gamma },e_q)$ denotes the outgoing slope of $\overline {\gamma }$ along the edge corresponding to q.

1.3 Rubber and multiscale differentials

Multiscale differentials were introduced to compactify strata of differentials over the Deligne–Mumford compactification of the moduli space of curves [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG⁺18]. We will focus on an equivalent approach based on logarithmic geometry [Reference Chen, Grushevsky, Holmes, Möller and SchmittCGH⁺22]. On a smooth curve, a holomorphic differential up to scaling is encoded in the location of its $2g-2$ zeroes, counted with multiplicity. Hence, we may think of strata of differentials as codimension g substacks of $\mathcal {M}_{g,n}$ . This naive approach fails over reducible curves, where the distribution of the markings does not reflect the multidegree of the dualising bundle (or any other bundle we may be interested in representing as a weighted sum of the markings). A possible solution is to twist by some components of the nodal curve (a vertical divisor which will be represented by a CL function on the dual graph up to translation), but even this may only work after restricting to a substack of a birational modification of $\overline {\mathcal M}_{g,n}$ . The following moduli problems were introduced in [Reference Marcus and WiseMW20] in order to extend the classical Abel-Jacobi section $\mathrm {aj}\colon \mathcal {M}_{g,n}\to \mathbf {Pic}_{g,n}$ beyond the compact-type locus. We fix an n-tuple $\mu =(m_1,\ldots ,m_n)$ of integers summing to $2g-2$ .

The following is one of the main results of [Reference Marcus and WiseMW20]:

Theorem 1.3. $\mathbf {Div}_{g,\mu }$ and $\mathbf {Rub}_{g,\mu }$ are represented by algebraic stacks with a log smooth log structure. In fact, they are both birational log ètale modifications of the stack of prestable curves:

$$\begin{align*}\mathbf{Rub}_{g,\mu}\to \mathbf{Div}_{g,\mu}\to\mathfrak{M}_{g,n}.\end{align*}$$

The Abel-Jacobi section extends to a finite and unramified morphism:

$$\begin{align*}\mathrm{aj}\colon\mathbf{Div}_{g,\mu}\to\mathbf{Pic}_{g,n}.\end{align*}$$

Definition 1.4. The moduli space of rubber differentials $\mathcal {H}^\dagger (\mu )$ is defined via the Cartesian diagram:

where the right vertical map associates to a stable curve C its dualising bundle $\omega _C$ .

Spelling out the definition, a rubber differential consists of a stable curve $C\to S$ and a conewise-linear function $\bar {\lambda }$ on $\Gamma =\operatorname {trop}(C)\to S$ up to translations by $\overline M_S$ , whose values at the vertices of $\Gamma $ are totally ordered, together with a specified isomorphism

(1.2) $$ \begin{align} \mathcal{O}_C(\bar{\lambda})=\omega_C,\text{ or equivalently }\omega_C(\lambda)=\mathcal{O}_C, \end{align} $$

where we have set $\lambda =-\bar {\lambda }$ .

We may thus think of $\Gamma $ as an ordered (or level) graph. Equation (1.2) may be thought of as a section of $\omega _C$ (i.e., a differential), vanishing on all but the top level. In fact, it is proved in [Reference Chen, Grushevsky, Holmes, Möller and SchmittCGH⁺22] that more information can be extracted from $\bar {\lambda }$ – namely, a collection $(\eta _i)_{i=0,\ldots ,-N}$ of meromorphic differentials, one on each level subcurve of C. The orders of the $\eta _i$ at markings and at the preimages of the nodes are determined by the slopes of $\bar {\lambda }$ at the corresponding legs and edges. Rubber differentials turn out to be equivalent to the generalised multiscale differentials from [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG⁺19]. To resemble the conventions adopted in the literature on multiscale differentials, we may fix a lift of $\bar {\lambda }$ to an honest CL function by setting $\max (\bar {\lambda })=0$ (equivalently, $\operatorname {min}(\lambda )=0$ ) and identify the values of $\bar {\lambda }$ with the set $\{0,-1,\ldots ,-N\}$ .

The vertical maps in the diagram of Definition 1.4 are strict; thus, a rubber differential on $C\to S$ induces the same minimal logarithmic structure on S as its tropicalisation $\bar {\lambda }$ . This is a locally free log structure parametrising the differences between consecutive values of $\bar {\lambda }$ . The moduli space of stable rubber differentials $\mathcal {H}^\dagger (\mu )$ is represented by a separated DM stack with log structure and is finite (in particular, representable) over the Hodge bundle over $\mathbf {Rub}$ (cf. [Reference Chen and ChenCC19, Corollary 3.2]). Typically, $\mathcal {H}^\dagger (\mu )$ is not irreducible. Limits of differentials on smooth curves are identified in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG⁺18] by means of an analytic condition called global residues.

1.4 Hyperelliptic admissible covers via roots and logs

Admissible covers were introduced by Harris and Mumford in their work on the Kodaira dimension of the moduli space of curves of high genus [Reference Harris and MumfordHM82] in order to compactify the locus of smooth k-gonal curves. Anticipating the developments of relative Gromov–Witten theory, their moduli were studied by S. Mochizuki by introducing logarithmic structures [Reference MochizukiMoc95], and by D. Abramovich, A. Corti and A. Vistoli by introducing orbifold structures [Reference Abramovich, Corti and VistoliACV03]. The two approaches are essentially equivalent in view of Olsson’s work on log twisted curves that has been recalled above. We refer the reader to the paper [Reference Schmitt and van ZelmSvZ20] (resp. the book [Reference Bertin and RomagnyBR11]) for a concise (resp. extensive) overview of the subject.

In this paper, we will focus on the hyperelliptic case. Fix a genus $g\geq 1$ and a number of markings $n\geq 0$ (later on, markings will record the zeroes of a differential). We first recall the orbifold point of view:

Pulling back the universal $\mathbb {Z}/2\mathbb {Z}$ -cover $\ast $ over $\mathcal {B}(\mathbb {Z}/2\mathbb {Z})$ , we obtain one $\psi ^\prime \colon C\to P^{\text {tw}}$ over the twisted curve. Since $\phi $ is representable and $\ast $ is a scheme, so is $C\to S$ : in fact, it is an ordinary nodal curve of genus g [Reference Abramovich, Corti and VistoliACV03, Lemma 2.2.1], and $\psi \colon C\to P$ is a $\mathbb {Z}/2\mathbb {Z}$ -admissible cover ramified exactly at the preimage of the gerby markings and nodes [Reference Abramovich, Corti and VistoliACV03, §4].

We may think of the necessity of gerby nodes as follows: when P is smooth, $\psi _*\mathcal {O}_C=\mathcal {O}_P\oplus \mathcal {O}_P(-\frac {1}{2}\mathbf b)$ makes sense because $\operatorname {deg}(\mathbf b)=2g+2$ is even and $\operatorname {Pic}(P)=\mathbb Z$ . When P is reducible, though, a problem occurs if $\mathbf b$ has odd degree on some components of P: in fact, removing any node q of P separates the latter into two connected components, and we call q even (resp. odd) if so is the degree of $\mathbf b$ on either of the two. Twisting along the odd nodes will allow us to find the root of $\mathcal {O}_P(-\mathbf b)$ , which we call $\mathcal {F}$ ; see, for instance, [Reference FedorchukFed14, §2]. Summing up, we have

where $\psi ^\prime $ is finite ètale with $\psi ^\prime _*\mathcal {O}_C=\mathcal {O}_{P^\prime }\oplus \mathcal {F}$ , and $\psi $ is finite but not flat with $\psi _*\mathcal {O}_C=\mathcal {O}_P\oplus \rho _*\mathcal {F}$ . Indeed, the character of the line bundle $\mathcal {F}$ at an odd node is nontrivial, and correspondingly, $\rho _*\mathcal {F}$ is a torsion-free, rank-one sheaf on P that is not locally free at the odd nodes.

Now, following [Reference MochizukiMoc95, Definition 3.5], we will consider families of hyperelliptic admissible covers adapted to the setting of log schemes. Parallel to [Reference MochizukiMoc95, Definition 3.4], we observe that the stack $\mathfrak {M}^{\mathrm {log}}_{g,n_1 + n_2}$ of prestable log curves of genus g with $n_1 + n_2$ markings admits a natural action of the symmetric group on $n_1$ letters $\mathfrak {S}_{n_1}$ permuting the first $n_1$ points. The stack quotient parametrises prestable log curves C with a simple divisor $\mathbf {r}$ of $n_1$ ‘symmetrised markings’, and $n_2$ ordered markings $u_1, \ldots , u_{n_2}$ (collectively $\mathbf {u}$ ).

We say that a log hyperelliptic admissible cover is stable if $(P,\mathbf b,\mathbf v)$ is Deligne–Mumford stable as a rational pointed curve. Notice that $(C,\mathbf r,\mathbf u)$ will be as well. We denote by $\mathcal {H}_{g,n}$ the moduli space of genus g, n-marked, stable log hyperelliptic admissible covers. Then, $\mathcal {H}_{g,n}$ is represented by a proper DM stack with log structure, which is furthermore (log) smooth [Reference MochizukiMoc95, §3.22].

Indeed, given a family of log hyperelliptic admissible covers over S, there is an associated minimal logarithmic structure on S, satisfying a similar universal property to the minimal log structure for pointed log curves. When $\psi $ is stable, this is the same as the log structure pulled back from $\mathcal {H}_{g,n}$ along the classifying map. More explicitly, it is a Kummer extension of the minimal log structure of P as a log smooth curve, introducing square roots of the smoothing parameters corresponding to the nodes over which $\psi $ is ramified. Indeed, $\psi $ can be factored as $C\to P^{\mathrm {tw}}\to P$ , where the first map is strict étale, and the second one is Kummer log étale and birational – albeit it is only representable by DM stacks. The minimal log structure for $\psi $ is the same as that for $P^{\text {tw}}$ as a log twisted curve, which is recalled in §1.1.

1.5 Hyperelliptic differentials

The paper [Reference Chen and ChenCC19] developed the theory of log differentials without imposing an alignment (so, replacing the stack $\mathbf {Rub}$ with the stack $\mathbf {Div}$ in Definition 1.4), focusing on the hyperelliptic and spin components. Here, we shall revisit [Reference Chen and ChenCC19, Definition 5.3]. Unlike [Reference Chen and ChenCC19], we do impose an alignment.

Definition 1.11. A hyperelliptic rubber differential over a log scheme S is the datum of

1. (i) a log hyperelliptic admissible cover $\psi \colon (C,\mathbf r,\mathbf u)\to (P,\mathbf b,\mathbf v)$ over S, and

2. (ii) a rubber differential $\eta $ on C,

such that

(1.3) $$ \begin{align} \iota^*(\eta)=-\eta, \end{align} $$

where $\iota $ is the hyperelliptic involution.

Given the latter condition, the vanishing orders of $\eta $ at the conjugate points $u_{i,1}$ and $u_{i,2}$ are the same. We may therefore denote the vanishing order of $\eta $ by an n-tuple $\mu =(m_1,\ldots ,m_n)\in \mathbb {N}^n_{g-1}$ of nonnegative integers summing to $g-1$ . We restrict our attention to strata of hyperelliptic differentials with no zeroes at the Weierstrass points.Footnote ⁴

1.6 Contraction data

Consider a hyperelliptic rubber differential $(\psi : C \to P, \bar {\lambda })$ . Denote by $\Gamma , T^\prime $ and T the tropicalizations of $C,P^{\text {tw}}$ and P, respectively. As $\rho : P^{\mathrm {tw}} \to P$ is a root stack, the induced map $T^\prime \to T$ is also a root stack, introducing halves of lengths of the edges of T corresponding to branching nodes of P. By condition (1.3), the CL function $\bar {\lambda }$ descends to a CL function $\bar {\lambda }_T$ on $T^\prime $ . We view $\bar {\lambda }_T$ as a CL function on T, except that it may have half-integral slopes on the branching edges; pulling back to $\Gamma $ multiplies the slope by $2$ precisely along these edges. Equation (1.2) descends then to

(1.4) $$ \begin{align} \omega_{P^{\mathrm{tw}}}(\mathbf b/2)=\mathcal{O}_{P^{\mathrm{tw}}}(\bar{\lambda}_T)\text{ or equivalently }\omega_{P^{\mathrm{tw}}}(\mathbf b/2+\lambda_T)=\mathcal{O}_{P^{\mathrm{tw}}}, \end{align} $$

where $\lambda _T = -\bar {\lambda }_T$ . Indeed, $\rho ^*\omega _P=\omega _{P^{\text {tw}}}$ [Reference ChiodoChi08, Proposition 2.5.1], and $\psi ^*\omega _P=\omega _C(-\mathbf r)$ by definition of the ramification divisor. Since $\rho $ induces an isomorphism of Picard groups up to torsion and P is rational, (a multiple of) condition (1.4) can be checked numerically. Equation (1.4) becomes

(1.5) $$ \begin{align} \text{val}(v)-2+\frac{1}{2}\deg(\mathbf b)(v)+\operatorname{div}(\lambda_T)(v)=0, \end{align} $$

for every vertex v of T. Here, $\text {val}$ denotes the edge valency, $\deg (\mathbf b)$ the multidegree (or tropicalization) of the branch divisor, and $\operatorname {div}(\lambda _T)(v)$ the sum of the outgoing slopes of $\lambda _T$ along all the edges adjacent to v.Footnote ⁵ In keeping with Remark 1.5, we could say that $\lambda _T$ is in the log canonical tropical linear series of $T^\prime $ . The same is true as well for its restriction to any subcurve of $T^\prime $ .

In order to produce a Gorenstein contraction of a log hyperelliptic cover, we do not need the whole data of a log hyperelliptic differential, but only its combinatorial shadow (i.e., its tropicalisation). In fact, if we are happy to allow nonreduced structures along components of the contraction, we do not need to know precisely at which points the zeroes of the differential are located, but only on which components. We extract this combinatorial information in the following:

Definition 1.13. Let $\psi \colon (C,\mathbf r,\mathbf u)\to (P,\mathbf b,\mathbf v)$ be a log hyperelliptic admissible cover. A contraction datum is a CL function $\lambda _T$ on $T^\prime =\operatorname {trop}(P^{\text {tw}})$ such that $\lambda _T(v)\geq 0$ for every vertex v of $T^\prime $ , and $\lambda _T$ is a member of the log canonical tropical linear series of the support of $\lambda _T$ (i.e., the coefficient

$$\begin{align*}D(v)=\text{val}(v)-2+\frac{1}{2}\deg(\mathbf b)(v)+\operatorname{div}(\lambda_T)(v)\end{align*}$$

is a nonnegative integer for every vertex v such that $\lambda _T(v)$ is strictly positive).

Example 1.14. To illustrate the combinatorics of contraction data, we consider the dual graphs of several log hyperelliptic admissible covers with $g = 2$ and $n = 1$ (i.e., two simple zeroes at conjugate points). We first consider the possible stable hyperelliptic admissible covers where $T^\prime $ has a single edge. We may attempt to construct a contraction datum supported on a single vertex v by computing the required slope of $\lambda _T$ at v using Equation (1.5), then shifting $\lambda _T$ so that its minimum value is $0$ . The possibilities are illustrated in Figure 1. We indicate the number of legs coming from $\mathbf {b}$ and the branching edges (nodes) in blue, the legs coming from markings in black ( $\lambda $ has slope $-1$ along them, since our sign convention implies that zeroes of the differential point down, and poles point up), and the positive slopes of $\lambda $ , $\lambda _T$ in green. We recall that pulling back $\lambda _T$ to $\lambda $ doubles slopes on branching edges. Since Equation (1.5) is stable under generization, we can deduce the slopes of contraction data on genus two admissible covers with more components by generizing to the single-edge graphs. See Figures 1 and 2.

Figure 1 Contraction data on hyperelliptic admissible covers with one edge.

Figure 2 A contraction datum on a larger log hyperelliptic admissible cover supported on the left 3 vertices. Unnumbered blue legs are single branch legs.

2.1 Contracting the rational curve

Consider the line bundle $\mathcal {L}:=\omega _{P^{\mathrm {tw}}}(\lambda _T)$ , whose pullback to C is $\omega _C(\lambda )$ . Note that $\mathcal {L}^{\otimes 2}$ descends to a line bundle $L^{\otimes 2}=\omega _P^{\otimes 2}(\mathbf b+2\lambda _T)$ of degree $2g-2$ and nonnegative multidegree on P. In particular, since P is rational, L has vanishing higher cohomology, $\pi _{P,*}L^{\otimes 2k}$ is a vector bundle of rank $2k(g-1)+1$ on S for any positive integer k (the relative Proj will therefore be flat by [Reference ProjectSta18, Tag 0D4C]), and $L^{\otimes 2}$ is semiample relative to the base (the contraction map $\tau $ is thus defined everywhere). Moreover, the formation of $\pi _{P,*}L^{\otimes 2k}$ commutes with arbitrary base-change by [Reference HartshorneHar77, Theorem III.12.11]. Let

$$\begin{align*}P\xrightarrow{\tau}\overline{P}:=\underline{\operatorname{Proj}}_S\left(\bigoplus_{k\geq 0}\pi_{P,*}L^{\otimes 2k}\right)\end{align*}$$

be the resulting contraction. The fibres of $\tau $ are connected components of the locus where $L^{\otimes 2}$ is trivial (equivalently, of multidegree $\underline 0$ ). In particular, they are connected, rational, nodal curves. The map $\mathcal {O}_P\to \mathcal {O}_{\overline {P}}$ induces an identification of the latter with $\operatorname {R}\!\tau _*\mathcal {O}_P$ , and $\overline {P}$ is a family of rational Cohen–Macaulay curves. Therefore, the singularities appearing in the fibres of $\overline {P}$ are ordinary m-fold points (the singularity of the coordinate axes in $\mathbb {A}^m$ ): $q\in \overline {P}$ is an m-fold point if $\tau ^{-1}(q)$ meets the rest of P in m nodes. These singularities are Gorenstein (even planar) if and only if they are smooth points or nodes (i.e., $m\leq 2$ ). Let denote the line bundle on $\overline {P}$ induced by $L^{\otimes 2}$ on P.

2.2 Square roots of bundles and curves

We will need a square root of . By mere multidegree considerations, such a line bundle does not always exist on $\overline {P}$ itself, but it does after reintroduction of some stack structure on P and $\overline {P}$ . An important observation is that this orbifold structure is only ever needed at nodes of P and $\overline {P}$ away from the exceptional locus of $\tau $ .

Lemma 2.1 (cf. [Reference FedorchukFed14, Lemma 3.5])

There is a commutative diagram

and unique line bundles $\mathcal {L}' \in \operatorname {\mathrm {Pic}}(P')$ , with the following properties:

1. 1. $P'$ is a partial coarsening of $P^{\mathrm {tw}}$ , while $\tau '$ is representable;

2. 2. is isomorphic to the pullback of ,

3. 3. , and so $(\mathcal {L}')^{\otimes 2}$ is isomorphic to the pullback of $L^{\otimes 2}$ ;

4. 4. $v_i^*(\mathcal {L}'|_{P^{\mathrm {tw}}}) \cong v_i^*\mathcal {L}$ for each $i = 1, \ldots , n$ .

2.3 The double cover: module structure

We will construct a Gorenstein double cover $\overline {\psi }\colon \overline {C}\to \overline {P}$ by first constructing the following commutative diagram of curve-line bundle pairs:

(2.1)

Then we will compose with the commutative square of Lemma 2.1.

For this, we will first construct an $\mathcal {O}_{\overline {P}'}$ -algebra , and then set

Let us denote by $\overline {\mathcal {F}}$ , and let $\overline {F}$ denote its pushforward to $\overline {P}$ . The latter is a rank one, torsion-free (i.e., depth one) sheaf on $\overline {P}$ , which fails to be a line bundle in two cases:

1. 1. when $\overline {P}$ has worse than nodal singularities, because $\omega _{\overline {P}}$ is not locally free there;

2. 2. and at the odd nodes of $\overline {P}$ , because both $\omega _{\overline {P}'}$ and are line bundles over those nodes, but the former comes from $\overline {P}$ and the latter does not, so $\overline {\mathcal {F}}$ has a nontrivial character.

As a consequence, $\overline {\psi }$ will fail to be flat at those points. However, since $\overline {P}$ is flat over the base, then so is $\overline {C}$ .

2.4 The double cover: algebra structure

In order to give $\mathcal {O}_{\overline {C}}$ an $\mathcal {O}_{\overline {P}'}$ -algebra structure, we need a cosection $\overline {\mathcal {F}}^{\otimes 2}\to \mathcal {O}_{\overline {P}'}$ . Twisting by , it is equivalent to find an $\mathcal {O}_{\overline {P}'}$ -module map . Notice that everything comes from $\overline {P}$ , so we could as well work on the coarse curve for this subsection. Since $\tau '$ is representable and it only contracts rational curves, its higher direct images vanish, and $\operatorname {R}\!\tau ^{\prime }_*\mathcal {O}_{P'}=\mathcal {O}_{\overline {P}'}$ . A direct application of Grothendieck duality yields

$$\begin{align*}\omega_{\overline{P}'}=\operatorname{R}\!\mathcal{H}om(\operatorname{R}\!\tau^{\prime}_*\mathcal{O}_{P'},\omega_{\overline{P}'})=\operatorname{R}\!\tau^{\prime}_*\operatorname{R}\!\mathcal{H}om(\mathcal{O}_{P'},(\tau')^!\omega_{\overline{P}'})=\tau^{\prime}_*\omega_{P'}. \end{align*}$$

Recall from §1.2 that the fundamental sequence of log geometry associates to every section $\bar \gamma $ of the characteristic monoid $\overline M^{\text {gp}}$ an $\mathcal {O}^*$ -torsor of lifts to the log structure $M^{\text {gp}}$ , whose associated line bundle we denote by $\mathcal {O}(-\bar \gamma )$ . Moreover, if $\overline \gamma \geq 0$ in the partial order of $\overline M^{\text {gp}}$ induced by $\overline M$ , the log structure map $\alpha \colon M\to \mathcal {O}^*$ endows $\mathcal {O}(-\bar \gamma )$ with a cosection, making it into a generalised Cartier divisor. Putting together the (vertical) divisor of $\lambda _T$ with the (horizontal) branch divisor $\mathbf {b}$ , adjunction gives us a map

(2.2) $$ \begin{align} (\tau')^*\omega_{\overline{P}'}^{\otimes 2}\to(\tau')^*\tau^{\prime}_*\omega_{P'}^{\otimes2}\to\omega_{P'}^{\otimes2}\to\omega_{P'}^{\otimes2}(2\lambda_T+\mathbf b)=(\mathcal{L}')^{\otimes 2} \end{align} $$

pushing forward to the desired

Note that there is an involution $\bar \iota $ of $\mathcal {O}_C$ over $\mathcal {O}_T$ acting as $-1$ on sections of $\overline {\mathcal {F}}$ .

Also, note that the fibres of $\overline {C}$ fail to be reduced whenever there is a component $P_1$ of P in the support of $\lambda _T$ such that the degree of $L^{\otimes 2}$ on $P_1$ is strictly positive. Indeed, the section $\mathcal {O}_{P'}\to \mathcal {O}_{P'}(2\lambda _T)$ is constantly $0$ along such a component, and $P_1$ is not contracted by $\tau '$ ; so the algebra structure of $\overline {C}$ over the generic point of $P_1$ is isomorphic to $\mathbf {k}(t)[\epsilon ]/(\epsilon ^2)$ .

2.5 The Gorenstein property

We argue that $\overline {C}$ is Gorenstein. More precisely, its dualising sheaf $\omega _{\overline {C}}$ can be identified with the line bundle . Duality for the finite morphism $\overline {\psi }'$ gives

Now, $\mathcal {O}_{\overline {P}'}\to \mathcal {H}om(\omega _{\overline {P}'},\omega _{\overline {P}'})$ is an isomorphism because everything is pulled back from the coarse curve $\overline {P}$ , and the statement is true there by [Reference HartshorneHar07, Corollary 1.7]. So we get a morphism , or equivalently, . Since $\overline {\psi }'$ is affine, and therefore $\overline {\psi }^{\prime }_*$ is exact, it is enough to show that this is an isomorphism after pushforward along $\overline {\psi }'$ , which follows from push-pull and the above:

2.6 The contraction morphism

We argue that there exists a morphism $\sigma \colon C\to \overline {C}$ covering $\tau '$ . Since $\overline {\psi }'$ is affine, it is enough to define an $\mathcal {O}_S$ -algebra map $\mathcal {O}_{\overline {C}}\to \sigma _*\mathcal {O}_C$ after pushing forward along $\overline {\psi }'$ . Since we want $\overline {\psi }^{\prime }_*\sigma _*=\tau ^{\prime }_*\psi ^{\prime }_*$ , by adjunction, it is enough to define a morphism $(\tau '\psi ')^*\overline {\psi }^{\prime }_*\mathcal {O}_{\overline {C}}\to \mathcal {O}_C$ . We focus on $\overline {\mathcal {F}}$ since the pullback of $\mathcal {O}_{\overline {P}'}$ is naturally identified with $\mathcal {O}_C$ . The map is induced by $(\tau ')^*\omega _{\overline {P}'}\to \omega _P'$ (see 2.4) and by the effective Cartier divisor $\mathbf r+\lambda $ on C as follows:

This an isomorphism in the complement of the ramification and contracted loci.

2.7 The arithmetic genus

Finally, we note that the arithmetic genus of $\overline {C}$ is the same as that of C. This follows from smoothing and the following important observation: the construction of Diagram (2.1) – in particular, of the contraction $\tau $ , the double cover $\overline {\psi }$ and the contraction $\sigma $ – commutes with arbitrary base-change.

The arithmetic genus of $\overline {C}$ can also be computed directly. Since $\overline {\psi }'$ is affine, it is enough to compute . Since $\omega _{\overline {P}'}$ is the pullback of $\omega _{\overline {P}}$ and since $\overline {P}$ is rational, $h^0(\overline {P}',\omega _{\overline {P}'})=0$ . However, , since $\mathcal {L}'$ is a nonnegative line bundle of total degree $g-1$ on the twisted rational curve $P'$ .

Summing up, we have proved the following:

3.1 Local equations

Suppose now that p is a point of $\overline {P}$ with $\ell $ branches. Write $s_i$ for a local parameter along the ith branch of P above p (we shall replace these by unit multiples if needed, taking advantage of $\mathbf {k}$ being algebraically closed). Local equations of $\overline {P}$ at p are

Now write q for the point of $\overline {C}$ above p. We obtain the local ring of $\overline {C}$ at q by adjoining a variable $u_i$ for every local generator of $\overline {F}$ (or $\omega _{\overline {P}}$ ) at p, subject to the relations expressing the multiplicative structure. Recall that the latter is determined by twisting with $2\lambda _T$ and $\mathbf b$ . Let $m_i$ be the (positive) slope of $2\lambda _T$ at the ith branch (note that this is not the slope of $\lambda $ at the non-ramified edges of $\operatorname {Supp}(\lambda )$ ). Nonreduced (double) components, also called split ribbons, arise in $\overline {C}$ when a component of $\operatorname {Supp}(\lambda _T)$ is not contracted under $\tau $ . We encode this by writing

$$\begin{align*}\delta_i = \begin{cases} 0 & \text{ if } \lambda_T> 0 \text{ on the}\ i\text{th branch} \\ 1 & \text{ else.} \end{cases} \end{align*}$$

Then, if $\ell \neq 1$ , we have the ring

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q} = A[u_2, \ldots, u_\ell] / I, \end{align*}$$

where $u_2, \ldots , u_\ell $ represent the differentials $\frac {ds_1}{s_1} -\frac {ds_2}{s_2}, \ldots , \frac {ds_1}{s_1}-\frac {ds_\ell }{s_\ell }$ , and I is generated by

1. 1. $s_1(u_i - u_j)$ for each $2 \leq i < j \leq \ell $ ;

2. 2. $s_iu_j$ for each $i \neq j$ , $2 \leq i,j \leq \ell $ ;

3. 3. $u_i^2 - \delta _1s_1^{m_1} - \delta _is_i^{m_i}$ for each $i = 2,\ldots , \ell $ ;

4. 4. $u_iu_j - \delta _1s_1^{m_1}$ for each $i \neq j$ with $2 \leq i,j \leq \ell $ ,

where the first two equations describe the A-module structure of $\omega _{\overline {P}}$ , and the last two describe the multiplication as in Equation (2.2).

If $\ell = 1$ and p is the image of a contracted component, the formula above needs a correction to account for the difference between $\omega _{\overline {P}}$ (being generated by $ds_1$ ) and (with $\omega _P$ being generated by $\frac {ds_1}{s_1}$ ). More precisely, if $\tilde p$ denotes the node of P over p, the natural map $\tau ^*\omega _{\overline {P}} \cong \tau ^*\tau _*\omega _{P} \to \omega _{P}$ is induced by twisting by $\tilde {p}$ . We thus have

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q} = A[u] / (u^2 - \delta_1s_1^{m_1 + 2}), \end{align*}$$

which is either a germ of an $A_{m_1 + 1}$ singularity (if $\delta _1\neq 0$ ) or a ribbon (if $\delta _1=0$ ).

Similarly (but easier), if $\ell = 1$ and p belongs to $\mathbf b$ , we have

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q} = A[u] / (u^2 - \delta_1s_1), \end{align*}$$

either a germ of a ribbon or a point of ramification of the cover; and if $\ell = 1$ and p does not belong to $\mathbf b$ , we have

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q} = A[u] / (u^2 - \delta_1), \end{align*}$$

either a germ of a ribbon or a trivial part of the double cover.

3.2 Examples

We recover some familiar examples of curve singularities of low genus.

Example 3.1. We construct a genus one singularity with six branches, cf. [Reference SmythSmy11, Reference BozleeBoz21].

In green, we write the slopes of $\lambda _T$ , and in blue instead, the number of branch points. The line bundle L is trivial on the central vertex of the tree, so the corresponding component is contracted into an ordinary (rational) $3$ -fold point. The sheaf $\omega _{\overline {P}}\otimes \mathcal {O}_{\overline {P}}(-1)$ has two generators $u_{2}$ and $u_{3}$ , corresponding to the generators $(\frac {\operatorname {d}s_1}{s_1},-\frac {\operatorname {d}s_2}{s_2},0)$ and $(\frac {\operatorname {d}s_1}{s_1},0,-\frac {\operatorname {d}s_3}{s_3})$ . The resulting singularity has local equations of the form

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q}=\mathbf{k}[\![s_1,s_2,s_3]\!][u_{2},u_{3}]/(s_is_j,u_{2}^2-s_1^2-s_2^2,u_{3}^2-s_1^2-s_3^2,s_1(u_2-u_3),s_2u_3,s_3u_2,u_2u_3-s_1^2),\end{align*}$$

which is isomorphic to $\mathbf {k}[\![x_1,\bar x_1,x_2,\bar x_2, x_3]\!]/I_6$ from [Reference SmythSmy11, Proposition A.3] via

$$\begin{align*}s_1=x_1-\bar x_1,s_2=x_2-\bar x_2,s_3=\frac{1}{2}x_3,u_{2}=x_1+\bar x_1-(x_2+\bar x_2), u_{3}=x_1+\bar x_1-x_3.\end{align*}$$

Example 3.2. We construct a genus two singularity ‘of type I” with three branches; cf. [Reference BattistellaBat22].

The line bundle L is trivial on the middle component of the chain, which is contracted to a node. The dualising sheaf $\omega _{\overline {P}}$ is itself a line bundle, with local generator u. We obtain

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q}=\mathbf{k}[\![s_1,s_2]\!][u]/(s_1s_2,u^2-s_1^2-s_2^3),\end{align*}$$

which is isomorphic to $\mathbf {k}[\![x_1,\bar x_1,x_2]\!]/(x_1x_2-\bar x_1x_2,x_1\bar x_1-x_2^3)$ [Reference BattistellaBat22, Equation (4) on p.11] via

$$\begin{align*}s_1=\frac{1}{2}(x_1-\bar x_1),s_2=x_2,u=\frac{1}{2}(x_1+\bar x_1).\end{align*}$$

Example 3.3. We construct a $(2)$ -tailed ribbon of genus two; cf. [Reference Battistella and CarocciBC23, Definition 2.21].

In this case, $\overline {P}$ is isomorphic to P. Local equations:

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q}=\mathbf{k}[\![s_1,s_2]\!][u]/(s_1s_2,u^2-s_1^2),\end{align*}$$

which is isomorphic to $\mathbf {k}[\![x_1,x_2,y]\!]/(x_1x_2,(x_1-x_2)y)$ [Reference Battistella and CarocciBC23, Example 2.20] via

$$\begin{align*}x_1=u+s_1,x_2=u-s_1,y=s_2.\end{align*}$$

Example 3.5. We construct a nonreduced singularity of genus three.

Again, $\overline {P}$ is isomorphic to $P=L\cup R$ . Local equations:

$$\begin{align*}\hat{\mathcal{O}}_{\overline{C},q}=\mathbf{k}[\![s_1,s_2]\!][u]/(s_1s_2,u^2-s_1^3).\end{align*}$$

The double structure on the right component R is given by the line bundle $\mathcal {O}_R(-\frac {1}{2}\mathbf b-\lambda _T)=\mathcal {O}_R(-2)$ ; hence, the ribbon $\overline {R}$ has genus $1$ . We can obtain the curve $\overline {C}$ by gluing the cuspidal curve $\overline {L}$ with the ribbon $\overline {R}$ along a length two subscheme, which checks out to give $p_a(\overline {C})=3$ .

3.3 Gluing

The preimage in $\overline {C}$ of a branch of $\overline {P}$ is either a double curve or an $A_m$ singularity. Here, we explain how to recover $\overline {C}$ by gluing them along some tangent directions.

The subscheme cut out by $s_i$ for $i \neq 1$ and $u_i - u_j$ for $i < j$ is isomorphic to

(3.1)

where $u_1$ is the common image of $u_2,\ldots , u_m$ . It is either a germ of a ribbon or an $A_{m_1 - 1}$ singularity.

Similarly, for each $j = 2,\ldots , \ell $ , the subscheme cut out by $s_i$ for $i \neq j$ and $u_i$ for $i \neq j$ is isomorphic to

which is again either a germ of a ribbon or an $A_{m_j - 1}$ singularity.

If we only restrict down to the subscheme cut out by $s_1$ , we find that we get

(3.2)

the transverse union of the singularities for $j = 2, \ldots , \ell $ above.

Our next claim is that the singularity at q is the result of gluing the tangent vector $\frac {\partial }{\partial u_1}$ of Spec of (3.1) with the tangent vector $\sum _{i = 2}^\ell \frac {\partial }{\partial u_i}$ of Spec of (3.2).

To see this, consider the sequence

To see that the first map is injective, observe that the kernel is contained in $(s_1) \cap (s_2,\ldots , s_\ell ) = 0$ . Note that both $\langle s_1, 0 \rangle $ and $\langle 0, s_i \rangle $ for $i = 2, \ldots , \ell $ are in the image of the first map, so Q is supported on $V(s_1,\ldots , s_\ell )$ . Restricting to this vanishing, we find

$$\begin{align*}0 \to k[u_2,\ldots, u_\ell]/(u_2,\ldots, u_\ell)^2 \to k[u_1]/u_1^2 \times \frac{k[u_2,\ldots, u_\ell]}{(u_2, \ldots, u_\ell)^2} \to Q \to 0. \end{align*}$$

The first map clearly admits a retract, so we conclude $Q \cong k[\epsilon ]/\epsilon ^2$ . This yields the claim.

3.4 Normalisation

The normalisation $\overline {C}^\nu $ of the germ of $\overline {C}$ at q can be computed as follows. Consider

From this, we see that it is enough to understand the normalisation of the $A_m$ -singularities and of ribbons, and put these formulae together. Assume that $\delta _1=1$ and $m_1$ is even; the other cases are left to the avid reader. Renumbering $\{2,\ldots ,\ell \}$ , we may assume that

• ○ for $i=2,\ldots ,h$ , we have $\delta _i=1$ and $m_i$ even,

• ○ for $i=h+1,\ldots ,k$ , we have $\delta _i=1$ and $m_i$ odd,

• ○ for $i=k+1,\ldots ,\ell $ , we have $\delta _i=0$ .

The normalisation is then given by the ring

with ring homomorphism

(3.3) $$ \begin{align} s_i\mapsto\left. \begin{cases} (a_i,b_i)&\text{for }i=1,\ldots,h\\ c_i^2&\text{for }i=h+1,\ldots,k\\ d_i&\text{for }i=k+1,\ldots,\ell \end{cases} \right. \qquad u_i\mapsto \left. \begin{cases} (a_1^{m_1/2},-b_1^{m_1/2},a_i^{m_i/2},-b_i^{m_i/2})&\text{for }i=2,\ldots,h\\ (a_1^{m_1/2},-b_1^{m_1/2},c_i^{m_i})&\text{for }i=h+1,\ldots,k\\ (a_1^{m_1/2},-b_1^{m_1/2},0)&\text{for }i=k+1,\ldots,\ell\\ \end{cases}\right. \end{align} $$

Notice that $\frac {m_i}{2}$ (resp. $m_i$ ) is precisely the slope of $\lambda $ on the edge corresponding to $q_i,\ i=1,\ldots ,h$ (resp. $i=h+1,\ldots ,k$ ).

3.5 Differentials

For this section, we assume that $\overline {C}$ is reduced. Recall that the conductor ideal of the normalisation $\nu \colon C^{\nu }\to \overline {C}$ is $\mathfrak {c}=\operatorname {Ann}(\nu _*\hat {\mathcal {O}}_{C^\nu ,q}/\hat {\mathcal {O}}_{\overline {C},q})$ ; it is the largest ideal of $\hat {\mathcal {O}}_{C,q}$ that is also an ideal of $\nu _*\hat {\mathcal {O}}_{C^\nu ,q}$ . It follows from the explicit parametrisation in the previous section that

(3.4) $$ \begin{align} \mathfrak{c}=\left(a_i^{m_i/2+1},b_i^{m_i/2+1},c_i^{m_i+1}\right). \end{align} $$

From this, we can verify that $\overline {C}$ is Gorenstein in a second way.

Proof. The following criterion is due to Serre (see, for instance, [Reference Altman and KleimanAK70, Proposition VIII.1.16]): $\overline {C}$ is Gorenstein if and only if $\dim _{\mathbf {k}}(\mathcal {O}_{C^\nu }/\mathfrak c)=2\delta $ . Now,

(3.5) $$ \begin{align} \delta=g(q)+2h+k-1, \end{align} $$

where $2h+k$ is the number of branches of q, and $g(q)$ is the genus of the singularity.

The latter is the same as the genus of the subcurve of C contracted to it. This corresponds to a connected component of the support of $\lambda _T$ . Since the following formulae are stable under edge contraction, we may as well assume that there is a single vertex v in the support of $\lambda _T$ , resp. irreducible component $C_v$ of C contracting to q. The genus of this component is determined by the Riemann–Hurwitz formula

(3.6) $$ \begin{align} 2g(C_v)+2=b+k, \end{align} $$

where b is the number of branch points supported on $C_v$ , and k the number of odd nodes (for $\mathbf b$ ) adjacent to v. However, balancing $\lambda _T$ at v as in Equation (1.5), we find

(3.7) $$ \begin{align} \operatorname{div}(\lambda_T)=\sum_{i=1}^{h+k}\frac{m_i}{2}=\text{val}(v)-2+\frac{b}{2}.\end{align} $$

Finally, from the above formula for the conductor, we find that

$$ \begin{align*} \dim_{\mathbf{k}}(\mathcal{O}_{C^\nu}/\mathfrak c)&=\sum_{i=1}^{h+k}m_i+2h+k &\text{by eq. }(3.4)\\ &=2\text{val}(v)-4+b+2h+k &\text{by eq. }(3.7)\\ &=b+k-4+2(2h+k) &\text{by a simple manipulation}\\ &=2(g+2h+k-1)=2\delta. &\text{by eq. }(3.6) \text{ and eq. }(3.5).\\[-38pt] \end{align*} $$

We can also describe the dualising bundle of $\overline {C}$ more explicitly.

Corollary 3.8. A local generator of $\omega _{\overline {C}}$ at q is given by

$$ \begin{align*} \eta&=\frac{\operatorname{d}\!s_1}{us_1}-\sum_{i=2}^{k}\frac{\operatorname{d}\!s_i}{u_is_i} \\ &=\frac{\operatorname{d}\!a_1}{a_1^{m_1/2+1}}-\frac{\operatorname{d}\!b_1}{b_1^{m_1/2+1}}-\sum_{i=2}^{h}\left(\frac{\operatorname{d}\!a_i}{a_i^{m_i/2+1}}-\frac{\operatorname{d}\!b_i}{b_i^{m_i/2+1}}\right)-\sum_{i=h+1}^{k}\frac{\operatorname{d}\!c_i}{c_i^{m_i+1}}, \end{align*} $$

and, if we write $(C^{\nu },q_i,\bar q_i,q_j)_{\substack {i=1,\ldots ,h\\j=h+1,\ldots ,k}}$ for the pointed normalisation of $\overline {C}$ at q, then

$$\begin{align*}\nu^*\omega_{\overline{C}}=\omega_{C^{\nu}}\left(\sum_{i=1}^h\left(\frac{m_i}{2}+1\right)\left(q_i+\bar q_i\right)+\sum_{i=h+1}^k (m_i+1)q_i\right).\end{align*}$$

Proof. Recall Rosenlicht’s theorem [Reference Altman and KleimanAK70, Proposition VIII.1.16]: for a reduced curve $\overline {C}$ , sections of the dualising sheaf $\omega _{\overline {C}}$ can be identified with meromorphic differentials $\eta $ on the normalisation $C^\nu $ such that, for all regular functions f on $\overline {C}$ , one has

(3.8) $$ \begin{align} \sum_{q_i\in\nu^{-1}(q)}\operatorname{Res}_{q_i}(f\eta)=0. \end{align} $$

This implies that the order of vanishing of the conductor at $q_i$ is an upper bound for the order of pole of sections of $\omega _{\overline {C}}$ at $q_i$ : if t is a local parameter of $C^\nu $ at $q_i$ , and $t^\mu $ is a section of $\mathfrak {c}$ (and in particular of $\hat {\mathcal {O}}_{\overline {C},q}$ ), no meromorphic differential with pole order $\mu +1$ or higher at $q_i$ can ever descend to $\omega _{\overline {C}}$ . Under this condition, Equation (3.8) is automatically satisfied for all $f\in \mathfrak {c}$ .

Since $\hat {\mathcal {O}}_{\overline {C},q}/\mathfrak {c}$ is generated by $\left \langle 1,s_i,\ldots ,s_i^{m_i/2},u_j\right \rangle _{\substack {i=1,\ldots ,k;\\j=2,\ldots ,k}}$ as a $\mathbf {k}$ -vector space, it is easy to check that the meromorphic differential $\eta $ from the statement descends to a local section of $\omega _{\overline {C}}$ .

Moreover, since the latter is a line bundle and $\eta $ has the highest possible pole order at every $q_i$ , it follows that $\eta $ is indeed a generator: pick a local generator $\eta ^\prime $ , and write $\eta =g\eta ^\prime $ for some $g\in \hat {\mathcal {O}}_{C,q}$ ; then the order of pole of $\eta $ at $q_i$ is lower than that of $\eta ^\prime $ (or $\eta $ vanishes on the entire branch containing $q_i$ , which it does not), so they have to be equal, so g has to be an invertible scalar.

The second claim follows from Noether’s formula [Reference CataneseCat82, Proposition 1.2]: $\omega _{C^\nu }=\nu ^*\omega _{\overline {C}}(\mathfrak {c}).$

5.1 Abelian differentials in general

The moduli space of Abelian differentials has at most three connected components (depending on the multiplicity $\mu $ of the zeroes) [Reference Kontsevich and ZorichKZ03]. In general, connected components of the space of multiscale differentials are not irreducible. The global residue condition (GRC) was introduced in [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG⁺18] to single out the smoothable differentials. Roughly speaking, it says that the sum of the residues at poles of level i that are joined by a connected subcurve at level $i+1$ must vanish, despite the possibility that the corresponding nodes belong to different subcurves at level i. The proof of necessity goes by cutting the generic fibre of a smoothing along the vanishing cycle corresponding to these nodes, and applying Stokes’ theorem to compute the integral of the abelian differential on the resulting surface with boundary. The proof of sufficiency is more complicated and based on a refined plumbing construction. With the logarithmic understanding of the moduli space of generalised multiscale differentials reached in [Reference Chen and ChenCC19, Reference Chen, Grushevsky, Holmes, Möller and SchmittCGH⁺22], the GRC remains the only ingredient of [Reference Bainbridge, Chen, Gendron, Grushevsky and MöllerBCG⁺19] relying on transcendental techniques. A purely algebraic description of smoothable differentials is contained in the following conjecture, originally due to Ranganathan and Wise.

Here, $\widetilde {C}_i$ is determined by $\eta $ as follows, in order to ensure that the twist of the canonical bundle be trivial on the upper levels, and to avoid nonreduced components in the contractions $\overline {C}_i$ . Indeed, ribbons appear when $\omega _C(\lambda _i)$ has positive degree on the support of $\lambda _i$ . This happens precisely when at least one zero of order $m\geq 1$ is contained in the support of $\lambda _i$ . In this case, since we have a nontrivial logarithmic structure of marking type at the zero, we can subdivide the corresponding leg at level i; classically, this means sprouting a new semistable rational component at the marking. In the natural coordinates $[x_0:x_1]$ with respect to the two special points, the differential $\eta $ can be extended uniquely to the new component $\tilde {v}$ by setting $\eta _{\tilde {v}}=x_0^{m}\operatorname {d}\!x_0$ ; the choice of a nonzero scalar is compensated by the automorphisms of the underlying curve. Notice that this differential does not contribute to the GRC, since $m+2>1$ . The mere existence of the nonzero differentials at levels higher than i guarantees that the twist of the canonical bundle by $\lambda _i$ will be trivial (not just numerically). We provide the following ad hoc example in the hope of acquainting the reader with the log modification procedure.

5.2 Hyperelliptic differentials

The connected component consisting of hyperelliptic differentials is already irreducible [Reference Chen and ChenCC19, Proposition 5.16]. This is proved by identifying the moduli space of hyperelliptic differentials with a moduli space of quadratic differentials on rational curves. We therefore view the following result as a first proof of concept for Conjecture 5.1.

Example 5.6. Let $(C,\eta )$ be a genus $3$ hyperelliptic multiscale differential whose level graph is the following:

Let $\eta $ restrict to $\operatorname {d}\!z$ on the two elliptic curves. Choose coordinates on the rational curves $R_i$ at level $-1$ in such a way that the nodes are $0$ and $\infty $ , and the zeroes a and b. Then $\eta $ restricts to

$$\begin{align*}\alpha_i(t-a)(t-b)\frac{\operatorname{d}\!t}{t^2}\end{align*}$$

on the rational curve $R_i,\ i=1,2.$ Here, $\iota $ -anti-invariance forces $\alpha _1=-\alpha _2$ . Contracting the subcurves at level $0$ , we obtain two rational curves joined at two tacnodes. There is a linear condition for a meromorphic differential with poles of order two on the pointed normalisation to descend to the tacnode (cf. [Reference SmythSmy11, §2.2]) which is analogous to the condition $\alpha _1=-\alpha _2$ from above. However, any choice of $\alpha _i$ gives rise to a generalised multiscale differential.

Remark 5.7. If $a=-b$ , the residues are zero. By varying the $\alpha _i$ , we thus get an example of a multiscale differential which satisfies the GRC but is not anti-invariant. This will be the limit of differentials on smooth, non-hyperelliptic curves. There are of course even more examples of non-hyperelliptic differentials on a hyperelliptic curve if we do not impose that the sets of zeroes and poles are invariant under the hyperelliptic involution.