2.1 Tropical curves and their moduli

In [Reference Cavalieri, Gross and Markwig6], moduli spaces of tropical curves are introduced as stacks over the category of tropical spaces. We review the relevant definitions.

A TPL-space is a pair $({ X}, {\mathrm {PL}}_{ X})$ consisting of a topological space ${ X}$ and a sheaf ${\mathrm {PL}}_{X}$ of continuous -valued functions on ${ X}$ that is locally isomorphic to $(U, {\mathrm {PL}}_U)$ , the sheaf of continuous, integral, piecewise linear functions of some open subset U of a rational polyhedral set P in . Here, two pairs $(Y, \mathcal F_Y)$ and $(Z, \mathcal F_Z)$ of topological spaces and sheaves of functions on them are isomorphic if there exists a homeomorphism $f\colon Y\to Z$ so that for a function $\phi $ on an open subset of Z we have that $\phi \circ f$ is a section of $\mathcal F_Y$ if and only if $\phi $ is a section of $\mathcal F_Z$ . For a TPL-space ${ X}$ we denote by ${\mathrm {PL}}^{\mathrm {fin}}_{ X}$ the subsheaf of -valued piecewise linear functions. A morphism of TPL-spaces is a continuous map that pulls piecewise linear functions (section of ${\mathrm {PL}}$ ) back to piecewise linear functions.

A tropical space is a pair $(\mathtt {X}, \operatorname {\mathrm {Aff}}_{\mathtt X})$ consisting of a TPL-space ${\mathtt X}$ and a subsheaf $\operatorname {\mathrm {Aff}}_{\mathtt X}\subseteq {\mathrm {PL}}^{\mathrm {fin}}_{\mathtt X}$ , the sheaf of affine functions, that contains the constant sheaf . A morphism of tropical spaces is morphism of TPL-spaces that pulls affine functions back to affine functions. In that case, we also say that the underlying morphism of TPL-spaces is linear. The tropical cotangent bundle is the quotient sheaf .

A tropical curve is a one-dimensional tropical space ${ \mathtt C}$ , together with a finitely supported genus function . The genus of a tropical curve is $\sum _{x\in { \mathtt C}} \gamma (x)+ \mathrm {b_1}({ \mathtt C})$ . A cycle rigidification of a genus-g tropical curve ${ \mathtt C}$ is a g-tuple of elements spanning and such that each element is either $0$ or coming from a circuit in ${ \mathtt C}$ . We denote by ${\overline {\mathcal V}_{g,n}}$ the set of isomorphism classes of cycle-rigidified stableFootnote ² n-marked tropical curves of genus g. Forgetting the metric structure (that is, only retaining the topological graph and the cycle rigidification, the so-called combinatorial type) stratifies ${\overline {\mathcal V}_{g,n}}$ into locally closed sets that can be naturally identified with open cones. For a combinatorial type $\Gamma $ we denote by $\sigma _\Gamma ^\diamond $ the corresponding open cone and by $\sigma _\Gamma $ its closure. The natural face morphisms among the closed cones gives ${\overline {\mathcal V}_{g,n}}$ the structure of an extended cone complex. Given an $n\sqcup \{\star \}$ -marked curve, one can forget the $\star $ -mark and stabilize. As stabilization is a contraction, a cycle rigidification on the original graph induces a cycle rigidification on the stabilized graph. We thus obtain a map

(2.1) $$ \begin{align} \mu_\star \colon {\overline {\mathcal V}_{g,n\sqcup\{\star\}}}\to {\overline {\mathcal V}_{g,n}} , \end{align} $$

the forgetful map, which is a morphism of extended cone complexes. The fiber of $\mu _\star $ over a point ${\overline {\mathcal V}_{g,n}}$ corresponding to a cycle-rigidified tropical curve ${ \mathtt C}$ is isomorphic to ${ \mathtt C}$ . In particular, there are n natural sections of $\mu _\star $ and a global genus function .

A family of genus- g, n-marked stable TPL-curves over a TPL-space ${ X}$ is a morphism $\pi \colon C\to X$ of TPL-spaces, together with n sections $s_i\colon { X}\to C$ , $1\leq i\leq n$ , and a genus function , such that every $x\in { X}$ has an open neighborhood U for which there exists a map $f\colon U\to {\overline {\mathcal V}_{g,n}}$ and an isomorphism $\pi ^{-1}U\cong {\overline {\mathcal V}_{g,n\sqcup \{\star \}}} \times _{\mu _\star ,{\overline {\mathcal V}_{g,n}},f} U$ over U that respects the sections and the genus function.

A family of genus- g, n-marked stable tropical curves over a tropical space ${\mathtt X}$ is a family of genus-g, n-marked TPL-curves $\Pi \colon \mathtt C\to {\mathtt X}$ satisfying the following conditions:

1. 1. for each $x\in {\mathtt X}$ , the fiber , equipped with the induced affine structure, is a stable tropical curve,

2. 2. for every $y\in \mathtt C$ , the sequence

   (2.2) $$ \begin{align} 0 \to \Omega^1_{{\mathtt X}, \Pi(y)} \to \Omega^1_{{\mathtt C}, y} \to \Omega^1_{{\mathtt C}_{\Pi(y)},y} \to 0 \end{align} $$
   induced by pull-back from ${\mathtt X}$ and restriction to ${\mathtt C}_{\Pi (y)}$ , is exact. This can be phrased equivalently as saying that affine functions that are constant on fibers are pull-backs of affine functions on the base.

There is a natural affine structure on the space ${\overline {\mathcal V}_{g,n}}$ induced by cross ratio functions. We describe these functions, starting with ${\overline {\mathcal V}_{0,4}}=\overline {\mathcal M}^{\mathrm {trop}}_{0,4}$ . Choosing an ordering, for example, $((p_1,p_2), (p_3,p_4))$ , of the four markings defines two unique minimal paths in a curve $[{ \mathtt C}]\in \overline {\mathcal M}^{\mathrm {trop}}_{0,4}$ , one from $p_1$ to $p_2$ and one from $p_3$ to $p_4$ . The value of the cross ratio function $\xi _{((p_1,p_2),(p_3,p_4))}$ at $[{ \mathtt C}]$ is the signed length of the intersection of these two paths. Let $\Gamma $ be a combinatorial type of cycle-rigidified n-marked genus-g curves, let $\widetilde \Gamma $ denote the universal covering space of $\Gamma $ , and let $T\subset \widetilde \Gamma $ be a connected open subset all of whose vertices have genus $0$ . Note that T is automatically a tree, that is $h^1(T)=0$ , and its leaves are open edge segments. For every combinatorial type $\Gamma '$ specializing to $\Gamma $ , the preimage $T'$ of T under the induced retraction $\widetilde \Gamma '\to \widetilde \Gamma $ of universal covers is a connected open subset of $\widetilde \Gamma '$ not containing higher-genus vertices, and the retraction induces a bijection between the leaves of $T'$ and the leaves of T. Forgetting the rest of the curve, this produces for every $\Gamma '$ specializing to $\Gamma $ a map $\sigma _{\Gamma '}^\diamond \to \overline {\mathcal M}^{\mathrm {trop}}_{0, m}$ , where m is the number of leaves of T. Given a quadruple $((p_1,p_2),(p_3,p_4))$ of distinct leaves of T, we can postcompose with the forgetful morphism $\overline {\mathcal M}^{\mathrm {trop}}_{0,m}\to \overline {\mathcal M}^{\mathrm {trop}}_{0,4}$ that forgets all but the marks $p_1,\ldots ,p_4$ . From $\overline {\mathcal M}^{\mathrm {trop}}_{0,4}$ we can then pull-back the cross ratio function $\xi _{((p_1,p_2),(p_3,p_4))}$ to obtain a function on $\sigma _{\Gamma '}$ . For different $\Gamma '\to \Gamma $ , these functions glue together and define a cross ratio function $\xi _{(T,(p_1,p_2),(p_3,p_4))}$ on a neighborhood of $\sigma _\Gamma ^\diamond $ . This is the cross ratio function $\xi _c$ associated to the cross ratio datum $c=(T,(p_1,p_2),(p_3,p_4))$ (the tuple c is not literally a cross ratio datum in the sense of [Reference Cavalieri, Gross and Markwig6, Definition 4.8], but there is a natural induced cross ratio datum). The affine structure of ${\overline {\mathcal V}_{g,n}}$ is the sheaf generated by cross ratio functions and constants. It is in fact generated by constants and the cross ratio functions associated to primitive cross ratio data: these correspond to the cases where T is a neighborhood of a single vertex v or of a single edge e. The four markings $p_1,\ldots ,p_4$ are induced by two pairs of flags $((f_1,f_2),(f_3,f_4))$ at v (resp. at the end points of e) and we write $c_{((f_1,f_2),(f_3,f_4))}$ (resp. $c_{(e,(f_1,f_2),(f_3,f_4))}$ ) for the cross ratio datum induced by $(T,(p_1,p_2),(p_3,p_4))$ in the two respective cases and $\xi _{((f_1,f_2),(f_3,f_4))}$ (resp. $\xi _{(e,(f_1,f_2),(f_3,f_4))})$ for the associated cross ratio functions.

With these affine structures, the forgetful map $\mu _\star \colon {\overline {\mathcal V}_{g,n\sqcup \{\star \}}}\to {\overline {\mathcal V}_{g,n}}$ is a morphism of tropical spaces, but not a family of tropical curves. What fails is the condition on the fibers: it is true that all functions on ${\overline {\mathcal V}_{g,n\sqcup \{\star \}}}$ are harmonic on the fibers, but not every harmonic function on a fiber is the restriction of an affine function.

2.2 Cone complexes and toroidal embeddings

We recall how to associate a cone complex to a toroidal embedding without self intersections. The original reference is [Reference Kempf, Faye Knudsen, Mumford and Saint-Donat18], where what we call cone complex was called conical polyhedral complex with integral structures, and we refer there and to [Reference Abramovich, Caporaso and Payne1, Reference Ulirsch25, Reference Kajiwara16, Reference Cavalieri, Chan, Ulirsch and Wise5] for a more detailed treatment.

A (abstract) cone is a pair $(\sigma , M^\sigma )$ consisting of a topological space $\sigma $ and a lattice $M^\sigma $ of real-valued continuous functions on $\sigma $ with the property that the natural map maps $\sigma $ homeomorphically onto its image, a strictly convex rational polyhedral cone in . If we denote $M^\sigma _+=\{m\in M^\sigma : m\geq 0\}$ , then the image of $\sigma $ in is precisely . The faces of $(\sigma ,M^\sigma )$ are the pairs $(\tau , \{m\vert _\tau :m\in M^\sigma \})$ , where $\tau $ is a face of $\sigma $ . The relative interior of a cone $\sigma $ is the complement of all its proper faces; we denote it by $\sigma ^\diamond $ . A cone complex is pair $(\vert \Sigma \vert ,\Sigma )$ consisting of a topological space $\vert \Sigma \vert $ and a collection $\Sigma $ of cones whose underlying sets are closed subsets of $\vert \Sigma \vert $ and such that $\vert \Sigma \vert =\bigsqcup _{\sigma \in \Sigma }\sigma ^\diamond $ . Moreover, the set $\Sigma $ is closed under taking faces and intersections. A continuous function $\phi $ on a subset $U\subseteq \Sigma $ is strict piecewise linear if for all $\sigma \in \Sigma $ there exists $m\in M^\sigma $ and with $\phi \vert _{\sigma \cap U}=m\vert _{\sigma \cap U}+c$ . We denote the group of strict piecewise linear functions on U by ${\mathrm {sPL}}_\Sigma (U)$ .

For a cone $\sigma $ , the extended cone $\overline \sigma $ is given by . The extended cones of the cones in a cone complex $\Sigma $ can be glued to an extended cone complex $\overline \Sigma $ . The extended cone complex $\Sigma $ has a natural stratification $\overline \Sigma =\bigsqcup _{\tau \prec \sigma \in \Sigma }\sigma /\tau $ . Any choice of generators of $M^\sigma _+$ induces an embedding of an extended cone $\overline \sigma $ into for some and thus $\overline \sigma $ obtains the structure of a TPL-space. Declaring a function on an extended cone complex to be piecewise linear if and only if its restrictions to all extended cones of the complex are piecewise linear defines a TPL-structure on any extended cone complex.

A toroidal embedding without self-intersection is a pair $(X_0,X)$ consisting of a variety X and an open subset $X_0\subseteq X$ that locally looks like the inclusion of the big torus into a toric variety. More precisely, for every point $x\in X$ there exists a toric chart, that is, an open neighborhood U of x, a toric variety Y with big open torus T, and an étale morphism $f\colon U\to Y$ with $f^{-1}T=U\cap X_0$ .Footnote ³ The toroidal embedding X has an associated cone complex $\Sigma _X$ , with each $\sigma \in \Sigma _X$ corresponding to a stratum $O(\sigma )$ in the stratification of X induced by the toric charts. If we denote by $V(\sigma )$ the closure of $O(\sigma )$ , the strata closures $V(\rho )$ for rays $\rho \in \Sigma _X(1)$ are the irreducible components of the boundary $X\setminus X_0$ , and all other strata are connected components of $\bigcap _{\rho \in I}V(\rho )\setminus \bigcup _{\rho \notin I}V(\rho )$ for subsets $I\subseteq \Sigma _X(1)$ . For a cone $\sigma \in \Sigma _X$ , the monoid $M^\sigma $ is naturally identified with the set of effective boundary Cartier divisors near $O(\sigma )$ . In particular, the functions in ${\mathrm {sPL}}(\Sigma )$ with trivial constant part are in natural bijection with boundary Cartier divisors on X.

A dominant toroidal morphism of toroidal embeddings X and Y is a dominant morphism $X\to Y$ of schemes that can be expressed as a morphism of toric varieties in toric charts. A toroidal morphism from X to Y is any morphism of schemes $X\to Y$ that factors through a dominant toroidal morphism $X\to V(\sigma )$ for some $\sigma \in \Sigma _Y$ .Footnote ⁴ A dominant toroidal morphism $f\colon X\to Y$ induces a morphism ${\mathtt {Trop}}(f)\colon \Sigma _X\to \Sigma _Y$ , that is a continuous map $\vert \Sigma _X\vert \to \vert \Sigma _Y\vert $ mapping the cones of $\Sigma _X$ linearly into cones of $\Sigma _Y$ . The map ${\mathtt {Trop}}(f)$ extends to a morphism on the extended cone complexes. For $\tau \in \Sigma _Y$ , we have a natural identification of $\Sigma _{V(\tau )}$ with the subcomplex $\bigcup _{\tau \subseteq \sigma \in \Sigma _Y} \sigma /\tau $ of $\overline \Sigma _Y$ . In particular, for a not necessarily dominant toroidal morphism $f\colon X\to Y$ there is an induced morphism ${\mathtt {Trop}}(f)\colon \overline \Sigma _X\to \overline \Sigma _Y$ .

2.3 Tropical cycles

We present the notion of tropical cycles and their intersection pairing with tropical divisors in the generality developed in [Reference Cavalieri, Gross and Markwig6].

A k-weight on a cone complex $\Sigma $ is an equivalence class of pairs $(\Delta ,c)$ , where $\Delta $ is a proper subdivision of $\Sigma $ and is a map. Here, $\Delta (k)$ denote the set of k-dimensional cones of $\Delta $ , and the equivalence relation is induced by compatibility under refinements of $\Delta $ . Given an affine structure $\operatorname {\mathrm {Aff}}_\Sigma $ , which we assume to satisfy the condition that $\Omega ^1_\Sigma $ is constant on all cones, we define the balancing condition for weights as follows: a $1$ -weight represented by $(\Delta , c)$ is balanced with respect to an affine function $\phi \in \operatorname {\mathrm {Aff}}_\Sigma (\Sigma )$ with $\phi (0_\Sigma )=0$ (here, $0_\Sigma $ refers to the cone point of $\Sigma $ ), if we have $\sum _{\rho \in \Delta (1)} \phi (u_\rho )\cdot c(\rho )=0$ , where $u_\rho $ denotes the primitive lattice generator of a ray $\rho $ . We say that c is balanced if it is balanced with respect to every $\phi \in \operatorname {\mathrm {Aff}}_\Sigma (\Sigma )$ with $\phi (0_\Sigma )=0$ . More generally, a k-weight represented by $(\Delta , c)$ is balanced if for every $(k-1)$ -dimensional cone $\tau \in \Delta $ and every affine function $\phi \in \operatorname {\mathrm {Aff}}_\Sigma (\Delta ^\tau )$ with $\phi \vert _\tau =0$ , the induced $1$ -weight $\overline c$ on $\Sigma ^\tau /\tau $ is balanced with respect to the induced function $\overline \phi $ . Note that the balancing condition is independent of the choice of representative $(\Delta ,c)$ . A balanced k-weight is called a tropical k-cycle.

We say a function $\phi \in {\mathrm {sPL}}_\Sigma (\Sigma )$ is combinatorially principal at the cone $\sigma $ , and write $\phi \in {\mathrm {CP}}(\sigma )$ if there exists $\chi \in \operatorname {\mathrm {Aff}}_\Sigma (\Sigma ^\sigma )$ with $\chi \vert _\sigma =\phi \vert _\sigma $ . We say $\phi $ is combinatorially principal if it is combinatorially principal at every cone $\sigma \in \Sigma $ , and denote the group of combinatorially principal functions on $\Sigma $ by ${\mathrm {CP}}(\Sigma )$ . For a function $\phi \in {\mathrm {sPL}}_\Sigma (\Sigma )$ and a tropical $1$ -cycle A represented by $(\Delta ,c)$ , the intersection product $\phi \cdot A$ is the unique $0$ -cycle whose weight at the origin is given by

(2.3) $$ \begin{align} -\sum_{\rho\in \Delta(1)} \mathrm{slope}_\rho(\phi)c(\rho) , \end{align} $$

where $-\mathrm {slope}_\rho (\phi )=-\phi (u_\rho )$ is the incoming slope of $\phi $ at the origin, as required by the $\min $ convention. This only depends on the class of $\phi \in {\mathrm {sPL}}_\Sigma (\Sigma )/\operatorname {\mathrm {Aff}}_\Sigma (\Sigma )$ . Let $\phi \in {\mathrm {CP}}(\Sigma )$ and let $A=[(\Delta ,c)]$ be a tropical k-cycle on $\Sigma $ . The intersection product $\phi \cdot A$ is the tropical cycle represented by the $(k-1)$ -weight whose weight on $\tau \in \Delta (k-1)$ is given by the weight at the origin of $\overline \phi \cdot \overline c$ , where $\overline c$ is the tropical $1$ -cycle on $\Delta ^\tau /\tau $ induced by c and $\overline \phi $ is the function on $\Delta ^\tau /\tau $ induced by $\phi -\chi $ for some $\chi \in \operatorname {\mathrm {Aff}}_\Sigma (\Sigma )$ with $\chi \vert _\tau =\phi \vert _\tau $ . By construction, $\phi \cdot A$ depends only on the class of $\phi $ in ${\mathrm {CP}}(\Sigma )/\operatorname {\mathrm {Aff}}_\Sigma (\Sigma )$ .

We just presented the local theory of tropical cycles, which is all we need in this paper. These concepts can be globalized: there exists a sheaf $Z_k$ of tropical k-cycles on tropical spaces, a sheaf ${\mathrm {CP}}$ (denoted by $\mathrm {Rat}$ in [Reference Cavalieri, Gross and Markwig6, Definition 6.3]) of combinatorially principal functions, and an intersection-pairing

(2.4)

that locally looks like the pairing in the case of cone complexes with affine structure. The sheaf ${\mathrm {CP}}/\operatorname {\mathrm {Aff}}$ is the sheaf of combinatorially principal tropical Cartier divisors, sitting inside the sheaf ${\mathrm {PL}}^{\mathrm {fin}}/\operatorname {\mathrm {Aff}}$ of tropical Cartier divisors. The k-th tropical Chow group $A_k({ \mathtt X})$ of a tropical space ${ \mathtt X}$ is given by

(2.5)

2.4 Tropical line bundles and psi classes

A tropical line bundle on a tropical space ${ \mathtt X}$ is an $\operatorname {\mathrm {Aff}}_{ \mathtt X}$ -torsor. Equivalently, it is a morphism ${ \mathtt L}\to { \mathtt X}$ whose fibers are -torsors and which locally trivializes: it is locally isomorphic to as tropical spaces, and the isomorphism is an isomorphism of -torsors on each fiber. One can translate the two notions into each other by associating to a line bundle ${ \mathtt L}\to { \mathtt X}$ the sheaf of isomorphisms with the trivial line bundle, which is an $\operatorname {\mathrm {Aff}}_{ \mathtt X}$ -torsor because .

By [Reference Jell, Rau and Shaw15, Lemma 4.5], every tropical line bundle ${ \mathtt L}$ has a piecewise linear section. Any such section defines a tropical Cartier divisor $\mathtt D$ whose associated tropical line bundle is isomorphic to ${ \mathtt L}$ . In the language of torsors, the existence of a piecewise linear section means that given any $\operatorname {\mathrm {Aff}}_{ \mathtt X}$ -torsor, the associated ${\mathrm {PL}}$ -torsor is trivial. However, the associated ${\mathrm {CP}}$ -torsor is not necessarily trivial, that is the line bundle does not necessarily have a combinatorially principal section (i.e., the tropical Cartier divisor $\mathtt D$ from above is not necessarily combinatorially principal). If a tropical line bundle ${ \mathtt L}$ on a tropical space ${ \mathtt X}$ has a combinatorially principal section, any choice of such section defines the same element in $({\mathrm {CP}}/\operatorname {\mathrm {Aff}})({ \mathtt X})/{\mathrm {CP}}({ \mathtt X})$ (a combinatorially principal tropical Cartier divisor modulo linear equivalence), and this element defines a map

(2.6) $$ \begin{align} c_1({ \mathtt L})\colon A_*({ \mathtt X})\to A_{*-1}({ \mathtt X}) \end{align} $$

via the intersection pairing. The map $c_1({ \mathtt L})$ is called the first Chern class of ${ \mathtt L}$ . Given $\alpha \in A_\ast ({ \mathtt X})$ , we denote $c_1({ \mathtt L})(\alpha )$ by $c_1({ \mathtt L}) \frown \alpha $ to avoid proliferation of nested parentheses.

We recall the construction of tropical $\psi $ classes from [Reference Cavalieri, Gross and Markwig6]. Given a family of n-marked, stable tropical curves $\overline { \mathtt C}\to \overline { \mathtt B}$ , for every $1\leq i\leq n$ and the i-th section ${\mathtt s}_i\colon \overline { \mathtt B}\to \overline { \mathtt C}$ defines a tropical line bundle $\operatorname {\mathrm {Aff}}_{\overline { \mathtt C}}(k {\mathtt s}_i)$ in the form of an $\operatorname {\mathrm {Aff}}_{\overline { \mathtt C}}$ -torsor on the total space $\overline { \mathtt C}$ . Namely, every point in the image of ${\mathtt s}_i$ is the infinite point of a leg in its fiber. Therefore, we can define $\operatorname {\mathrm {Aff}}_{\overline { \mathtt C}}(k{\mathtt s}_i)$ as the subsheaf of $\iota _*\operatorname {\mathrm {Aff}}_{\overline { \mathtt C}\setminus {\mathtt s}_i(\overline { \mathtt B})}$ , where $\iota \colon \overline { \mathtt C}\setminus {\mathtt s}_i({ \mathtt B})\to \mathtt C$ is the inclusion, of all functions whose slope on the leg approaching ${\mathtt s}_i$ is $-k$ . The i-th tropical cotangent line bundle is the line bundle on $\overline { \mathtt B}$ defined as ${\mathtt s}_i^*\operatorname {\mathrm {Aff}}_{\overline { \mathtt C}}(-{\mathtt s}_i)$ .

If is the i-the cotangent bundle of a marked family of tropical curves, we call the first Chern class of ${ \mathtt L}$ the i-th $\psi $ class and denote it by

(2.7)

3.1 Affine structures on cone complexes

Let $\Sigma $ be a cone complex. A subcomplex S of $\Sigma $ is a union

(3.1) $$ \begin{align}S = \bigcup_{\sigma\in I}\sigma^\diamond,\end{align} $$

where I is a subset of cones of $\Sigma $ . We say that S is an open (resp. closed) subcomplex of $\Sigma $ if it defines an open (resp. closed) set.

For a cone $\sigma \in \Sigma $ , we define the open star $\Sigma ^\sigma $ of $\sigma $ to be the open subcomplex

(3.2) $$ \begin{align} \Sigma^\sigma= \bigcup_{\sigma\subseteq \tau} \tau^\diamond , \end{align} $$

given by the union of the relative interiors of all cones containing $\sigma $ . The usual star of $\sigma $ , consisting of the union of all cones containing $\sigma $ , is the closure of the open star, and we denote it by $\overline {\Sigma ^\sigma }.$ Footnote ⁵

Let X be a toroidal variety satisfying $(\ast )$ , and $\Sigma _X$ its cone complex as described in Section 2.2. Every $\sigma \in \Sigma _X$ corresponds to a stratum $O(\sigma )$ .

For a cone $\sigma \in \Sigma _X$ we denote by $X_\sigma $ the open neighborhood of $V(\sigma )$ given by

(3.3) $$ \begin{align} X_\sigma=\bigcup_{\tau\in \overline{\Sigma_X^\sigma}} O(\tau) . \end{align} $$

A strict piecewise linear function $\phi $ on $\Sigma _X^\sigma $ determines a Cartier divisor, and consequently an invertible sheaf on $X_\sigma $ : extend $\phi $ to $\overline {\Sigma _X^\sigma }$ by linearity, and for any ray $\rho \in \overline {\Sigma _X^\sigma }$ denote the slope of $\phi |_{\rho }$ by $\mathrm {slope}_{\rho }(\phi )$ . We define:

(3.4) $$ \begin{align} D_{X_\sigma}(\phi) = \sum_{\rho\in \overline{\Sigma_X^\sigma}} \mathrm{slope}_{\rho}(\phi) V(\rho) , & & \mathcal O_{X_{\sigma}}(\phi)= \mathcal O_{X_{\sigma}}\left(D_{X_\sigma}(\phi)\right). \end{align} $$

As shown in the following lemma, affine functions in the sense of Definition 3.1 are closed under restriction, hence we are not generating any new sections of $\operatorname {\mathrm {Aff}}_\Sigma $ by restriction.

Proof. Since $V(\sigma )$ is contained in $V(\tau )$ , $\mathcal O_{X_\tau }(\phi )\vert _{V(\tau )}$ being trivial implies that the bundle $\mathcal O_{X_\tau }(\phi )\vert _{V(\sigma )}$ is trivial as well. The assertion now follows from the fact that

(3.5) $$ \begin{align} \mathcal{O}_{X_\tau}(\phi)\vert_{X_\sigma}=\mathcal{O}_{X_\sigma}(\phi\vert_{\Sigma_\sigma}) . \end{align} $$

Remark 3.4. This notion of affine functions is related to the linear functions on Cartwright’s tropical complexes [Reference Cartwright4]. To compare the two notions, consider the situation where we are given a proper morphism such that X is smooth and connected and $\pi ^{-1}\{0\}$ is a reduced simple normal crossings divisor. In particular, the pair defines a toroidal embedding and $\pi $ is a dominant toroidal morphism. We obtain an induced morphism of cone complexes and the linear functions of [Reference Cartwright4] are defined on the fiber . To be linear in the sense of Cartwright, a function $\chi $ on $\Delta $ has to satisfy the condition that for each $\sigma \in \Sigma $ with $V(\sigma )$ a curve, there exists a function $\phi \in {\mathrm {sPL}}(\Sigma ^\sigma )$ with

(3.6) $$ \begin{align} \phi\vert_{\Delta\cap \Sigma^\sigma} &=\chi\vert_{\Delta\cap\Sigma^\sigma}, \qquad\text{and} \end{align} $$

(3.7) $$ \begin{align} \deg(\mathcal O_{X_\sigma}(\phi)\vert_{V(\sigma)}) &= 0. \end{align} $$

So Cartwright’s linearity is only checked locally around $\sigma \in \Sigma $ with $V(\sigma )$ a curve, and our condition that $\mathcal O_{X_\sigma }(\phi )\vert _{V(\sigma )}$ be trivial is weakened to it having degree $0$ . In other words, our notion of affine functions agrees with Cartwright’s locally around $\sigma \in \Sigma $ with , and differs from it otherwise.

We now show that the assignment of tropical spaces $\Sigma _X$ to toroidal varieties X is functorial.

Proposition 3.5. Let $f: X\to Y$ be a dominant morphism of toroidal varieties. Then the natural piecewise linear map

(3.8)

is a morphism of tropical spaces.

Proof. Let $\tau \in \Sigma _Y$ and let $\phi \in \operatorname {\mathrm {Aff}}_{\Sigma _Y}(\Sigma ^\tau _Y)$ be an affine function at $\tau $ . Let $\sigma $ be a cone of $F^{-1}\Sigma _Y^\tau $ . We want to show that $F^\ast (\phi )\vert _{\Sigma _X^\sigma }$ is an affine function at $\sigma $ , that is, $F^\ast (\phi )\vert _{\Sigma _X^\sigma }\in \operatorname {\mathrm {Aff}}_{\Sigma _X}(\Sigma ^\sigma _X)$ . After potentially restricting $\phi $ to the star of a cone containing $\tau $ , we may assume by Lemma 3.2 that $F(\sigma ^\diamond )\subseteq \tau ^\diamond $ . Then the morphism f maps $V(\sigma )$ into $V(\tau )$ and thus induces a morphism $f\vert _{V(\sigma )}\colon V(\sigma )\to V(\tau )$ . We have

(3.9) $$ \begin{align} \mathcal O_{X_\sigma}(F^*(\phi))\vert_{V(\sigma)} \cong \left(f\vert_{X_\sigma}^* \mathcal O_{Y_\tau}(\phi)\right)\vert_{V(\sigma)} \cong f\vert_{V(\sigma)}^* \left( \mathcal O_{Y_\tau}(\phi)\vert_{V(\tau)} \right) \cong f\vert_{V(\sigma)}^* \left(\mathcal O_{V(\tau)}\right)\cong \mathcal O_{V(\sigma)} , \end{align} $$

where the second to last isomorphism follows from $\phi $ being affine. Hence $F^\ast (\phi )$ is affine in a neighborhood of $\sigma $ .

In the proof of the invariance of the affine structure under modifications, we use the following lemma.

Proof. Let $\theta \in \Sigma _X$ be an inclusion-maximal cone with $\tau \subseteq \theta \subseteq \sigma $ , note that $\theta $ and $\sigma $ have the same dimension. The induced morphism $\Sigma _X^\theta /\theta \to \Sigma _Y^\sigma /\sigma $ is a subdivision, and the corresponding logarithmic modification agrees with the restriction $f_\theta := f\vert _{V(\theta )} \colon V(\theta )\to V(\sigma )$ . By Corollary 3.3, the function $\phi \vert _{\Sigma _X^\theta }$ is affine at $\theta $ and hence

(3.10) $$ \begin{align} \mathcal O_{V(\theta)}=\mathcal O_{X_\theta}(\phi\vert_{\Sigma_X^\theta})\vert_{V(\theta)} \cong f_\theta^*\left(\mathcal O_{Y_\sigma}(\phi)\vert_{V(\sigma)} \right) \end{align} $$

Taking first Chern classes and using the projection formula, we conclude that

(3.11) $$ \begin{align} \mathcal O_{Y_\sigma}(\phi)\vert_{V(\sigma)} \cong \mathcal O_{V(\sigma)} , \end{align} $$

which is equivalent to $\phi \in \operatorname {\mathrm {Aff}}_{\Sigma _Y}(\Sigma _Y^\sigma )$ .

Proposition 3.8. If $f: X\to Y$ is a log modification then the induced morphism

(3.12) $$ \begin{align} F \colon \vert \Sigma_X\vert\to \vert \Sigma_Y\vert \end{align} $$

is an isomorphism of tropical spaces.

Proof. Since f is a log modification, we know F is a bijective function, and by Proposition 3.5 affine functions pull-back to affine functions. It remains to show that any affine function on $\Sigma _X$ can be obtained by pull-back from an affine function of $\Sigma _Y$ . Let $\tau \in \Sigma _X$ and let $\sigma \in \Sigma _Y$ be the unique cone with $\tau ^\diamond \subseteq \sigma ^\diamond $ . We must prove that every function $\phi \in \operatorname {\mathrm {Aff}}_{ \Sigma _X}( \Sigma _X^\tau )$ is the restriction of a function in $\operatorname {\mathrm {Aff}}_{\Sigma _Y}( \Sigma _Y^\sigma )$ .

Figure 2

The setup used in showing that a function $\phi \in \operatorname {\mathrm {Aff}}_{ \Sigma _X}( \Sigma _X^\tau )$ is the restriction of a function in $\operatorname {\mathrm {Aff}}_{\Sigma _Y}( \Sigma _Y^\sigma )$ . The key point is that the map $f_\gamma $ is a $\mathbb {P}^1$ bundle. In the illustration we draw $\theta = \sigma $ for lack of dimensions: $\sigma $ is the minimal cone of $\Sigma _Y$ containing $\tau $ , and $\theta $ a maximal cone with such property.

First we show that $\phi $ is the restriction of a function in ${\mathrm {sPL}}_{ \Sigma _Y}(\Sigma _Y^\sigma )$ , see Figure 2 for the setup. Let $\theta \in \Sigma _Y$ be a maximal cone containing $\sigma $ and let $\delta \in \Sigma _X$ be a maximal cone of $ \Sigma _X$ with $\tau \subseteq \delta \subseteq \theta $ . Since the cones $\delta $ and $\theta $ have the same dimension, the restriction $\phi \vert _\delta $ extends uniquely to a linear function $m_\theta ^\delta $ on the cone $\theta $ . We now show that this function does not depend on the choice of $\delta $ . Let $\delta '\in \Sigma _X$ be a second maximal cone with $\tau \subseteq \delta '\subseteq \theta $ . Because $\delta $ can be reached from $\delta '$ by passing through codimension- $1$ cones of $ \Sigma _X$ that contain $\tau $ and are contained in $\theta $ , we may assume that $\delta $ and $\delta '$ are adjacent cones, and has codimension $1$ . Let $\varphi =\phi \vert _{\Sigma _X^\gamma }-m_\theta ^\delta \vert _{\Sigma _X^\gamma }$ . Then

(3.13) $$ \begin{align} \mathcal O_{X_\gamma}(\phi\vert_{\Sigma_X^\gamma}) \cong \mathcal O_{X_\gamma}(\varphi) \otimes f\vert_{X_\gamma}^* \mathcal O_{Y_\theta}(m_\theta^\delta) . \end{align} $$

Denote by $f_\gamma \colon V(\gamma )\to V(\theta )$ the restriction of f. Restricting (3.13) to $V(\gamma )$ and using the hypothesis of $\phi $ being affine we obtain

(3.14) $$ \begin{align} \mathcal O_{V(\gamma)} \cong \mathcal O_{X_\gamma}(\varphi)\vert_{V(\gamma)} \otimes f_\gamma^* \mathcal O_{Y_\theta}(m_\theta^\delta)\vert_{V(\theta)} . \end{align} $$

The function $\varphi $ vanishes on $\gamma $ and therefore induces a function $\overline \varphi $ on the star $ \Sigma _X^\gamma /\gamma =\Sigma _{V(\gamma )}$ . Since $\gamma $ has codimension $1$ in $\theta $ , the star $\Sigma _X^\gamma /\gamma $ is a union of two rays, corresponding to the divisors $V(\delta )$ and $V(\delta ')$ in $V(\gamma )$ , which both map isomorphically onto $V(\theta )$ . That makes $f_\gamma $ a -bundle on $V(\theta )$ with two distinct sections. The function $\overline \varphi $ has slope $0$ on the ray corresponding to $V(\delta )$ , we show that it must have slope $0$ along the other ray as well. If k is the slope of $\overline \varphi $ on the ray corresponding to $V(\delta ')$ , then

(3.15) $$ \begin{align} \mathcal O_{X_\gamma}(\varphi)\vert_{V(\gamma)} \cong \mathcal O_{V(\gamma)}(\overline \varphi) \cong \mathcal O_{V(\gamma)}(V(\delta'))^{\otimes k} \end{align} $$

and hence (3.14) gives

(3.16) $$ \begin{align} \mathcal O_{V(\gamma)} \cong \mathcal O_{V(\gamma)}(V(\delta'))^{\otimes k} \otimes f_\gamma^* \mathcal O_{Y_\theta}(m_\theta^\delta)\vert_{V(\theta)} . \end{align} $$

Since by virtue of the projective bundle formula, it follows that $k=0$ , that is, $\varphi \vert _{\delta '}=0$ . From this it follows immediately that $m^\delta _\theta =m^{\delta '}_\theta $ .

Figure 3

Illustration for the second part of the argument in Proposition 3.8. Here by lack of dimensions we have drawn $\sigma = \gamma $ . Observe that $\tilde {\gamma } \subset \gamma $ , and the cones $\theta $ and $\theta '$ are the two-dimensional triangles bounded by the black edges.

For $\theta \in \Sigma _Y$ a maximal cone containing $\sigma $ we can now define $m_\theta =m^\delta _\theta $ for some maximal cone $\delta \in \Sigma _X$ with $\tau \subseteq \delta \subseteq \theta $ ; this is independent of the choice of $\delta $ . Given a second maximal cone $\theta '\in \Sigma _Y$ containing $\sigma $ intersecting $\theta $ in , we claim that

(3.17) $$ \begin{align} m_\theta\vert_\gamma=m_\theta'\vert_\gamma . \end{align} $$

Referring to Figure 3, if $\widetilde \gamma \in \Sigma _X$ is a maximal cone in $\gamma $ containing $\tau $ , then there exist maximal cones $\delta ,\delta '\in \Sigma _X$ containing $\widetilde \gamma $ with $\delta \subseteq \theta $ and $\delta '\subseteq \theta '$ . Then we have

(3.18) $$ \begin{align} m_\theta\vert_{\widetilde\gamma}=m_\theta^\delta\vert_{\widetilde\gamma}=\phi\vert_{\widetilde\gamma}=m_{\theta'}^{\delta'}\vert_{\widetilde\gamma}=m_{\theta'}\vert_{\widetilde\gamma} . \end{align} $$

As $\widetilde \gamma $ is full-dimensional in $\gamma $ , this shows the desired equality. We conclude that the functions $m_\theta $ glue to give a function $\widetilde \phi \in {\mathrm {sPL}}_{ \Sigma _Y}(\Sigma _Y^\sigma )$ , and by construction we have $\widetilde \phi \vert _{\Sigma _X^\tau }=\phi $ . By Lemma 3.7, we have $\widetilde \phi \in \operatorname {\mathrm {Aff}}_{ \Sigma _Y}(\Sigma _Y^\sigma )$ , completing the proof.

3.2 The affine structure at infinity

In this section, we extend the previous constructions to the context of extended cone complexes. Given a cone complex $\Sigma $ , its extended cone complex is set-theoretically described as a disjoint union of quotients of cone complexes indexed by the cones of $\Sigma $ :

(3.19) $$ \begin{align} \overline\Sigma = \bigsqcup_{\sigma\in \Sigma}\Sigma^\sigma/\sigma . \end{align} $$

Observe that if $0_\Sigma $ is the cone point of $\Sigma $ , then $\Sigma ^{0_\Sigma }/0_\Sigma = \Sigma $ . We call sets of the form $\sigma /\tau \subseteq \overline \Sigma $ for some $\sigma \supseteq \tau \in \Sigma $ the cells of the extended cone complex. The open star of a cell of an extended cone complex is

(3.20) $$ \begin{align} \overline\Sigma^{\sigma/\tau}= \bigcup_{ \substack{\scriptscriptstyle\tilde\sigma \supseteq \sigma\\ \scriptscriptstyle\tilde\tau \subseteq \tau} } (\tilde\sigma/\tilde\tau)^\diamond . \end{align} $$

Taking $\tau = 0_\Sigma $ , (3.20) agrees with the earlier notion of open star of a cone in a cone complex (3.2). We will denote by ${\mathrm {sPL}}(\overline \Sigma ^{\sigma /\tau })$ the subgroup of ${\mathrm {sPL}}(\Sigma ^\sigma )$ consisting of those functions whose continuous extensions to $\overline {\Sigma ^\sigma }$ are constant on $\tau $ . Equivalently, the functions in ${\mathrm {sPL}}(\overline \Sigma ^{\sigma /\tau })$ are precisely those functions in ${\mathrm {sPL}}(\Sigma ^\sigma )$ that extend continuously to $\overline \Sigma ^{\sigma /\tau }$ .

Definition 3.10. Let X be a toroidal variety, and let $\overline {\Sigma }_X$ denote the extended cone complex associated to X. We extend the affine structure on $\Sigma _X$ to $\overline {\Sigma }_X$ by defining

(3.21) $$ \begin{align} \operatorname{\mathrm{Aff}}_{\overline\Sigma_X}= \iota_*\operatorname{\mathrm{Aff}}_{\Sigma_X}\cap {\mathrm{PL}}^{\mathrm{fin}}_{\overline\Sigma_X} , \end{align} $$

where $\iota \colon \Sigma _X\to \overline \Sigma _X$ denotes the inclusion, and the intersection is taken in $\iota _*{\mathrm {PL}}^{\mathrm {fin}}_{\Sigma _X}$ . In other words, the affine functions on $\overline \Sigma _X$ are precisely the continuous extension of the affine functions on $\Sigma _X$ .

An important feature of extended tropicalization is that at infinity one can witness a stratification by the ${\Sigma }^\sigma /\sigma $ ’s, which is parallel (rather than opposite) to the original toroidal stratification. The idea is that the geometry at infinity in the direction of $\tau $ should reflect the geometry of $V(\tau )$ . For a cell at infinity of the the form $\sigma /\tau $ , we define $V(\sigma /\tau )=V(\sigma )$ , which we think of as a closed stratum inside $V(\tau )$ . We show that affine structures are compatible with this perspective.

Proof. As $\phi $ is constant on $\tau $ , the support of the Cartier divisor $D_{X_\sigma }(\phi )$ is does not contain $V(\tau )$ , hence its restriction to $V(\tau )$ is transversal:

(3.22) $$ \begin{align} D_{X_\sigma}(\phi)\vert_{V(\tau)_{\sigma/\tau}}=D_{V(\tau)_{\sigma/\tau}}(\chi) . \end{align} $$

It follows that $\mathcal O_{X_\sigma }(\phi )\vert _{V(\tau )_{\sigma /\tau }}$ agrees with $\mathcal O_{V(\tau )_{\sigma /\tau }}(\chi )$ , and, in particular, their restriction to $V(\sigma )$ agree, from which the assertion follows immediately.

Proposition 3.12. Let X be a toroidal variety with tropicalization $\Sigma _X$ , and let $\tau \in \Sigma _X$ . With the natural identification $ \Sigma _X^\tau /\tau = \Sigma _{V(\tau )}$ , we have an isomorphism

(3.23) $$ \begin{align} \operatorname{\mathrm{Aff}}_{\overline\Sigma_X}\vert_{ \Sigma_X^\tau/\tau} \cong \operatorname{\mathrm{Aff}}_{\Sigma_{V(\tau)}} \end{align} $$

Corollary 3.13. Let $f: X\to Y$ be a (not necessarily dominant) toroidal morphism of toroidal varieties. Then the natural piecewise linear map

(3.24) $$ \begin{align} {\mathtt{Trop}}(f) \colon \vert \overline \Sigma_X\vert\to \vert \overline \Sigma_Y\vert \end{align} $$

is a morphism of tropical spaces.

3.3 Affine structures on families of curves

We apply the constructions in Section 3.1 to families of curves and their tropicalizations. We first observe, in what seems like a groundhog day moment but is a necessary verification, that in genus zero the affine structures we obtain from tropicalization agree with those from [Reference Cavalieri, Gross and Markwig6] and with those obtained from any prior incarnation of $\mathcal M_{0,n}^{\mathtt {trop}}$ . Next we study the tropicalization of families of curves of arbitrary genus, and observe that while tropicalization gives a map of TPL-spaces with affine structures, this is not always a family of tropical curves with affine structures in the sense of [Reference Cavalieri, Gross and Markwig6]. However we identify combinatorial conditions describing a locus where the affine structures agree. We begin with some technical definitions that are useful in developing these ideas.

Definition 3.14. Let $\Sigma $ be a pure-dimensional cone complex, let $\sigma \in \Sigma $ , and let $H^\sigma $ be a subgroup of ${\mathrm {sPL}}(\Sigma ^\sigma )$ such that $[\Sigma ^\sigma ]$ (the weight on $\Sigma ^\sigma $ which is $1$ on all top-dimensional cones) is balanced in the affine structure induced by $H^\sigma $ . Let ${\mathrm {CP}}_{H^\sigma }(\Sigma ^\sigma )$ denote the subgroup of ${\mathrm {sPL}}(\Sigma ^\sigma )$ consisting of those piecewise linear functions $\phi $ for which on every cone $\delta $ of $\Sigma ^\sigma $ there exists a function $h\in H^\sigma $ with $\phi \vert _\delta =h\vert _\delta $ . We define the closure $\overline {H^\sigma }$ of $H^\sigma $ as

(3.25) $$ \begin{align} \overline{H^\sigma}=\{\phi\in {\mathrm{CP}}_{H^\sigma}(\Sigma^\sigma) \mid \phi\cdot [\Sigma^\sigma]=0 \} \end{align} $$

and say that $H^\sigma $ is normal if $H^\sigma =\overline {H^\sigma }$ . When no superscript is added, we assume that we are choosing $\sigma $ to be the cone point $0_\Sigma $ .

Example 3.15. Consider a cone complex $\Sigma $ with one vertex and four rays denoted $\rho _{i}$ , for $i = 1,\ldots , 4$ . Denote by $\varphi _{i}$ the piecewise linear function with slope one along $\rho _i$ and zero along all other rays. For $\sigma $ equal to the vertex $0_\Sigma $ , if we choose

(3.26)

then $\Sigma ^\sigma = \Sigma $ is balanced at $\sigma $ in the affine structure induced by H. In this case ${\mathrm {CP}}_H(\Sigma ) = {\mathrm {sPL}}(\Sigma )$ , as one can find on each ray a function in H with any given integral slope.

The piecewise linear function $\varphi _3-\varphi _4$ belongs to $\overline {H}\smallsetminus H$ , thus showing that H is not normal. One can verify that adding this function one spans all of $\overline {H}$ , thus

(3.27)

See Figure 4 for a geometric illustration of this example.

In the next proposition, we show that $\overline {H^\sigma }$ is the largest subgroup of ${\mathrm {CP}}_{H^\sigma }(\Sigma ^\sigma )$ containing $H^\sigma $ in which $\Sigma ^\sigma $ is balanced at $\sigma $ .

Proposition 3.16. Let X be a toroidal variety with associated cone complex $\Sigma _X$ , let $\sigma \in \Sigma _X$ , and let $H^\sigma \subseteq \operatorname {\mathrm {Aff}}_{\Sigma _X}(\Sigma _X^\sigma )$ be a subgroup such that $[\Sigma _X^\sigma ]$ is balanced. Then we have

(3.28) $$ \begin{align} \operatorname{\mathrm{Aff}}_{\Sigma_X}(\Sigma_X^\sigma)\cap {\mathrm{CP}}_{H^\sigma}(\Sigma_X^\sigma)\subseteq \overline{H^\sigma} . \end{align} $$

In particular, if ${\mathrm {CP}}_{H^\sigma }(\Sigma _X^\sigma )={\mathrm {sPL}}(\Sigma _X^ \sigma )$ , we have $\operatorname {\mathrm {Aff}}_{\Sigma _X}(\Sigma _X^\sigma ) \subseteq \overline {H^\sigma } $ .

Proof. By Definition 3.10, we have

(3.29) $$ \begin{align} \operatorname{\mathrm{Aff}}_{\overline\Sigma_{\overline{\mathcal M}_{0,n}}}=\iota_*\operatorname{\mathrm{Aff}}_{\Sigma_{\overline{\mathcal M}_{0,n}}} \cap {\mathrm{PL}}^{\mathrm{fin}}_{\overline\Sigma_{\overline{\mathcal M}_{0,n}}} , \end{align} $$

where $\iota \colon \Sigma _{\overline {\mathcal M}_{0,n}}\to \overline \Sigma _{\overline {\mathcal M}_{0,n}}$ is the inclusion. By [Reference Cavalieri, Gross and Markwig6, Lemma 4.16] we also have

(3.30) $$ \begin{align} \operatorname{\mathrm{Aff}}_{\overline{\mathcal M}_{0,n}^{\mathtt{trop}}}= \kappa_*\operatorname{\mathrm{Aff}}_{\mathcal M_{0,n}^{\mathtt{trop}}}\cap {\mathrm{PL}}^{\mathrm{fin}}_{\overline{\mathcal M}_{0,n}^{\mathtt{trop}}} , \end{align} $$

where $\kappa \colon \mathcal M_{0,n}^{\mathtt {trop}}\to \overline {\mathcal M}_{0,n}^{\mathtt {trop}}$ denotes the inclusion. Since $ {\mathrm {PL}}^{\mathrm {fin}}_{\overline \Sigma _{\overline {\mathcal M}_{0,n}}} \hspace {-.4cm}= \ {\mathrm {PL}}^{\mathrm {fin}}_{\overline {\mathcal M}_{0,n}^{\mathtt {trop}}}$ , it suffices to show that the affine structures on $\Sigma _{\overline {\mathcal M}_{0,n}}$ and $\mathcal M_{0,n}^{\mathtt {trop}}$ coincide.

Figure 4

The primitive ray generators for the cone complex $\Sigma $ , depicted in red, are embedded into via the chosen generators of H, and into via the generators of $\overline {H}$ . The two balanced fans are related by a linear function. The failure of normality of H detects the fact that the embedding of $\Sigma $ as a plane balanced fan is not as linearly independent as possible.

By [Reference Cavalieri, Gross and Markwig6, Theorem 4.32], affine functions on $\mathcal M_{0,n}^{\mathtt {trop}}$ are given by cross ratios [Reference Cavalieri, Gross and Markwig6, Definition 4.8]. We first show that the tropicalization of a cross ratio on $\mathcal M_{0,n}$ is a tropical cross ratio. All cross ratios are pull-backs of cross ratios on $\mathcal M_{0,4}$ via the forgetful map, both algebraically and tropically, and tropicalization respects the $S_4$ -action on $\mathcal M_{0,4}$ . Thus, by the functoriality of taking cone complexes of Proposition 3.5, it suffices to show that the cross ratio $(p_1,p_2;p_3,p_4)$ on $\mathcal M_{0,4}$ tropicalizes to a cross ratio. We have

(3.31) $$ \begin{align} (\infty,0;1, z) = z , \end{align} $$

whose associated divisor is $(0)-(\infty )$ . The point $0$ corresponds to the nodal curve $(p_2p_4 \vert p_1p_3)$ , whereas the point $\infty $ corresponds to the curve $(p_1p_4\vert p_2p_3)$ . Therefore, the tropicalization of the cross ratio $(p_1,p_2;p_3,p_4)$ is the function with slope $1$ on the ray $\rho _{(13\vert 24)}$ , slope $-1$ on the ray $\rho _{(14\vert 23)}$ , and slope $0$ on the ray $\rho _{(12\vert 34)}$ . This is precisely the tropical cross ratio function $\xi _{((1,2),(3,4))}$ corresponding to the primitive vertex cross ratio datum $c_{(((1,2),(3,4))}$ at the unique vertex of the tropical curve in the minimal cone of $\mathcal M_{0,4}^{\mathtt {trop}}$ .

We have shown that the natural identification

(3.32) $$ \begin{align} \Sigma_{\overline{\mathcal M}_{0,n}}\to \mathcal M_{0,n}^{\mathtt{trop}} \end{align} $$

is linear. It remains to show that there are not more functions on $\Sigma _{\overline {\mathcal M}_{0,n}}$ . The subgroup $H = \operatorname {\mathrm {Aff}}_{\mathcal M_{0,n}^{\mathtt {trop}}}(\Sigma _{\overline {\mathcal M}_{0,n}})$ is naturally identified with the dual to the integral lattice in . Since $\mathcal M_{0,n}^{\mathtt {trop}}$ is a tropical linear space [Reference Maclagan and Sturmfels20, Theorem 6.4.12], we have that $H = \overline {H}$ by [Reference Francois10, Proposition 4.5]. In addition ${\mathrm {CP}}_H(\mathcal M_{0,n}^{\mathtt {trop}})={\mathrm {sPL}}(\mathcal M_{0,n}^{\mathtt {trop}})$ , because $\mathcal M_{0,n}^{\mathtt {trop}}$ is embedded in by its global linear functions. We can thus apply Proposition 3.16 to conclude that $\operatorname {\mathrm {Aff}}_{\Sigma _{\overline {\mathcal M}_{0,n}}}(\Sigma _{\overline {\mathcal M}_{0,n}})\subseteq \overline {H} = \operatorname {\mathrm {Aff}}_{\mathcal M_{0,n}^{\mathtt {trop}}}(\Sigma _{\overline {\mathcal M}_{0,n}})$ .

Given any cone $\sigma \in \mathcal M_{0,n}^{\mathtt {trop}}$ , the same argument applies to $\Sigma _{\overline {\mathcal M}_{0,n}}^\sigma $ , because $\Sigma _{\overline {\mathcal M}_{0,n}}^\sigma -\sigma $ is again a tropical linear space [Reference Maria Feichtner and Sturmfels9, Proposition 2.5]. This shows that $\operatorname {\mathrm {Aff}}_{\Sigma _{\overline {\mathcal M}_{0,n}}}(\Sigma ^\sigma _{\overline {\mathcal M}_{0,n}}) = \operatorname {\mathrm {Aff}}_{{\mathcal M_{0,n}^{\mathtt {trop}}}}(\Sigma ^\sigma _{\overline {\mathcal M}_{0,n}})$ , thus concluding the proof.

We turn our attention to more general families of curves and their tropicalization.

Example 3.20. While the strata $O(\sigma )$ and their closures $V(\sigma )$ are always irreducible, condition (3) in Definition 3.18 is not superfluous. It can happen that the fibers of $V(\sigma )\to \pi (V(\sigma ))$ are reducible, but because of the monodromy action on the irreducible components, the domain $V(\sigma )$ is still irreducible. As an example consider the family of plane curves over given by a union of three lines as

(3.34) $$ \begin{align} &\hspace{-50pt} \{(t-1)x+(t-\zeta)y+ (t-\zeta^2)z=0\} \quad\cup \nonumber \\ &\qquad \cup \quad\{(t-\zeta)x+(t-\zeta^2)y+ (t-1)z=0\} \quad \cup \nonumber \\ &\hspace{100pt} \quad \cup \quad \{(t-\zeta^2)x+(t-1)y+ (t-\zeta)z=0\} , \end{align} $$

where denotes a primitive third root of unity. Traveling from $t=1$ to $t=\zeta $ permutes the three lines cyclically, and in particular the family descends to an irreducible family over whose fibers are unions of three lines in a plane that do not meet in a point. The construction of this example already shows how to fix families in which condition (3) is violated: one has to pass to an étale cover that trivializes the monodromy action on the components of the fibers.

Proof. We first show the linearity of f on the interior ${ \mathtt B}$ of $\overline { \mathtt B}$ . Let $b_0\in { \mathtt B}$ and let $\sigma $ be the unique cone of ${ \mathtt B}$ with $b_0\in \sigma ^\diamond $ . Let G be the cycle-rigidified, n-marked, genus-g stable graph such that $f(b_0) \in \sigma _G^\diamond $ . As the combinatorial type of the fibers of ${\Pi }$ remains constant on $\sigma ^\diamond $ , we have $f(\sigma ^\diamond )\subseteq \sigma _G^\diamond $ and f and $f_\star $ induce an identification of the combinatorial type of $\overline { \mathtt C}_b$ with G for every $b\in \sigma ^\diamond $ . Since linear functions on ${\overline {\mathcal V}_{g,n}}$ are generated by vertex and edge primitive cross ratios ([Reference Cavalieri, Gross and Markwig6, Proposition 4.13]), we show linearity of f by showing that the pullback of a primitive cross ratio is linear on ${ \mathtt B}$ .

Let $c=c_{((f_s,f_e),(f_{-1},f_1))}$ be a primitive cross ratio of vertex type at a rational vertex v of G with associated function $\xi _c$ on $({\overline {\mathcal V}_{g,n}})^{\sigma _G}$ . We show that $f^*\xi _c$ is linear on $\mathtt B^\sigma $ .

We may assume, possibly after subdividing some cones of ${ \mathtt B}$ , that ${ \mathtt B}^\sigma $ is simply connected. We may also assume, possibly after subdividing some cones of $\overline { \mathtt C}$ , that the four flags $f_s, f_e, f_{-1}, f_1$ belong to distinct edges in each fiber. Then the vertex v defines a component $V(v)$ of $\mathcal C_{V(\sigma )}$ . The four tangent directions $f_s$ , $f_e$ , $f_{-1}$ , and $f_1$ determine four sections of $V(v)$ and thus a morphism

(3.35) $$ \begin{align} V(\sigma)\to \overline{\mathcal M}_{0,4}; \end{align} $$

precomposing its tropicalization with the map quotienting $\sigma $ , we obtain ${ \mathtt B}^\sigma \to \mathcal M_{0,4}^{\mathtt {trop}}$ . Since $f^*\xi _c$ agrees with the pullback of the appropriate cross ratio from $\mathcal M_{0,4}^{\mathtt {trop}}$ , linearity follows from Theorem 3.17.

Now let $c=c_{(e, (f_s,f_e), (f_{-1},f_1))}$ be a primitive cross ratio of edge type on G at an edge e. First assume that there exists a face $\tau $ of $\sigma $ such that $f(\tau ^\diamond )\subseteq \sigma _{\overline G}$ for some edge contraction $\overline G$ of G in which e got contracted to a genus- $0$ vertex v. Then $f_s$ , $f_e$ , $f_{-1}$ , and $f_1$ induce flags at v and in particular a primitive cross ratio $\overline c=c_{((f_s,f_e),(f_1,f_1))}$ of vertex type on $\overline G$ . By the vertex type case above, the function $f^*\xi _{\overline c}$ is linear on $\mathtt B^\tau $ , and by Lemma 3.2 the restriction $f^*\xi _{\overline c}\vert _{\mathtt {B}^\sigma }$ is linear on $\mathtt B^\sigma $ . But $f^*\xi _{\overline c}\vert _{\mathtt B^\sigma }$ coincides with $f^*\xi _c$ on $\mathtt B^\sigma $ , concluding the proof in this case.

Next, assume that for any face of $\sigma $ where the edge e gets contracted, a vertex of positive genus is formed. We can view $\mathcal C$ as the base of a family of $(n\sqcup \{\star \})$ -marked stable curves

(3.36) $$ \begin{align} \pi_\star\colon \mathcal C_\star\to \mathcal C. \end{align} $$

The cycle rigidification of ${\Pi }$ induced by the maps f and $f_\star $ from diagram (3.33) produces a cycle rigidification of the family

(3.37)

that is a Cartesian TPL-diagram

(3.38)

Let $\sigma _e\in \Sigma $ be the cone over $\sigma $ whose fibers corresponds to e. The stratum $V(\sigma _e)$ is the node corresponding to e in every fiber of $\mathcal C_{V(\sigma )}\to V(\sigma )$ and thus the restriction of $\pi $ defines an isomorphism $V(\sigma _e)\to V(\sigma )$ . The pull-back $\mu _\star ^*\xi _c$ is a cross ratio on ${\overline {\mathcal V}_{g,n\sqcup \{\star \}}}$ , although not a primitive one. More precisely, if $G_\star $ denotes the rigidified $(n\sqcup \{\star \})$ -marked graph, obtained by introducing a vertex v on e in G and attaching a $\star -$ marked leg $l_\star $ at v, then $f_\star (\sigma _e)\subseteq \sigma _{G_\star }$ and c induces a cross ratio datum on $G_\star $ by ignoring the fact that e has been subdivided. If $e_1$ and $e_2$ denote the two edges adjacent to v in the order seen when traversing e according to c, and $g_1$ and $g_2$ are the corresponding half-edges starting at v, as shown in Figure 5, then we have

(3.39) $$ \begin{align} \mu_\star^*\xi_c = \xi_{(e_1,(f_s,l_\star),(f_{-1},g_2))}+ \xi_{(e_2,(l_\star,f_e),(g_2,f_1))} \end{align} $$

The two edges $e_1$ and $e_2$ are contracted on the two faces of $\sigma _e$ that map isomorphically onto $\sigma $ and correspond to the two vertices of e. Therefore, for each of the two primitive cross ratio summands in (3.39) we are in the special situation from before: the edge of the edge-type primitive cross ratio is contracted to a genus zero vertex. Algebraically, this yields

(3.40) $$ \begin{align} \mathcal O_{V(\sigma_e)}&\cong \left(\mathcal O_{\mathcal C_{\sigma_e}}(f_\star^*\xi_{(e_1,(f_s,l_\star),(f_{-1},g_2))})\otimes \mathcal O_{\mathcal C_{\sigma_e}}(f_\star^*\xi_{(e_2,(l_\star,f_e),(g_2,f_1))})\right)\vert_{V(\sigma_e)} \cong \nonumber \\ &\quad \cong \mathcal O_{\mathcal C_{\sigma_e}}(f_\star^*\mu_\star^*\xi_c)\vert_{V(\sigma_e)}\cong \mathcal O_{\mathcal C_{\sigma_e}}({\Pi }^*f^* \xi_c\vert_{\sigma_e})\vert_{V(\sigma_e)}\cong \left(\pi^*\mathcal O_{\mathcal B_{\sigma}}(f^*\xi_c)\right)\Big\vert_{V(\sigma_e)} . \end{align} $$

As we have noticed before, $\pi $ induces an isomorphism $V(\sigma _e)\to V(\sigma )$ , so we conclude that

(3.41) $$ \begin{align} \mathcal O_{\mathcal B_{\sigma}}(f^*\xi_c)\vert_{V(\sigma)}\cong \mathcal O_{V(\sigma)} , \end{align} $$

which is equivalent to $f^\ast \xi _c$ being linear on $\mathtt B^\sigma $ .

Figure 5

The subdivision of the edge e in $G_\star $ .

To show that f is also linear on the boundary of $\mathtt B$ it suffices to note that on both source (by Definition 3.10) and target (by [Reference Cavalieri, Gross and Markwig6, Lemma 4.16]), the affine structure near the boundary is induced by the affine structure on the interior in the sense that the affine functions at the boundary are precisely the continuous extensions of affine functions on the interior.

To show that $f_\star $ is linear one applies the same arguments to the map $\pi _\star $ and the rigidification displayed in (3.38).

Proposition 3.22. With the set-up and notation of Proposition 3.21, let $x\in {\overline { \mathtt C}}$ . If x has genus $0$ , then all functions in $\operatorname {\mathrm {Aff}}_{{{ \mathtt C}},x}$ are linear on the fibers of ${\Pi }$ close to x. Moreover, we have an exact sequence

(3.42) $$ \begin{align} 0 \to \operatorname{\mathrm{Aff}}_{\overline{ \mathtt B},{\Pi }(x)}\to \operatorname{\mathrm{Aff}}_{{\overline{ \mathtt C}},x} \to \Omega^1_{{{\overline{ \mathtt C}}}_{{\Pi }(x)},x}\to 0 . \end{align} $$

If x is a vertex (of genus $0$ ) in its fiber, or is on an edge adjacent to a vertex of genus $0$ , then $\Omega ^1_{{{\overline { \mathtt C}}}_{{\Pi }(x)},x}$ consists of the differentials of all harmonic functions.

Proof. The injectivity of the pull-back is obvious, so is the surjectivity of restriction to the fiber (we consider the fiber with the induced affine structure). We show the harmonicity of affine functions on fibers and the exactness in the middle simultaneously.

We first prove the proposition in the case when $x\in {{{ \mathtt C}}}$ . Suppose x is on an edge in its fiber. In that case a function $\phi \in \operatorname {\mathrm {Aff}}_{{{{ \mathtt C}}},x}$ is linear, and hence harmonic, on all fibers in a neighborhood of x. If $\phi \vert _{{\Pi }^{-1}\{{\Pi }(x)\}}$ is constant near x, then $\phi $ is constant on fibers on a small neighborhood of x. Let $\sigma \in {{{ \mathtt C}}}$ be the unique cone with $x\in \sigma ^\diamond $ and let $\tau ={\Pi }({{\sigma }})$ . By definition of the affine structure on ${{{ \mathtt C}}}$ , $\phi $ defines a piecewise linear function on ${{{ \mathtt C}}}^\sigma $ , and $\phi $ is constant on all the fibers of the induced map ${{{ \mathtt C}}}^\sigma \to { \mathtt B}^\tau $ . Taking any section ${ \mathtt B}^\tau \to {{{ \mathtt C}}}^\sigma $ of this map (for example a section corresponding to one of the two vertices of the edge on which x sits), we obtain a piecewise linear function $\chi $ on ${ \mathtt B}^\tau $ with ${\Pi }^*\chi =\phi $ . The stratum $V(\sigma )$ is a node in each fiber, and the projection $\pi \vert _{V(\sigma )} \colon V(\sigma )\to V(\tau )$ is an isomorphism. We have

(3.43) $$ \begin{align} \pi\vert_{V(\sigma)}^* \mathcal O_{\mathcal B_\tau}(\chi)\vert_{V(\tau)}\cong \mathcal O_{\mathcal C_\sigma}({\Pi }^*\chi)\vert_{V(\sigma)} \cong \mathcal O_{\mathcal C_\sigma}(\phi)\vert_{V(\sigma)} \cong \mathcal O_{V(\sigma)} , \end{align} $$

where the last isomorphism follows because $\phi $ is affine; we conclude that $\chi $ is affine on ${ \mathtt B}$ .

Now suppose x is a vertex of genus $0$ , let $\sigma $ and $\tau $ be as above, and let $\phi \in \operatorname {\mathrm {Aff}}_{{ \mathtt C},x}$ . Let e be an edge in the fiber of x that is adjacent to x. After subtracting a suitable cross ratio, which is harmonic on fibers and affine by Proposition 3.21, we may assume that $\phi $ is constant on e. So by the case where x was on an edge, there exists an affine function $\chi \in \operatorname {\mathrm {Aff}}_{{ \mathtt B},{\Pi }(x)}$ such that $\phi -{\Pi }^*\chi $ vanishes on e. In particular, by replacing $\phi $ with this difference of affine functions, we may assume that $\phi \vert _{\sigma }=0$ . By Proposition 3.12, $\phi $ is induced by an affine function $\overline \phi $ on ${{{ \mathtt C}}}^\sigma /\sigma ={{{ \mathtt C}}}_{V(\sigma )}$ . Let $T_x({{{ \mathtt C}}}_{{\Pi }(x)})$ denote the set of half-edges starting at x in its fiber. For each $h\in T_x({{{ \mathtt C}}}_{{\Pi }(x)})$ let $m_h$ be the slope of $\phi $ in the direction of h, and let $\sigma _h$ be the cone of ${{{ \mathtt C}}}$ spanned by $\sigma $ and h. Then the normalization $\widetilde V(\sigma )$ of the stratum $V(\sigma )$ is a family of $T_x({{{ \mathtt C}}}_{{\Pi }(x)})$ -marked rational stable curves over $V(\tau )$ , and $V(\sigma _h)$ is the section of this family corresponding to $h\in T_x({{{ \mathtt C}}}_{{\Pi }(x)})$ . The fact that $\phi $ is affine means that the divisor

(3.44) $$ \begin{align} \sum_{h\in T_x({{{ \mathtt C}}}_{{\Pi }(x)})} m_h V(\sigma_h) \end{align} $$

is rationally equivalent to $0$ on $\widetilde V(\sigma )$ . Since the fiber of $\widetilde V(\sigma )\to V(\tau )$ over any $y\in O(\tau )$ is isomorphic to , we conclude that $\sum _h m_h=0$ , because every divisor on rationally equivalent to $0$ has degree $0$ . This shows that $\phi $ is harmonic at x. Onto exactness in the middle of (3.43), if x is a vertex and $\phi \vert _{{\Pi }^{-1}\{{\Pi }(x)\}}$ is constant near x, then it is constant on an edge adjacent to x; by the case where x was on an edge we obtain an affine function $\chi \in \operatorname {\mathrm {Aff}}_{{ \mathtt B},{\Pi }(x)}$ with $\phi ={\Pi }^*\chi $ .

Let x be either a rational vertex or adjacent to a genus $0$ vertex in its fiber. By the linearity of $f_\star $ (Proposition 3.21), all (pull-backs of) cross ratios are linear on ${{{ \mathtt C}}}$ . By the condition on x, by [Reference Cavalieri, Gross and Markwig6, Proposition 4.24] there are enough cross ratio functions defined at x to realize all possible differentials of harmonic functions on the fiber.

It remains to analyze the case where $x\in \overline { \mathtt C}\setminus { \mathtt C}$ . Let $\delta $ be the unique cone of ${ \mathtt C}$ with $x\in { \mathtt C}^\delta /\delta $ and let $\theta ={\Pi }(\delta )$ , which is a cone in ${ \mathtt B}$ . We can either have $\dim (\delta )=\dim (\theta )+1$ or $\dim (\delta )=\dim (\theta )$ . In the first case, the point x is either the endpoint of a leg or a node at infinity in its fiber (see [Reference Cavalieri, Gross and Markwig6, p. 14]). It follows that $\Omega ^1_{{\Pi }(x),x}=0$ and we need to show that the pull-back $\operatorname {\mathrm {Aff}}_{\overline { \mathtt B},\Pi (X)}\to \operatorname {\mathrm {Aff}}_{\overline { \mathtt C},x}$ is an isomorphism. The morphism $V(\delta )\to V(\theta )$ induced by $\pi $ is an isomorphism because $V(\delta )$ either consists of a single marked point or a single node in each fiber over $V(\theta )$ . So $\pi $ also induces an isomorphism $\Sigma _{V(\delta )}\to \Sigma _{V(\theta )}$ of tropical spaces and we are done by Proposition 3.12. In the second case, that is, when $\delta $ and $\theta $ have the same dimension, a neighborhood of x in $\overline { \mathtt C}_{\Pi (x)}$ is contained in ${ \mathtt C}^\delta /\delta $ . It therefore suffices to prove the assertion for the family

$$\begin{align*}\Sigma_{V(\delta)}\cong { \mathtt C}^\delta/\delta\to { \mathtt B}^\theta/\theta\cong \Sigma_{V(\theta)} , \end{align*}$$

where the canonical isomorphisms come from Proposition 3.12. This family is the tropicalization of the induced family of curves $V(\delta )\to V(\theta )$ and x is in the interior of $\Sigma _{V(\delta )}$ , so we have reduced to the case that we already proved.

Remark 3.23. Consider the set ${ \mathtt U}\subseteq {{{ \mathtt C}}}$ of all genus- $0$ points that are either vertices or on an edge adjacent to a genus- $0$ vertex. The locus ${ \mathtt U}$ is closely related to the locus $\mathcal V^{\mathrm {good}}_{g,n}$ introduced in [Reference Cavalieri, Gross and Markwig6, Definition 4.22]. More precisely, we have

(3.45) $$ \begin{align} f^{-1}\mathcal V^{\mathrm{good}}_{g,n}= \{y \mid {{{ \mathtt C}}}_y\subseteq { \mathtt U}\} \, \end{align} $$

where ${{{ \mathtt C}}}_y$ denotes the fiber over y. One can view ${ \mathtt U}$ as the locus where ${{{ \mathtt C}}}\to { \mathtt B}$ is a family of tropical curves in the sense of [Reference Cavalieri, Gross and Markwig6]. The affine structure on ${ \mathtt U}$ is completely determined by that on ${ \mathtt B}$ . Equivalently, the inclusion

(3.46) $$ \begin{align} { \mathtt U}\to { \mathtt B}\times_{{\overline {\mathcal V}_{g,n}}}{\overline {\mathcal V}_{g,n\sqcup \{\star\}}} , \end{align} $$

where the fiber product is taken in the category of tropical spaces, is an open immersion.

5.1 Tropicalization of the cotangent line bundle

In [Reference Cavalieri, Gross and Markwig6, Definition 6.16], tropical $\psi $ classes are defined as first Chern classes of certain torsors over the sheaf of affine functions of the base of a family of tropical curves. In this section we develop language and describe suitable conditions that allow us to compare the tropicalization of the i-th cotangent line bundle with the torsors above.

Definition 5.1. Let $\Sigma $ be a cone complex with a sheaf of affine functions $\operatorname {\mathrm {Aff}}_\Sigma $ , let ${ \mathtt U}\subseteq \overline \Sigma $ be an open subcomplex, and let $\phi \in {\mathrm {sPL}}(\Sigma \cap { \mathtt U})$ . We define the sheaf $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ by

(5.1)

for open subsets ${ \mathtt V}\subseteq { \mathtt U}$ . Note that $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ is a pseudo-torsor over $\operatorname {\mathrm {Aff}}_{ \mathtt U}$ , that is it is a torsor whenever it is nonempty. If $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ is a torsor, it defines a tropical line bundle, which we also denote by $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ by abuse of notation.

Proposition 5.2. Let X be a toroidal variety, let $\phi \in {\mathrm {sPL}}(\Sigma _X)$ and let ${ \mathtt U}\subseteq \overline \Sigma _X$ . If $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ is a tropical line bundle, then ${\mathtt {Trop}}(\mathcal O_X(\phi ))\vert _{ \mathtt U}$ is a tropical line bundle as well and

(5.2) $$ \begin{align} {\mathtt{Trop}}(\mathcal O_X(\phi))\vert_{ \mathtt U} \cong \operatorname{\mathrm{Aff}}_{ \mathtt U}(\phi) \end{align} $$

Proof. By Proposition 4.4, ${\mathtt {Trop}}(\mathcal O_X(\phi ))\vert _{ \mathtt U}$ is a tropical line bundle if and only if for every cell $\sigma /\tau $ of ${ \mathtt U}$ there exists $\phi _{\sigma /\tau }\in {\mathrm {sPL}}(\overline \Sigma ^{\sigma /\tau }_X)$ with

(5.3) $$ \begin{align} \mathcal O_{X_\sigma}(\phi_{\sigma/\tau}+\phi)\vert_{V(\sigma)}\cong \mathcal O_{V(\sigma)} \; \end{align} $$

the collection of all the $-\phi _{\sigma /\tau }$ ’s generates the pseudo-torsor ${\mathtt {Trop}}(\mathcal O_X(\phi ))\vert _{{ \mathtt U}}(\overline \Sigma ^{\sigma /\tau }_X)$ of local affine sections. By definition of $\operatorname {\mathrm {Aff}}_{\overline \Sigma _X}$ , equation (5.3) means that $\phi _{\sigma /\tau }+\phi \in \operatorname {\mathrm {Aff}}_{\overline \Sigma _X}(\overline \Sigma _X^{\sigma /\tau }\cap \Sigma _X)$ , which is equivalent to $-(\phi _{\sigma /\tau }+\phi )\in \operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )(\overline \Sigma _X^{\sigma /\tau })$ . The $\operatorname {\mathrm {Aff}}_{{ \mathtt U}}(\overline \Sigma ^{\sigma /\tau }_X)$ -equivariant map

(5.4) $$ \begin{align} {\mathtt{Trop}}(\mathcal O_X(\phi))\vert_{{ \mathtt U}}(\overline\Sigma^{\sigma/\tau}_X) \to \operatorname{\mathrm{Aff}}_{ \mathtt U}(\phi)(\overline\Sigma_X^{\sigma/\tau}), \;\; -\phi_{\sigma/\tau} \mapsto \chi = -\phi_{\sigma/\tau} -\phi \end{align} $$

is an isomorphism of pseudo-torsors. Therefore, ${\mathtt {Trop}}(\mathcal O_X(\phi ))\vert _{ \mathtt U}$ being a line bundle is equivalent to $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ being a torsor, and the collection of maps (5.4) for $\sigma /\tau \subseteq { \mathtt U}$ defines an isomorphism between ${\mathtt {Trop}}(\mathcal O_X(\phi ))\vert _{ \mathtt U}$ and $\operatorname {\mathrm {Aff}}_{ \mathtt U}(\phi )$ .

We now apply these constructions to our main objects of interest: families of pointed curves and their tropicalizations.

Proposition 5.3. Let $\mathcal C\to \mathcal B$ be a tropicalizable family of n-marked stable curves with associated family of tropical curves $ \overline { \mathtt C}\to \overline { \mathtt B}$ , and let $1\leq i \leq n$ . Let $\phi $ be the piecewise linear function that has slope $1$ on the ray $\rho _i$ corresponding to the i-th section $s_i$ and is $0$ on all cones not containing $\rho _i$ . Then there exists an isomorphism of pseudo $\operatorname {\mathrm {Aff}}_{\overline { \mathtt C}}$ -torsors

(5.5) $$ \begin{align} \operatorname{\mathrm{Aff}}_{\overline{\mathtt C}}({\mathtt s}_i)\cong \operatorname{\mathrm{Aff}}_{\overline {\mathtt C}}(\phi) . \end{align} $$

Definition 5.4. Let $\mathcal C \to \mathcal B$ be a tropicalizable family of n-marked stable curves with tropicalization $\overline {{ \mathtt C}}\to \overline { \mathtt B}$ . If $\operatorname {\mathrm {Aff}}_{\overline {{ \mathtt C}}}({\mathtt s}_i)$ is a tropical line bundle, we define the i-th tropical cotangent bundle analogously to [Reference Cavalieri, Gross and Markwig6, Definition 6.16]:

(5.6)

where ${\mathtt s}_i={\mathtt {Trop}}(s_i)$ is the tropicalization of the i-th section

Theorem 5.5. With setup and notation as in Definition 5.4, assume $\operatorname {\mathrm {Aff}}_{\overline {{ \mathtt C}}}({\mathtt s}_i)$ is a tropical line bundle. Then tropicalizes to a tropical line bundle and we have

(5.7)

Proof. Let $\phi $ be the function on $\mathtt C$ that has slope $1$ on the ray $\rho _i$ corresponding to $s_i$ and is $0$ on all cones of $\mathtt C$ that do not contain $\rho _i$ . By Proposition 5.3, we have $\operatorname {\mathrm {Aff}}_{\overline {\mathtt C}}(\phi )\cong \operatorname {\mathrm {Aff}}_{\overline {\mathtt C}}({\mathtt s}_i)$ . Using the assumption that $\operatorname {\mathrm {Aff}}_{\overline {\mathtt C}}({\mathtt s}_i)$ is a tropical line bundle, we conclude that $\operatorname {\mathrm {Aff}}_{\overline {\mathtt C}}(\phi )$ is a tropical line bundle as well. Applying Proposition 5.2, we see that the bundle $\mathcal O_{\mathcal C}(s_i)\cong \mathcal O_{\mathcal C}(\phi )$ is tropicalizable and that the second isomorphism in the chain of isomorphisms

(5.8) $$ \begin{align} {\mathtt{Trop}}(\mathcal O_{\mathcal C}(s_i))\cong {\mathtt{Trop}}(\mathcal O_{\mathcal C}(\phi)) \cong \operatorname{\mathrm{Aff}}_{\overline { \mathtt C}}(\phi)\cong \operatorname{\mathrm{Aff}}_{\overline { \mathtt C}}({\mathtt s}_i) \end{align} $$

exists, the other ones being evident. Dualizing and applying Proposition 4.5 to the section $s_i$ then shows that is defined and that we have a sequence of isomorphisms

(5.9)

5.2 Tropicalizations of Cycles and Tropical Cycles

In order to translate Theorem 5.5 into a statement about $\psi $ classes, we need to take the first Chern class of a tropical line bundle. In this section we present the necessary bits of tropical intersection theory for this task. Similar definitions and constructions appeared in [Reference Cavalieri, Gross and Markwig6, Section 6]; here we present them in the slightly larger generality needed for the current context.

Definition 5.6. Let X be a complete toroidal variety with associated complex $\Sigma _X$ , and let $c\in A^\ast (X)$ . Then the tropicalization of c is given by the weight

(5.10)