2.1 Cone complexes associated to logarithmic stacks

2.1.4 The tropicalization of a logarithmic scheme

Now let $X$ be a Zariski fine saturated (fs) log scheme of finite type. For the generic point $\eta$ of a stratum of $X$, its characteristic monoid $\overline{\mathcal {M}}_{X,\eta }$ defines a dual monoid $(\overline{\mathcal {M}}_{X,\eta })^{\vee } := {\operatorname {Hom}}(\overline{\mathcal {M}}_{X,\eta }, \mathbb {N})$ lying in the group $(\overline{\mathcal {M}}_{X,\eta })^* := {\operatorname {Hom}}(\overline{\mathcal {M}}_{X,\eta }, \mathbb {Z})$, see § 1.2, and hence a dual cone

(2.1)\begin{equation} \sigma_{\eta}:=((\overline{\mathcal{M}}_{X,\eta})^{\vee}_\mathbb{R},\,(\overline{\mathcal{M}}_{X,\eta})^*). \end{equation}

If $\eta$ is a specialization of $\eta '$, then there is a well-defined generization map $\overline{\mathcal {M}}_{X,\eta } \longrightarrow \overline{\mathcal {M}}_{X,\eta '}$ since we assumed $X$ is a Zariski logarithmic scheme. Dualizing, we obtain a face morphism $\sigma _{\eta '} \longrightarrow \sigma _{\eta }$. This gives a diagram of cones indexed by strata of $X$ with face morphisms, and hence gives a generalized cone complex $\Sigma (X)$. We call this the tropicalization of $X$, following [Reference Gross and SiebertGS13, Appendix B].Footnote ¹ For $\sigma \in \Sigma (X)$ we denote by

\[ X_\sigma\subset X \]

the closure of the corresponding stratum of $X$, endowed with the reduced induced scheme structure. We refer to these subschemes with reduced induced structure as closed strata of $X$.

This construction is functorial: given a morphism of log schemes $f:X \longrightarrow Y$, the map $f^{\flat }:f^{-1}\overline{\mathcal{M}}_Y \longrightarrow \overline{\mathcal{M}}_X$ induces a map of generalized cone complexes $\Sigma (f):\Sigma (X) \longrightarrow \Sigma (Y)$.

Definition 2.1 [GS13, Definition B.2] We say $X$ is monodromy free if $X$ is a Zariski log scheme and for every $\sigma \in \Sigma (X)$, the natural map $\sigma \longrightarrow |\Sigma (X)|$ is injective on the interior of any face of $\sigma$. We say $X$ is simple if the map is injective on every $\sigma$.

Here is an example of a Zariski log structure that is monodromy free, but not simple. Take $X$ to be the Neron $2$-gon, the fibred sum of two copies of $\mathbb {P}^1$ joined at two pairs of points. Thus $X$ has two irreducible components $X_1,X_2$ and two nodes $q_1,q_2$. Take a log structure $\mathcal {M}_X$ on $X$ with $\overline{\mathcal{M}}_X$ constant with fibres $\mathbb {N}^2$ along $X_1$, with fibers $\mathbb {N}$ on $X_2\setminus \{q_1,q_2\}$ and with generization maps $\overline{\mathcal{M}}_{X,q_i}=\mathbb {N}^2\to \mathbb {N}$ to the generic point of $X_2$ the two projections. See also [Reference Gross and SiebertGS13, Expl. B.1] for another example.

Simplicity is, however, true in the Zariski log smooth case over a trivial log point. Such log schemes can in fact be viewed as toroidal pairs without self-intersections and the following statement follows readily from the classical treatment in [Reference Kempf, Knudsen, Mumford and Saint-DonatKKMS73, pp. 70–72].

Proposition 2.2 Let $X$ be a Zariski log scheme, log smooth over $\operatorname {Spec} \Bbbk$ with the trivial log structure. Then $X$ is simple.

As remarked in [Reference Gross and SiebertGS13], more generally we can define the generalized cone complex associated with a finite type logarithmic stack $X$, in particular allowing for logarithmic schemes $X$ in the étale topology. In fact, one can always find a cover $X' \longrightarrow X$ in the smooth topology with $X'$ a union of simple log schemes, and with $X''=X'\times _{X} X'$; then define $\Sigma (X)$ to be the colimit of $\Sigma (X'')\rightrightarrows \Sigma (X')$. The resulting generalized cone complex is independent of the choice of cover. This process is explicitly carried out in [Reference Abramovich, Caporaso and PayneACP15, Reference UlirschUli15].

Examples 2.3 (1) If $X$ is a toric variety with the canonical toric logarithmic structure, then $\Sigma (X)$ is abstractly the fan defining $X$. It is missing the embedding of $|\Sigma (X)|$ as a fan in a vector space $N_{\mathbb {R}}$, and should be viewed as a piecewise linear object.

(2) Let $\Bbbk$ be a field and $X=\operatorname {Spec}(\mathbb {N} \longrightarrow \Bbbk )$ the standard log point with $\mathcal {M}_X=\Bbbk ^{\times } \times \mathbb {N}$. Then $\Sigma (X)$ consists of the ray $\mathbb {R}_{\ge 0}$.

(3) Let $C$ be a curve with an étale logarithmic structure with the property that $\overline{\mathcal {M}}_C$ has stalk $\mathbb {N}^2$ at any geometric point, but has monodromy of the form $(a,b)\mapsto (b,a)$, so that the pull-back of $\overline{\mathcal {M}}_C$ to an unramified double cover $C' \longrightarrow C$ is constant but $\overline{\mathcal {M}}_C$ is only locally constant. Then $\Sigma (C)$ can be described as the quotient of $\mathbb {R}_{\ge 0}^2$ by the automorphism $(a,b)\mapsto (b,a)$. If we use the reduced presentation, $\Sigma (C)$ has three cones, one each of dimension $0$, $1$ and $2$.

2.2 Artin fans

Let $X$ be a fine and saturated algebraic log stack. We are quite permissive with algebraic stacks, as delineated in [Reference OlssonOls03, (1.2.4)–(1.2.5)], since we need to work with stacks with non-separated diagonal. An Artin stack logarithmically étale over $\operatorname {Spec}\Bbbk$ is called an Artin fan.

The logarithmic structure of $X$ is encoded by a morphism $X \longrightarrow {\operatorname {Log}}$ to Olsson's stack ${\operatorname {Log}}$ of fine log structures, see [Reference OlssonOls03]. One crucial idea developed in the context of the present paper is a refinement of the stack ${\operatorname {Log}}$ by an Artin fan that takes into account the stratification of $X$ defined by $\overline{\mathcal {M}}_X$. Following preliminary notes written by two of us (Chen and Gross), the paper [Reference Abramovich and WiseAW18] introduces a canonical Artin fan $\mathcal {A}_X$ associated to a logarithmically smooth fs log scheme $X$. This was generalized in [Reference Abramovich, Chen, Marcus and WiseACMW17, Proposition 3.1.1] as follows.

Note that $\mathcal {A}_X$ in the theorem is indeed an Artin fan because ${\operatorname {Log}}$ is logarithmically étale over $\operatorname {Spec}\Bbbk$.

If $X$ is a Deligne–Mumford stack, $\mathcal {A}_X$ can be constructed from the cone complex $\Sigma (X)$ as follows. For any cone $\sigma \subset N_{\mathbb {R}}$, let $P=\sigma ^{\vee }\cap M$ be the corresponding monoid. We write

(2.2)\begin{equation} \mathcal{A}_\sigma=\mathcal{A}_P:=[\operatorname{Spec}\Bbbk[P]/\operatorname{Spec} \Bbbk[P^{{\mathrm{gp}}}]]. \end{equation}

This stack carries the standard toric logarithmic structure induced by descent from the global chart $P \longrightarrow \Bbbk [P]$. Then $\mathcal {A}_X$ is the colimit

(2.3)\begin{equation} \mathcal{A}_X=\varinjlim_{\sigma\in\Sigma(X)} \mathcal{A}_\sigma, \end{equation}

in the category of sheaves over ${\operatorname {Log}}$.

Our next aim is to prove functoriality of the formation of Artin fans for maps with Zariski log smooth domains, stated as Proposition 2.8 below. We need two lemmas.

Proposition 2.8 Let $X \longrightarrow Y$ be a morphism of log schemes. Suppose $X$ is log smooth with Zariski log structure. Then there is a canonical morphism $\mathcal {A}_X \longrightarrow \mathcal {A}_Y$ such that the following diagram commutes.

Using Proposition 2.8 we can also define a relative notion of Artin fan for maps with log smooth domains.

Definition 2.9 The relative Artin fan for a morphism $X \longrightarrow B$ of log schemes with $X$ log smooth with Zariski log structure is defined as the fibre product

\[ \mathcal{X}= B\times_{\mathcal{A}_B} \mathcal{A}_X. \]

While not, strictly speaking, needed for this paper, we end this subsection with the instructive result that giving a log morphism to the Artin fan $\mathcal {A}_X$ of a log scheme $X$ is combinatorial in nature, captured entirely by the induced map of cone complexes.

Proposition 2.10 Let $X$ be a Zariski fs log scheme log smooth over $\operatorname {Spec}\Bbbk$. Then for any fs log scheme $T$ there is a canonical bijection

\[ {\operatorname{Hom}}_{\mathrm{fs}}(T,\mathcal{A}_X) \longrightarrow {\operatorname{Hom}}_{\mathbf{Cones}}(\Sigma(T),\Sigma(X)), \]

which is functorial in $T$.

2.3 Stable logarithmic maps and their moduli

This section reviews the theory of stable logarithmic maps developed in [Reference Gross and SiebertGS13, Reference ChenChe14, Reference Abramovich and ChenAC14], emphasizing the tropical language from [Reference Gross and SiebertGS13]. Most references in the following are therefore to [Reference Gross and SiebertGS13], but of course all results have analogues in [Reference ChenChe14, Reference Abramovich and ChenAC14] under the slightly stronger assumption of global generatedness of $\overline{\mathcal {M}}_X$. Note that the restriction on global generatedness has been removed in [Reference Abramovich, Chen, Marcus and WiseACMW17] by base changing to a refinement of the Artin fan $\mathcal {A}_X$ of $X$.

2.3.1 Definition

We fix a log morphism $X \longrightarrow B$ with the logarithmic structure on $X$ being defined in the Zariski topology. Recall the following from [Reference Gross and SiebertGS13, Definition 1.6].

Definition 2.12 A stable logarithmic map $(C/S,\mathbf {p}, f)$ is a commutative diagram

(2.4)

where the following hold.

1. (i) $\pi :C \longrightarrow S$ is a proper, logarithmically smooth and integral morphism of log schemes together with a tuple of sections ${\mathbf {p}}=(p_1,\ldots ,p_{k})$ of $\underline {\pi }$ such that every geometric fibre of $\pi$ is a reduced and connected curve, and if $U\subset \underline {C}$ is the non-critical locus of $\underline {\pi }$ then $\overline{\mathcal{M}}_C|_U \simeq \underline {\pi }^*\overline{\mathcal{M}}_S\oplus \bigoplus _{i=1}^{k}p_{i*}\mathbb {N}_S$.

2. (ii) For every geometric point $\bar s \longrightarrow \underline {S}$, the restriction of $\underline {f}$ to $\underline {C}_{\bar s}$ together with ${\mathbf {p}}$ is an ordinary stable map.

2.3.4 Structure of $\psi$

The homomorphism $\psi$ is an isomorphism when restricted to the complement of the special (nodal or marked) points of $C$. The sheaf $\overline{\mathcal{M}}_C$ has stalks $Q'\oplus \mathbb {N}$ and $Q'\oplus _{\mathbb {N}}\mathbb {N}^2$ at marked points and nodal points, respectively. The latter fibred sum is determined by a map

(2.5)\begin{equation} \mathbb{N} \longrightarrow Q',\quad 1\longmapsto \rho_q \end{equation}

and the diagonal map $\mathbb {N} \longrightarrow \mathbb {N}^2$, see [Reference Gross and SiebertGS13, Definition 1.5]. The map $\psi$ at these special points is given by the inclusion $Q' \longrightarrow Q'\oplus \mathbb {N}$ and $Q' \longrightarrow Q'\oplus _{\mathbb {N}}\mathbb {N}^2$ into the first component for marked and nodal points, respectively.

2.3.5 Structure of $\varphi$

For $\bar x\in C$ a geometric point with underlying scheme-theoretic point $x$, the map $\varphi$ induces maps $\varphi _{\bar x}:P_{x} \longrightarrow \overline{\mathcal{M}}_{C,\bar x}$ for

\[ P_x:=\overline{\mathcal{M}}_{X,\underline{f}(\bar x)}. \]

Note that $\overline{\mathcal{M}}_{X,\underline{f}(\bar x)}$ is independent of the choice of $\bar x \longrightarrow x$ since the logarithmic structure on $X$ is Zariski. Following Discussion 1.8 of [Reference Gross and SiebertGS13], we have the following behaviour at three types of points on $C$.

1. (i) $x=\eta$ is a generic point, giving a local homomorphismFootnote ² of monoids

   (2.6)\begin{equation} \varphi_{\bar\eta}:P_{\eta}\longrightarrow Q'. \end{equation}

2. (ii) $x=p$ is a marked point, giving the composition

   (2.7)\begin{equation} u_p:P_p\stackrel{\varphi_{\bar p}}{\longrightarrow} Q'\oplus\mathbb{N}\stackrel{\operatorname{pr}_2}{\longrightarrow} \mathbb{N}. \end{equation}
   The element $u_p\in P_p^{\vee }$ is called the contact order at $p$.

3. (iii) $x=q$ is a node contained in the closures of $\eta _1$, $\eta _2$. If $\chi _i:P_q \longrightarrow P_{\eta _i}$ are the generization maps there exists a homomorphism

   \[ u_q:P_q \longrightarrow \mathbb{Z}, \]
   called contact order at $q$, such that
   (2.8)\begin{equation} \varphi_{\bar\eta_2}(\chi_2(m))-\varphi_{\bar \eta_1}(\chi_1(m) )=u_q(m)\cdot \rho_q, \end{equation}
   with $\rho _q\neq 0$ given in (2.5), see [Reference Gross and SiebertGS13, (1.8)]. The maps $\varphi _{\bar \eta }\circ \chi _i$ and $u_q$ are equivalent to providing the local homomorphism $\varphi _{\bar q}:P_q \longrightarrow Q'\oplus _{\mathbb {N}} \mathbb {N}^2$.

The choice of ordering $\eta _1,\eta _2$ for the branches of $C$ containing a node is called an orientation of the node. We note that reversing the orientation of a node $q$ (by interchanging $\eta _1$ and $\eta _2$) results in reversing the sign of $u_q$.

2.3.7 The basic monoid

Given a combinatorial type of a stable logarithmic map $(C/S,\mathbf {p},f)$, we define a monoid $Q$ by first defining its dual

(2.9)\begin{equation} Q^{\vee}= \Big\{ ((V_{\eta})_{\eta}, (e_q)_q ) \in \bigoplus_{\eta} P_{\eta}^{\vee}\oplus \bigoplus_q\mathbb{N} \mid \forall q: V_{\eta_2}-V_{\eta_1}=e_qu_q \Big\}. \end{equation}

Here the sum is over generic points $\eta$ of $C$ and nodes $q$ of $C$. Readers with background in tropical geometry should recognize this monoid as the moduli cone of tropical curves of fixed combinatorial type, as will be discussed in § 2.5. We then set

\[ Q:={\operatorname{Hom}}(Q^{\vee},\mathbb{N}). \]

It is shown in [Reference Gross and SiebertGS13, § 1.5], that $Q$ is a sharp monoid, fine and saturated by construction as the dual of a finitely generated submonoid of a free abelian group. Note also that $Q$ indeed only depends on the combinatorial type of $(C/S,\mathbf {p},f)$.

Given a stable logarithmic map $(C'/S',\mathbf {p}',f')$ over $S'=\operatorname {Spec}(Q' \longrightarrow \Bbbk )$ of the same combinatorial type, we obtain a canonically defined map

(2.10)\begin{equation} Q \longrightarrow Q' \end{equation}

which is most easily defined as the transpose of the map

\[ (Q')^{\vee} \longrightarrow Q^{\vee}\subset \bigoplus_{\eta} P_{\eta}^{\vee} \oplus\bigoplus_q \mathbb{N},\quad m\longmapsto ((\varphi_{\bar\eta}^t(m))_{\eta}, (m(\rho_q))_q), \]

with $\varphi _{\bar \eta }$ and $\rho _q$ defined in (2.6) and (2.5), respectively.

Definition 2.14 (Basic maps) Let $(C/S, \mathbf {p}, f)$ be a stable logarithmic map. We say $f$ is basic if at every geometric point $\bar s$ of $S$, the map $Q \longrightarrow Q'=\overline{\mathcal {M}}_{S,\bar s}$ from (2.10) defined by the restriction $(C_{\bar s}/\bar s, \mathbf {p}_{\bar s}, f|_{C_{\bar s}})$ is an isomorphism.

2.3.8 Degree data and class

In what follows, $H_2^+(X)$ denotes a semigroup carrying degree data for curves in $X$, which are locally constant in flat families, such as effective $1$-cycles on $X$ modulo algebraic or numerical equivalence or, working over $\mathbb {C}$, classes in singular homology $H_2(X,\mathbb {Z})$ pairing non-negatively with a Kähler form. We require that the moduli spaces of ordinary stable maps of fixed curve class, genus and number of marked points are of finite type.

Definition 2.15 A class $\beta$ of stable logarithmic maps to $X$ consists of the following:

1. (i) the data $\underline {\beta }$ of an underlying ordinary stable map, i.e. the genus $g$, a curve class $A\in H_2^+(X)$, and the number of marked points $k$;

2. (ii) integral elements $u_{p_1},\ldots ,u_{p_k}\in |\Sigma (X)|$.Footnote ⁴

We say a stable logarithmic map $(C/S,\mathbf {p},f)$ is of class $\beta$ if two conditions are satisfied. First, the underlying ordinary stable map must be of type $\underline{\beta} =(g,A,k)$. Second, define the closed subset $\underline{Z}_i\subset \underline{X}$ to be the union of strata with generic points $\eta$ such that $u_{p_i}$ lies in the image of $\sigma _{\eta } \longrightarrow |\Sigma (X)|$. Then for any $i$ we have $\operatorname {im}(\underline {f}\circ p_i)\subset \underline {Z}_i$ and for any geometric point $\bar s \longrightarrow \underline {S}$ such that $p_i(\bar s)$ lies in the stratum of $X$ with generic point $\eta$, there exists $u\in \sigma _\eta ={\operatorname {Hom}}(\overline{\mathcal {M}}_{X,\bar \eta },\mathbb {N})$ mapping to $u_{p_i}\in |\Sigma (X)|$ making the following diagram commute.

Here $\chi$ is the generization map. In particular, $s_i$ specifies the contact order $u_{p_i}$ at the marked point $p_i(\bar s)$ as defined in (2.7).

We emphasize that the class $\beta$ does not specify the contact orders $u_q$ at nodes.

Definition 2.16 Let $\mathscr {M}(X/B,\beta )$ denote the stack of basic stable logarithmic maps of class $\beta$. This is the category whose objects are basic stable logarithmic maps $(C/S,\mathbf {p},f)$ of class $\beta$, and whose morphisms $(C/S,\mathbf {p},f) \longrightarrow (C'/S',\mathbf {p}',f')$ are commutative diagrams

with the left-hand square cartesian, $S \longrightarrow S'$ strict, and $f=f'\circ g$, $g\circ {\mathbf {p}}= {\mathbf {p}}'\circ h$.

Theorem 2.17 If $X \longrightarrow B$ is proper, then $\mathscr {M}(X/B,\beta )$ is a proper Deligne–Mumford stack. If furthermore $X \longrightarrow B$ is logarithmically smooth, then $\mathscr {M}(X/B,\beta )$ carries a perfect obstruction theory, defining a virtual fundamental class $[\mathscr {M}(X/B,\beta )]^{{\operatorname {virt}}}$ in the rational Chow group of $\mathscr {M}(X/B,\beta )$.

Proof. Under the given assumption that $X$ is a Zariski log scheme, [Reference Gross and SiebertGS13, Theorem 2.4] proves that $\mathscr {M}(X/B,\beta )$ is a Deligne–Mumford stack. Properness was shown in [Reference Gross and SiebertGS13, Theorem 2.4] under a technical assumption, and in general in [Reference Abramovich, Chen, Marcus and WiseACMW17].

The existence of a perfect obstruction theory when $X \longrightarrow B$ is logarithmically smooth was proved in [Reference Gross and SiebertGS13, § 5].

2.4 Stacks of pre-stable logarithmic curves

For the obstruction theory in Theorem 2.17 one works over the Artin stack $\mathfrak {M}_B$ of pre-stable logarithmically smooth curves defined over $B$. Since this stack will be important later on, let us briefly recall its construction. First, working over a field $\Bbbk$, there is a stack ${\mathbf {M}}$ of pre-stable basic logarithmic curves over $\operatorname {Spec}\Bbbk$, essentially constructed by Kato in [Reference KatoKat00]. Endowing ${\mathbf {M}}$ with its basic log structure, the fibre product ${\mathbf {M}}\times _{\operatorname {Spec}\Bbbk } B$ in the category of log stacks is a fs log stack. We can then define $\mathfrak {M}_B$ using Olsson's stack over ${\mathbf {M}}\times B$:

\[ \mathfrak{M}_B:={\operatorname{Log}}_{{\mathbf{M}}\times B}. \]

Indeed, an object in this stack is a log scheme $T$ with two morphisms $T\to {\mathbf {M}}$ and $T \longrightarrow B$. The corresponding pre-stable log smooth curve over $T$ is the logarithmic pull-back to $T$ of the universal pre-stable curve over ${\mathbf {M}}$.

We also consider the following refinements of $\mathbf {M}$ introduced in [Reference Behrend and ManinBM96, Definition 2.6] and further discussed in [Reference BehrendBeh97, p. 603]. Let $G$ be a graph decorated by a map

\[ \mathbf{g}: V(G) \longrightarrow \mathbb{N}, \]

associating to each vertex its genus. Then there is an algebraic stack

(2.11)\begin{equation} \mathbf{M}(G,\mathbf{g})\quad \textrm{of} \quad(G,\mathbf{g})\text{-marked pre-stable curves} \end{equation}

with objects over a $B$-scheme $S$ given by:

1. (1) for each $v\in V(G)$, a family of pre-stable curves $C_v\to S$ of genus $\mathbf {g}(v)$, together with marked sections $x_L:S\to C_v$ defined by the legs $L\in L(G)$ with $v\in L$;

2. (2) for each edge $E\in E(G)$ with vertices $v,w$, a pair of marked sections $y_v, y_w$ of $C_v\to S$, $C_w\to S$, respectively.

All marked sections are required to be mutually disjoint and to have image in the non-critical locus of $\coprod _v C_v \longrightarrow S$. Taking the fibred sum of $\coprod _v C_v$ along the pairs of marked sections associated to the edges, we may as well view the objects of $\mathbf {M}(G,\mathbf {g})$ as families of marked nodal curves

(2.12)\begin{equation} (C \longrightarrow S,x); \end{equation}

from this point of view, each edge $E$ defines a nodal section $y_E:S\to C$ and each vertex a closed embedding $C_v\to C$ of a family of pre-stable curves of genus $g(v)$ and with image a union of irreducible components. Thus we have a morphism of algebraic stacks

(2.13)\begin{equation} \textbf{{M}}(G,\mathbf{g}) \longrightarrow \textbf{{M}}, \end{equation}

turning $\textbf {M}(G,\mathbf {g})$ into a logarithmic algebraic stack by pulling back the log structure from $\textbf {M}$. Note that on the level of the underlying stacks, (2.13) induces the identification of the stack quotient $[\mathbf {M}(G,\mathbf {g})/ \operatorname {Aut}(G,\mathbf {g})]$ with the normalization of a closed substack of $\mathbf {M}$, defining the well-known stratified structure of $\mathbf {M}$. See [Reference Arbarello, Cornalba and GriffithsACG11, XII, § 10] for a detailed discussion. Now define

(2.14)\begin{equation} \mathfrak{M}_B(G,\mathbf{g}):={\operatorname{Log}}_{\mathbf{M}(G,\mathbf{g})\times B},\quad \mathfrak{C}_B(G,\mathbf{g}):=\mathfrak{M}_B(G,\mathbf{g})\times_{\mathbf{M}(G,\mathbf{g})} \mathbf{C}(G,\mathbf{g}). \end{equation}

An important feature of the collection of stacks $\mathbf {M}(G,\mathbf {g})$ and in turn of $\mathfrak {M}_B(G,\mathbf {g})$ is their functorial behaviour under contraction morphisms of decorated graphs

(2.15)\begin{equation} \phi:(G,\mathbf{g}) \longrightarrow (G',\mathbf{g}'), \end{equation}

that is, an isomorphism of $G'$ with the graph $G/E_\phi$ contracting a subset of edges $E_\phi \subset E(G)$ such thatFootnote ⁵

\[ \mathbf{g}'(v')= b_1(\phi^{-1}(v'))+ \sum_{v\in V(\phi)^{-1}(v')} \mathbf{g}(v) \]

holds for all $v'\in V(G')$ [Reference Behrend and ManinBM96, Definition 1.3]. Here $V(\phi ):V(G) \longrightarrow V(G')$ is the surjection on the set of vertices defined by $\phi$, and we have in addition a compatible inclusion $E(\phi ): E(G')\smash {\mathop {\longrightarrow }\limits ^{\simeq }} E(G)\setminus E_\phi \subset E(G)$ of the sets of edges and a bijection $L(\phi ):L(G')\to L(G)$ on the sets of legs. This notion of morphism captures the behaviour of the combinatorial type of pre-stable curves under generization and is indeed compatible with the finite maps (2.13) to $\mathbf {M}$.

Proposition 2.18 For any contraction morphism $(G,\mathbf {g}) \longrightarrow (G',\mathbf {g}')$ of genus-decorated graphs, there are finite unramified morphisms of ordinary stacks $\mathbf {M}(G,\mathbf {g}) \longrightarrow \mathbf {M}(G',\mathbf {g}')$ and

\[ \mathfrak{M}_B(G,\mathbf{g}) \longrightarrow \mathfrak{M}_B(G',\mathbf{g}'). \]

We emphasize that Proposition 2.18 is purely on the level of stacks with no log structures involved. Incorporating log structures in the picture is more subtle and is part of the gluing formalism developed in [Reference Abramovich, Chen, Gross and SiebertACGS20].

2.5 The tropical interpretation

The basic monoid $Q$ was originally derived from its tropical interpretation, which will play an important role here. We review this in our general setting. Given a stable logarithmic map $(C/S,\mathbf {p},f)$, we obtain an associated diagram of cone complexes.

(2.16)

This diagram can be viewed as giving a family of tropical curves mapping to $\Sigma (X)$, parameterized by the cone complex $\Sigma (S)$. Indeed, a fibre of $\Sigma (\pi )$ is a graph and the restriction of $\Sigma (f)$ to such a fibre can be viewed as a tropical curve mapping to $\Sigma (X)$. We make this precise.

To avoid difficulties in notation, we shall assume that $X$ is simple (Definition 2.1). This is not a restrictive assumption in this paper since we assume $X$ to be log smooth over the trivial log point $\operatorname {Spec}\Bbbk$, and as $X$ is assumed to be Zariski in any event, it follows that $X$ is simple (Proposition 2.2). We use the reduced presentation of $\Sigma (X)$ from 2.1.2. Then simplicity implies that if $\tau ,\sigma \in \Sigma (X)$ and the image of $\tau$ in $|\Sigma (X)|$ is a face of the image of $\sigma$, then there is a unique face map $\tau \longrightarrow \sigma$ in the diagram.

The left-hand vertical arrow of (2.16) is a family of abstract tropical curves according to the following definition, cf. also [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, Definition 3.2].

Definition 2.19 A (family of) tropical curves $(G,\mathbf {g},\ell )$ over a cone $\omega \in \mathbf {Cones}$ is a connected graph $G$ together with a bijection $L(G) \longrightarrow \{1,\ldots ,k\}$ (leg ordering) and two maps

\[ \mathbf{g}: V(G) \longrightarrow \mathbb{N},\quad \ell: E(G) \longrightarrow {\operatorname{Hom}}(\omega\cap N_\omega,\mathbb{N})\setminus\{0\}. \]

For $v\in V(G)$ and $E\in E(G)$ we call $g(v)$ the genus of $v$ and $\ell (E)$ the length function of $E$.

The genus of a family of tropical curves $(G,\mathbf {g},\ell )$ is defined by

(2.17)\begin{equation} |\mathbf{g}|= b_1(G) + \sum_{v\in V(G)} \mathbf{g}(v). \end{equation}

Note that given a tropical curve $(G,\mathbf {g},\ell )$ over a cone $\omega$ and $s\in \omega$ not contained in any proper face, then $s\circ \ell$ assigns a strictly positive real number to each edge. Together with the convention that legs are infinite length, $(G,s\circ \ell )$ therefore specifies a metric graph, reproducing the traditional definition of an abstract tropical curve. Hence our definition makes precise the notion of a family of abstract tropical curves parameterized by $\omega \in \mathbf {Cones}$.

Construction 2.20 We suppress the genus decoration in the notation $(G,\mathbf {g},\ell )$ and conflate $(G,\mathbf {g},\ell )$ with its associated morphism of cone complexes

(2.18)\begin{equation} \Gamma=\Gamma(G,\ell)\stackrel{\pi_\Gamma}{\longrightarrow} \omega, \end{equation}

constructed as follows. For each $v\in V(G)$ take one copy $\omega _v$ of $\omega$, while for each $E\in E(G)$ take the cone

(2.19)\begin{equation} \omega_E=\{ (s,\lambda)\in\omega\times\mathbb{R}_{\ge0}\,|\, \lambda\le \ell(E)(s)\}. \end{equation}

The cone $\omega _E$ has two facets, each isomorphic to $\omega$ via projection to the first factor. The corresponding inclusions,

\[ s\longmapsto (s,0),\quad s\longmapsto (s,\ell(s)), \]

define face morphisms $\omega _v,\omega _{v'} \longrightarrow \omega _E$ for the two vertices $v,v'$ adjacent to $E$. Note this definition is independent of the chosen labellings $v,v'$ and works also for graphs with loops. Finally, for each $L\in L(G)$ with adjacent vertex $v$ take $\omega _L=\omega \times \mathbb {R}_{\ge 0}$ with face morphism $\omega _v\to \omega _L$ defined by the facet $\omega \times \{0\}\subset \omega _L$. Then $\Gamma$ is the generalized cone complex defined by this directed system in $\mathbf {Cones}$. The morphism to $\omega$ is defined on each $\omega _E$ by the projection to the first factor.

By construction, each vertex $v\in V(G)$ defines a section of $\pi _\Gamma : \Gamma \to \omega$ denoted as follows:

(2.20)\begin{equation} \omega \longrightarrow \Gamma,\quad s\longmapsto v(s)\in \omega_v. \end{equation}

Then for $s\in \omega$ not contained in a proper face, the fibre ${\pi _\Gamma }^{-1}(s)$ is the metric graph $(G,s\circ \ell )$ previously defined.

It is also not hard to replace individual cones as base spaces for families of tropical curves by cone complexes. See [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20, § 3] for an elaboration of such ideas.

Definition 2.21 A (family of) tropical maps (from a tropical curve) to $\Sigma (X)$ over a cone $\omega \in \mathbf {Cones}$ is a tropical curve $(G,\mathbf {g},\ell )$ over $\omega$ (Definition 2.19) with associated cone complex $\Gamma =\Gamma (G,\ell )$ (Construction 2.20), together with a morphism of cone complexes

\[ h:\Gamma\longrightarrow \Sigma(X). \]

Remark 2.22 There are a number of discrete data that we can extract from a tropical map $h:\Gamma \longrightarrow \Sigma (X)$ over $\omega \in \mathbf {Cones}$ which are of importance in the sequel.

(1) Image cones. For a vertex, edge, or leg $x$ of $G$, let $\omega _x\in \Gamma$ be the cone associated to $x$. Define

(2.21)\begin{equation} \boldsymbol{\sigma}: V(G) \cup E(G) \cup L(G) \longrightarrow \Sigma(X) \end{equation}

by mapping $x$ to the minimal cone $\tau \in \Sigma (X)$ containing $h(\omega _x)$. Note that if $E$ is a leg or edge incident to a vertex $v$, then there is an inclusion of faces $\boldsymbol {\sigma }(v)\subset \boldsymbol {\sigma }(E)$ in the (reduced) presentation of $\Sigma (X)$.

(2) Contact orders at edges. Let $E_q\in E(G)$ be an edge with a chosen order of vertices $v,v'$ (orientation). Then by the definition of the cone $\omega _{E_q}$ of $\Gamma$ associated to $E_q$ in (2.19), the image of $(0,1)\in N_{\omega _{E_q}}= N_\omega \times \mathbb {R}$ under $h$ defines $u_q\in N_{\boldsymbol {\sigma }(E_q)}$ such that in $N_{\boldsymbol {\sigma }(E_q)}$,

(2.22)\begin{equation} h(v(s))-h(v'(s))=\ell(E_q)(s)\cdot u_q \end{equation}

holds for any $s\in \omega _{E_q}$. Here $v(s)\in \Gamma$ is the section of $\Gamma \to \omega$ defined in (2.20). Reversing the orientation of $E_q$ results in replacing $u_q$ by $-u_q$.

(3) Contact orders at marked points. Similarly, for a leg $L_p\in L(G)$, the image of $(0,1)\in N_{\omega _{L_p}}= N_\omega \times \mathbb {R}$ defines $u_p \in N_{\boldsymbol {\sigma }(L_p)}\cap \boldsymbol {\sigma }(L_p)$ with $h(\operatorname {Int}(\omega _{L_p}))\subset \operatorname {Int}(\boldsymbol {\sigma }(L_p))$.

Definition 2.23 (1) The type of a family of tropical maps $h:\Gamma \longrightarrow \Sigma (X)$ over $Q^\vee _\mathbb {R}\in \mathbf {Cones}$ is the quadruple $\tau =(G,\mathbf {g},\boldsymbol {\sigma },\mathbf {u})$ consisting of the associated genus decorated graph $(G,\mathbf {g})$, the map $\boldsymbol {\sigma }$ from (2.21) recording the strata and the contact orders $\mathbf {u}=\{ u_p,u_q\}$ as defined in Remark 2.22. Note that we are suppressing the leg numbering, viewing the set $L(G)$ as identical with $\{1,\ldots ,k\}$.

(2) For a type $\tau$ of a family of tropical maps, $\operatorname {Aut}(\tau )$ denotes the subset of automorphisms of $G$ commuting with the maps $\mathbf {g},\boldsymbol {\sigma },\mathbf {u}$.

(3) Given a type $\tau$ of a family of tropical maps, the associated basic monoid $Q(\tau )$ is the dual of the monoid $Q^\vee$ defined in (2.9), depending only on $G,\boldsymbol {\sigma }$ and $\mathbf {u}$.

(4) If in addition we have given a map

\[ \mathbf{A}: V(G) \longrightarrow H_2^+(X), \]

we call ${\boldsymbol{\tau }}=(\tau ,\mathbf {A})$ the decorated type of a family of tropical maps, the pair $(h,\mathbf {A})$ a decorated family of tropical maps and

\[ |\mathbf{A}| = \sum_{v\in V(G)} \mathbf{A}(v) \]

the total curve class of $\mathbf {A}$.

Generalizing (2.15) we have a notion of contraction morphism for (decorated) types of families of tropical maps needed below.

2.5.1 Families of tropical curves from logarithmically smooth curves

Now suppose

\[ S=\operatorname{Spec}(Q \longrightarrow\Bbbk) \]

for some monoid $Q$ and $(C/S,\textbf {p})$ is a family of marked log smooth curves, as in Definition 2.12(i).

Proposition 2.25 The tropicalization

(2.23)\begin{equation} \Sigma(\pi):\Sigma(C)\longrightarrow \Sigma(S)=Q^{\vee}_{\mathbb{R}} \end{equation}

of $(C/S,\textbf {p})$ naturally has the structure of a family of tropical curves $(G,\mathbf {g},\ell )$ over $Q^\vee _\mathbb {R}$.

Proof. Take for $G$ the dual intersection graph of $C$. If $\eta$ is a generic point of $C$, then $\omega _{\eta }=Q^{\vee }_{\mathbb {R}}$ and $\Sigma (\pi )|_{\omega _{\eta }}$ is the identity. Thus each fibre of $\Sigma (\pi )|_{\omega _{\eta }}$ is a point $v$. We take the weight $\mathbf {g}(v) = g(C(v))$, the geometric genus of the component $C(v)$ with generic point $\eta$. The cone of $\Sigma (C)$ defined by a node $q$ of $C$ is

\[ \omega_q={\operatorname{Hom}}(Q\oplus_{\mathbb{N}} \mathbb{N}^2,\mathbb{R}_{\ge 0})=Q_{\mathbb{R}}^{\vee} \times_{\mathbb{R}_{\ge 0}} \mathbb{R}^2_{\ge 0}, \]

where the maps $Q^{\vee }_{\mathbb {R}} \longrightarrow \mathbb {R}_{\ge 0}$ and $\mathbb {R}^2_{\ge 0} \longrightarrow \mathbb {R}_{\ge 0}$ are given by evaluation at $\rho _q\in Q\setminus \{0\}$ and by $(a,b)\mapsto a+b$, respectively. The projection $\mathbb {R}^2_{\ge 0} \longrightarrow \mathbb {R}_{\ge 0}$ to, say, the first factor, defines an isomorphism

\[ Q_{\mathbb{R}}^{\vee} \times_{\mathbb{R}_{\ge 0}} \mathbb{R}^2_{\ge 0} \longrightarrow \{ (m,\lambda)\in Q_\mathbb{R}^\vee\times\mathbb{R}_{\ge0}\, |\, \lambda\le m(\rho_q)\}. \]

Thus defining $\ell (E_q)= \rho _q$, we have a canonical isomorphism $\omega _q\simeq \omega _{E_q}$ with $\omega _{E_q}$ defined in (2.19). For a marked point $p_i\in C$, we have $\omega _{p_i}=Q^{\vee }_{\mathbb {R}} \times \mathbb {R}_{\ge 0}$, and $\Sigma (\pi )|_{\omega _{p_i}}$ is the projection onto the first component, again compatible with the definition of $\Gamma =\Gamma (G,\ell )$ in Construction 2.20.

2.5.2 Families of tropical maps to $\Sigma (X)$ from stable logarithmic maps

We continue working over a logarithmic point $S=\operatorname {Spec}(Q \longrightarrow \Bbbk )$ and assume in addition given an fs log scheme $X$, which is simple in the sense of Definition 2.1.

Proposition 2.26 The tropicalization of a stable logarithmic map $(C/S,\textbf {p},f)$ over the logarithmic point $S=\operatorname {Spec}(Q \longrightarrow \Bbbk )$ defines a family of tropical maps to $\Sigma (X)$ over $Q_\mathbb {R}^\vee$.

Proof. In view of Proposition 2.25 the statement follows readily from the definitions.

Remark 2.27 An element $x\in V(G)\cup E(G)\cup L(G)$ corresponds to a point $x\in C$ – either a generic point, a double point, or a marked point. The cone $\boldsymbol {\sigma }(x)$ introduced in Remark 2.22 is

\[ \boldsymbol{\sigma}(x)=(P_x)^{\vee}_{\mathbb{R}}={\operatorname{Hom}}(\overline{\mathcal{M}}_{X,\underline{f}(\bar x)},\mathbb{R}_{\ge 0}) \in \Sigma(X), \]

for any geometric point $\bar x$ mapping to $x$. With this identification of cones understood, it is a matter of unravelling the definitions that the other discrete data introduced in Remark 2.22, the contact orders $u_{L_p},u_{E_q}$, agree with $u_p,u_q$ defined in § 2.3.5. Note in particular how (2.22) appears as the tropical manifestation of (2.8). Thus the type of the tropicalization of a stable logarithmic map, as a family of tropical maps (Definition 2.23), agrees with its combinatorial type from Definition 2.13.

2.5.3 Traditional tropical maps: the relative situation

A situation of particular interest arises when working over the standard log point $b_0=\operatorname {Spec}(\mathbb {N}\to \Bbbk [\mathbb {N}]))$. Then all generalized cone complexes come with a morphism $\pi$ to $\Sigma (b_0)=\mathbb {R}_{\ge 0}$. Taking the fiber of $\pi$ over $1\in \mathbb {R}_{\ge 0}$ then produces a generalized polyhedral complex as introduced in § 2.1.3. Conversely, let $\pi :\Sigma \to \mathbb {R}_{\ge 0}$ be a map of generalized cone complexes such that no maximal cone of $\Sigma$ maps to $0\in \mathbb {R}_{\ge 0}$. Then $\Sigma$ and $\pi$ can be recovered from the generalized polyhedral complex $\pi ^{-1}(1)$ by replacing each polyhedron $\sigma =(\sigma _\mathbb {R},N)$ by the closure of $\mathbb {R}_{\ge 0}(\sigma \times \{1\})$ in $N_\mathbb {R}\times \mathbb {R}$.

If $X$ is a finite type logarithmic stack over the standard log point $b_0$ with associated tropicalization $\pi :\Sigma (X) \longrightarrow \Sigma (B_0)=\mathbb {R}_{\ge 0}$, we now write

\[ \Delta(X)= \pi^{-1}(1) \subset \Sigma(X) \]

for the associated polyhedral complex.

In particular, this discussion applies to the logarithmic scheme $X_0$ and logarithmically smooth morphism $X_0 \longrightarrow b_0$ from the main theorem in this paper, Theorem 1.2. Let $(C/S,\mathbf {p},f)$ be a stable log map to $X_0$ with $S=\operatorname {Spec}(Q\to \Bbbk )$ a log point as in § 2.5.2, but now coming with a map to $b_0$. Let $\pi _S:Q^\vee \to \mathbb {N}$ be the tropicalization of $S \longrightarrow b_0$. Then the family of tropical maps $\Sigma (C) \longrightarrow \Sigma (X_0)$ over $Q^\vee$ carries the same information as its restriction to the fiber over $1\in \mathbb {R}_{\ge 0}$, a family of maps from metric graphs to $\Delta (X)$ parameterized by the polyhedron $\pi _S^{-1}(1)\subset Q^\vee$.

The transition from cone complexes to polyhedral complexes provides the link to more traditional tropical language. In the remainder of this paper we use cone complexes for most of the general results and polyhedral complexes for explicit computations. With regards to using both cones and polyhedra as parameter spaces for families of tropical maps, note that there is no conflict of language: a family of tropical maps to $\Sigma (X_0)$ over a cone $\sigma$ can be viewed as a family of maps of metric graphs to $\Sigma (X_0)$ interpreted as a polyhedral complex, now parameterized by $\sigma$ as a polyhedron.Footnote ⁶

As a matter of notation, we indicate the transition from cone complexes to polyhedral complexes by overlining. Thus a family of tropical maps $h:\Gamma \longrightarrow \Sigma (X_0)$ over a cone $\sigma$ with a map $\pi _S: \sigma \longrightarrow \mathbb {R}_{\ge 0}$ induces the family of tropical maps

(2.24)\begin{equation} \bar h:\bar \Gamma \longrightarrow \bar\Sigma(X)=\Delta(X) \end{equation}

over the polyhedron $\bar \sigma =\pi _S^{-1}(1)$.

2.5.4 Basic maps and tropical universal families

Basicness of a stable logarithmic map $(C/S,\mathbf {p},f)$ over a logarithmic point can then be recast as follows.

Proposition 2.28 Let $(C/S,\mathbf {p},f)$ be a stable logarithmic map over a logarithmic point $S=\operatorname {Spec}(Q \longrightarrow \Bbbk )$ and $\tau$ its combinatorial type (Definition 2.13). Then $(C/S,\mathbf {p},f)$ is basic if and only if the family of tropical maps in Proposition 2.26 is universal among families of tropical maps to $\Sigma (X)$ of type $\tau$.

Proof. The definition of the dual of the basic monoid $Q^{\vee }$ precisely encodes the data of a family of tropical maps to $\Sigma (X)$ over $\sigma =\mathbb {R}_{\ge 0}$ of type $\tau$ (Definition 2.23). Indeed, let $G_C$ be the dual intersection graph of $C$ from § 2.3.6, with vertices $v_\eta$, edges $E_q$ and legs $L_p$. Then a tuple $((V_{\eta })_{\eta }, (e_q)_q)\in \operatorname {Int}(Q^{\vee }_{\mathbb {R}})$ specifies a family of tropical maps

\[ h:\Gamma(G,\ell) \longrightarrow \Sigma(X) \]

over $\mathbb {R}_{\ge 0}$ of the given type, by defining $\ell (E_q)=e_q$ and $h|_{\omega _v}$ by mapping $1\in \mathbb {R}_{\ge 0}=\omega _{v_\eta }$ to $V_{\eta }\in \Sigma (X)$. The type also determines $h$ on each leg $L_p$. It is shown in [Reference Gross and SiebertGS13, Proposition 1.9] that if one such tropical map to $\Sigma (X)$ of a certain type exists then there exists one over $Q_\mathbb {R}^\vee$; moreover, any other tropical map of the same type, say over $\sigma \in \mathbf {Cones}$, is obtained from this one by pull-back via a homomorphism $\sigma \to Q_{\mathbb{R}}^\vee$.

Remark 2.29 Note that if $S$ is not a log point, the diagram (2.16) still exists, but the fibres of $\Sigma (\pi )$ may not be the expected ones. In particular, if $\bar s$ is a geometric point of $S$, there is a functorial diagram

but this diagram need not be Cartesian due to monodromy in the family $S$. For example, it is easy to imagine a situation where $C_{\bar s}$ has two irreducible components and two nodes for every geometric point $\bar s$, but the nodal locus of $C \longrightarrow S$ is irreducible, as there is monodromy interchanging the two nodes. Then a fibre of $\Sigma (C) \longrightarrow \Sigma (S)$ may consist of two vertices joined by a single edge, while a fibre of $\Sigma (C_{\bar s}) \longrightarrow \Sigma (\bar s)$ will have two vertices joined by two edges. Similarly, there may be monodromy interchanging irreducible components, and hence a fibre of $\Sigma (C) \longrightarrow \Sigma (S)$ may have fewer vertices than $C_{\bar w}$ has irreducible components. This issue can be resolved by redefining moduli of tropical curves as stacks, following [Reference Cavalieri, Chan, Ulirsch and WiseCCUW20].

2.5.5 Decorated tropical maps from stable logarithmic maps

In the situation of Proposition 2.26, the tropical map $h:\Sigma (C)\to \Sigma (X)$ comes with the natural decoration

(2.25)\begin{equation} \mathbf{A}: V(G) \longrightarrow H_2^+(X),\quad v\longmapsto [f(C(v))]. \end{equation}

Here $C(v)\subset C$ is the irreducible component corresponding to the vertex $v$ and $[f(C(v))]$ is the class of $f(C(v))$ in $H_2^+(X)$. The decoration by curve classes is compatible with the contraction morphisms of decorated graphs (Definition 2.24) defined by generization.

Lemma 2.30 Let $(C/S,\mathbf {p},f)$ be a stable logarithmic map to $X$ over some logarithmic scheme $S$ and $(\tau _{\bar s},\mathbf {A}_{\bar s})$ with $\tau _{\bar s}= (G_{\bar s},\mathbf {g}_{\bar s},\boldsymbol {\sigma }_{\bar s},\mathbf {u}_{\bar s})$ its decorated type at the geometric point $\bar s\to S$ according to Definition 2.23 and (2.25). Then if $\bar s,\bar s'\to S$ are two geometric points with $\bar s$ a generization of $\bar s'$, the induced map

\[ (\tau_{\bar s'},\mathbf{A}_{\bar s'}) \longrightarrow (\tau_{\bar s},\mathbf{A}_{\bar s}) \]

is a contraction morphism (Definition 2.24).

Proof. [Reference Gross and SiebertGS13, Lemma 1.11] says that $\tau _{\bar s'}\to \tau _{\bar s}$ is a contraction morphism. To check the statement on the curve classes, recall that the preimage of $v\in G_{\bar s}$ in $G_{\bar s}'$ consists of those vertices $v'\in G_{\bar s'}$ with $C_{\bar s'}(v')$ contained in the closure of $C_{\bar s}(v)$. Since this closure defines a flat family of curves, invariance of classes in $H_2^+(X)$ in flat families then implies

\[ [f(C_{\bar s}(v))] = \sum_{v'\mapsto v} [f(C_{\bar s'}(v'))], \]

as claimed.

2.6 Stacks of stable logarithmic maps marked by tropical types

We now put ourselves in the situation of the main result in this paper, Theorem 1.2, and assume $X_0 \longrightarrow b_0$ is logarithmically smooth and $X_0$ is simple. In particular, curve classes are understood to take values in $H_2^+(X_0)$.

Similar to $\mathbf {M}(G,\mathbf {g})$, we can now define stacks of stable logarithmic maps to $X_0$ over $b_0$ with restricted decorated types of tropicalizations.

Given a decorated type ${\boldsymbol{\tau }}=(\tau ,\mathbf {A})$ of tropical maps, we define

(2.26)\begin{equation} \mathscr{M}(X_0,{\boldsymbol{\tau}}) \end{equation}

as the stack with objects over a scheme $S$ basic stable logarithmic maps $(C/S,\mathbf {p},f)$ over $b_0$ marked by the decorated type ${\boldsymbol{\tau }}$. We emphasize $\mathscr {M}(X_0,{\boldsymbol{\tau }})$ is a moduli space of stable maps over $b_0$, but we suppress $/b_0$ in the notation for simplicity. Similarly, we henceforth write $\mathscr {M}(X_0,\beta )$ instead of $\mathscr {M}(X_0/b_0,\beta )$.

For later use let us also show here that the monoid ideals in Definition 2.31(4) define a coherent sheaf of ideals [Reference OgusOgu18, Proposition II.2.6.1] in $\mathcal {M}_{\mathscr {M}(X_0,{\boldsymbol{\tau }})}$.

Let $\beta =(g,A,u_{p_1},\ldots ,u_{p_k})$ with $g=|\mathbf {g}|$, $A=|\mathbf {A}|$, $k=|L(G)|$.

Proof. By condition (1) in Definition 2.31, we have a morphism of stacks

\[ \mathscr{M}(X_0,{\boldsymbol{\tau}}) \longrightarrow \mathscr{M}(X_0,\beta)\times_\mathbf{M} \mathbf{M}(G,\mathbf{g}) = \mathscr{M}(X_0,\beta)\times_\mathfrak{M} \mathfrak{M}(G,\mathbf{g}). \]

Conditions (2) and (4) in Definition 2.31 defines a closed substack of the fibre product on the right-hand side. Prescribing the contact orders $u_p, u_q$ at $p\in L(G)$, $q\in E(G)$ and the curve classes for the subcurves of $C$ defined by each $v\in V(G)$ imposes locally constant conditions, and hence select a union of connected components of this closed substack. Thus $\mathscr {M}(X_0,{\boldsymbol{\tau }})$ is isomorphic to a closed substack of the algebraic stack $\mathscr {M}(X_0,\beta )\times _\mathbf {M} \mathbf {M}(G,\mathbf {g})$, proving (1). The second statement follows since $\mathfrak {M}(G,\mathbf {g}) \longrightarrow \mathfrak {M}$ is finite and unramified (Proposition 2.18).

3.1 Decomposition in the log smooth case

The decomposition formula is based on the following simple fact in toric geometry. Let $\pi : W \longrightarrow \mathbb {A}^1$ be a morphism of toric varieties with $\Sigma _\pi :\Sigma _W\to \Sigma _{\mathbb {A}^1}$ the corresponding morphism of fans, defined by a homomorphism $N \longrightarrow N_{\mathbb {A}^1}$ of co-character lattices. We identify $\Sigma _W$ with the cone complex $\Sigma (W)$ associated to $W$ with its toric log structure, by forgetting the embedding of $|\Sigma _W|$ into $N_\mathbb {R}$, and similarly for $\Sigma _{\mathbb {A}^1}$. For a ray $\gamma \in \Sigma _W$ denote by $D_\gamma \subset W$ the corresponding toric divisor and by $m_\gamma \in \mathbb {N}$ the generator of the image of

\[ \mathbb{Z}\simeq N_\gamma\stackrel{\Sigma(\pi)}{\longrightarrow} N_{\mathbb{A}^1}\simeq\mathbb{Z}. \]

Proposition 3.1 We have the following equality of Weil divisors on $W$:

\[ \pi^*(\{0\}) = \sum_{\gamma} m_\gamma D_\gamma. \]

Proposition 3.1 can equivalently be stated as a decomposition of the fundamental class of $W_0=\pi ^{-1}(0)$. Our decomposition theorem is based on the generalization of this statement to a log smooth morphism $W_0 \longrightarrow b_0$ of logarithmic algebraic stacks locally of finite type. Note first that in this situation, $W_0$ is locally pure-dimensional by log smoothness over $b_0$. Thus it makes sense to define the fundamental cycle $[W_0]$ as locally finite formal linear combination of locally top-dimensional integral substacks.

Next, to define the multiplicities $m_\gamma$, consider the morphism of generalized cone complexes $\Sigma (W_0) \to \Sigma (b_0)$ associated to $W_0 \longrightarrow b_0$ as defined after Proposition 2.2. We have $\Sigma (b_0) \simeq \mathbb {R}_{\geq 0}$ with the lattice $N_{b_0} \simeq \mathbb {Z}$. Working in charts, there is still a correspondence between rays $\gamma \in \Sigma (W_0)$ and integral substacks $W_\gamma \subset W_0$, now locally of top dimension. Note that if $\sigma \in \Sigma (W_0)$ and $\gamma \to \sigma$ is a morphism in $\Sigma (X)$, then the pull-back of $W_\gamma$ to a chart for $W_0$ at a geometric point of the stratum $W_0(\sigma )$ is contained in the union of all toric divisors for rays $\gamma '\subset \sigma$ with $\gamma \simeq \gamma '$ in $\Sigma (W_0)$. Hence $W_\gamma$ may not be locally irreducible if $\Sigma (W_0)$ has cones with self-identifications. But since we work with cone complexes with reduced presentations, such rays $\gamma ,\gamma '\subset \sigma$ define the same one-dimensional cone in $\Sigma (W_0)$. These rays may be identified by self-maps of $\sigma$ or simply correspond to several maps $\sigma ^\vee \to \gamma ^\vee$ defined by generization in $\overline{\mathcal{M}}_{W_0}$.

For a ray $\gamma$ with integral lattice $N_{\gamma }$, we have $\gamma \cap N_{\gamma }\simeq \mathbb {N}$, and the homomorphism $\mathbb {Z}\simeq N_\gamma \longrightarrow N_{b_0}\simeq \mathbb {Z}$ is multiplication by an integer $m_\gamma$.

For the following statement recall also the notion of idealized log structures and idealized log smoothness from [Reference OgusOgu18, III.1.3 and IV.3]. In a nutshell, this notion is designed to treat strata of logarithmic spaces, by adding sheaves of ideals $\mathcal {K}\subset \mathcal {M}_X$ defining these strata as part of the data.

Corollary 3.2 Let $\pi :W_0 \longrightarrow b_0$ be a log smooth morphism locally of finite type from a logarithmic algebraic stack to the standard log point $b_0$. Denote by $[W_0]$ the fundamental cycle of $W_0$, well-defined since $W_0$ is locally pure-dimensional. Then the following formula holds

\[ [W_0] = \sum_{\gamma} m_\gamma [W_\gamma] \]

in the group of locally top-dimensional algebraic cycles on $W_0$ [Reference KreschKre99]. The sum runs over the one-dimensional cones in the generalized cone complex $\Sigma (W_0)$ of $W_0$.

Moreover, $W_\gamma$ is idealized log smooth over $b_0$ for some sheaf of ideals $\mathcal {K}_\gamma \subset \mathcal {M}_{W_\gamma }$.

Proof. The claimed equality of cycles can be checked on a cover by smooth charts. We may thus assume that $W_0$ is covered by a neat chart, that is, that we have a commutative diagram

where (i) $h$ is an étale surjection, (ii) $\operatorname {Spec}\Bbbk \longrightarrow \mathbb {A}^1$ is the inclusion of the origin and $g:U\to \operatorname {Spec}\Bbbk \times _{\mathbb {A}^1} V= \pi _V^{-1}(0)$ is smooth, (iii) $V$ is the affine toric variety $\operatorname {Spec}\Bbbk [\sigma ^\vee \cap N^*]$ defined by $(\sigma _\mathbb {R},N)\in \Sigma (W_0)$ and $\pi _V:V\to \mathbb {A}^1$ is a toric morphism. Thus we have

\[ h^*[W_0]=[U]=g^*([V_0]) \]

via flat pull-back, where $V_0=\pi ^{-1}(0)$. Now Proposition 3.1 describes $[V_0]$ in terms of the toric divisors $D_{\gamma '}\subset V$ defined by the rays $\gamma '\subset \sigma$. Thus

(3.1)\begin{equation} h^*[W_0]= \sum_{\gamma'\subset\sigma} m_{\gamma'} g^*(D_{\gamma'}), \end{equation}

with $m_{\gamma '}$ the generator of the image of $\mathbb {Z}\simeq N_{\gamma '}\to N_{\mathbb {A}^1}=\mathbb {Z}$. Each such $\gamma '$ defines a one-dimensional cone $\gamma \in \Sigma (W_0)$ with $m_\gamma =m_{\gamma '}$. Moreover, for two different rays $\gamma ',\gamma ''\subset \sigma$, the geometric generic points of $D_{\gamma '}$, $D_{\gamma ''}$ map to the same geometric generic point of $W_0$ if and only if there exists a one-dimensional cone $\gamma \in \Sigma (W_0)$ and morphisms $\gamma \to \gamma '$ and $\gamma \to \gamma ''$. Since $\Sigma (x)$ is the colimit of such $\sigma$ appearing in neat charts of $W_0$, the equality (3.1) in a chart verifies the claimed equation of cycles.

The claim on idealized log smoothness of $W_\gamma$ follows from the local description as a union of toric strata and the criteria in [Reference OgusOgu18, IV.3.1.21 and IV.3.1.22].

3.2 Logarithmic maps to the relative Artin fan $\mathcal {X}_0$

To lift the decomposition result Corollary 3.2 to the moduli space $\mathscr {M}(X_0,\beta ) =\mathscr {M}(X_0/b_0,\beta )$ of stable logarithmic maps in Theorem 1.2, we factor the map $\mathscr {M}(X_0,\beta ) \to \mathfrak {M}_{b_0}$ forgetting the logarithmic map to $X_0$ via an intermediate log stack that is log étale over $\mathfrak {M}_{b_0}=\mathfrak {M}_B\times _B b_0$. This intermediate log stack is the stack $\mathfrak {M}(\mathcal {X}_0,\beta ')$ of basic logarithmic maps to the relative Artin fan $\mathcal {X}_0 = b_0\times _B \mathcal {X}$ of $X_0$ over $b_0$ (Definition 2.9). Since curve classes do not make sense on $\mathcal {X}_0$, we have no stability in $\mathfrak {M}(\mathcal {X}_0,\beta ')$ and

\[ \beta'=(g,u_{p_1},\ldots,u_{p_k}) \]

only keeps the genus and the contact orders at the marked points from $\beta =(g,A,u_{p_1},\ldots ,u_{p_k})$. The point is that $\mathfrak {M}(\mathcal {X}_0,\beta ')$ is pure-dimensional, has unobstructed deformations and captures the tropical geometry of the situation, while the decomposition according to Corollary 3.2 has a simple tropical interpretation on this stack.

Note that Proposition 3.3(2) also shows that $\mathfrak {M}(\mathcal {X}_0,\beta ')$ is log smooth over $b_0$, because $\mathfrak {M}_{b_0}$ is, and that the obstruction theory of $\mathscr {M}(X_0,\beta )$ over $\mathfrak {M}_{b_0}$ induces an obstruction theory for $\mathscr {M}(X_0,\beta )$ over $\mathfrak {M}(\mathcal {X}_0,\beta ')$.

We also need the $\tau$-marked refinements $\mathfrak {M}(\mathcal {X}_0,\tau )$ of $\mathfrak {M}(\mathcal {X}_0,\beta ')$, similar to $\mathscr {M}(X_0,{\boldsymbol{\tau }})$ for $\mathscr {M}(X_0,\beta )$. Omitting the curve class, $\tau$ is now a type of tropical map to $\Sigma (X_0)$ of total genus $g$ and with $k$ legs (Definition 2.23(1)). Then

\[ \mathfrak{M}(\mathcal{X}_0,\tau) \]

is defined as in Definition 2.31 with $\mathcal {X}_0$ replacing $X_0$ and disregarding the curve classes in Definition 2.31(3). Analogous to $\mathcal {K}_{\boldsymbol{\tau }}$ for $\mathscr {M}(X_0,{\boldsymbol{\tau }})$ constructed in Lemma 2.33, we have a sheaf of ideals

(3.2)\begin{equation} \mathcal{K}_\tau\subset \mathcal{M}_{\mathfrak{M}(\mathcal{X}_0,\tau)}. \end{equation}

We first observe the following analogue of Proposition 2.34.

We are now in position to apply Corollary 3.2 to $\mathfrak {M}(\mathcal {X}_0,\beta ')\to b_0$. The key is the description of the components $W_\gamma$ in this corollary in terms of rigid tropical maps.

In the language of polyhedral complexes, being rigid is equivalent to saying that the restriction $\bar h: \bar \Gamma \longrightarrow \Delta (X)$ of $h$ to the fiber over $1\in \mathbb {R}_{\ge 0}=\Sigma (b_0)$ cannot be deformed as a map of generalized polyhedral complexes. In other words, as a traditional tropical map, any deformation of $\bar h$ keeping the combinatorial data (i.e. of constant type) is trivial.

The following decomposition of the Artin stack $\mathfrak {M}(\mathcal {X}_0,\beta ')$ according to rigid tropical curves is the main result of this section.

Proof. The logarithmic stack $\mathfrak {M}(\mathcal {X}_0,\beta ')$ is logarithmically smooth over $b_0$ by Proposition 3.3 and since $\mathfrak {M}_{b_0}/b_0$ is logarithmically smooth. Up to a smooth factor, the map

\[ \mathfrak{M}(\mathcal{X}_0,\beta') \longrightarrow b_0 \]

is locally given by base change to the central fibre of the map of toric varieties $\operatorname {Spec} \Bbbk [Q] \longrightarrow \operatorname {Spec} \Bbbk [\mathbb {N}]$ with $Q$ the basic monoid of a tropical map to $\Sigma (X)$ of some type $\tau '$ and $\mathbb {N}\to Q$ induced by the structure map

\[ \Sigma(\pi): \Sigma(X) \longrightarrow \Sigma(B)=\mathbb{R}_{\ge0}. \]

Locally the subschemes $W_\gamma$ are defined by the toric divisors in $\operatorname {Spec}\Bbbk [Q]$, which are in bijection to extremal rays in $Q_\mathbb {R}^\vee$. Each extremal ray defines a rigid tropical map, say of type $\tau$. Any localization map of the associated basic monoids $Q_{\tau '} \longrightarrow Q_\tau =\mathbb {N}$ is the contraction of the codimension one face dual to the one-dimensional cone in $Q_{\tau '}^\vee$ defined by $\tau$. By the definition of $\mathcal {K}_\tau$, the monoid ideal defining the corresponding toric prime divisor agrees with the ideal in $Q_{\tau '}$ given by $\mathcal {K}_\tau$. Since this description is compatible with the restriction of charts, the first statement follows.

Corollary 3.2 also shows that $W_\gamma \to b_0$ is idealized log-smooth. The corresponding sheaf of ideals has just been checked to agree with $\mathcal {K}_\tau$ locally along $W_\gamma$.

Corollary 3.8 We have the following equality of top-dimensional algebraic cycles in the pure-dimensional algebraic stack $\mathfrak {M}(\mathcal {X}_0,\beta ')$:

\[ [\mathfrak{M}(\mathcal{X}_0,\beta')] = \sum_\tau m_\tau\cdot [\iota_\tau(\mathfrak{M}(\mathcal{X}_0,\tau))]. \]

The sum is over all types $\tau$ of rigid tropical maps to $\Sigma (X)$ and $m_\tau \in \mathbb {N}\setminus \{0\}$ is the projection of the generator of the dual basic monoid $Q_\tau ^\vee \simeq \mathbb {N}$ to $\Sigma (b_0)=\mathbb {R}_{\ge 0}$.

3.3 Proof of the decomposition theorem

To prove the main theorem, Theorem 1.2, it remains to apply the virtual bivariant machinery developed by Costello [Reference CostelloCos06] and Manolache [Reference ManolacheMan12]. We need two lemmas.

Lemma 3.9 The degree of the finite map

\[ \iota_\tau:\mathfrak{M}(\mathcal{X}_0,\tau) \longrightarrow \iota_\tau(\mathfrak{M}(\mathcal{X}_0,\tau))\subset\mathfrak{M}(\mathcal{X}_0,\beta') \]

from Proposition 3.5(2) over any irreducible component of the image is $|\operatorname {Aut}(\tau )|$.

As an intermediate object we define the stack of basic stable logarithmic maps marked by a tropical type $\tau$ by

(3.3)\begin{equation} \mathscr{M}_\tau(X_0,\beta):= \mathfrak{M}(\mathcal{X}_0,\tau)\times_{\mathfrak{M}(\mathcal{X}_0,\beta')}\mathscr{M}(X_0,\beta). \end{equation}

Compared to $\mathscr {M}(X_0,{\boldsymbol{\tau }})$, this stack keeps the total curve class $A$ from $\beta =(g,A,u_{p_1},\ldots ,u_{p_k})$, but drops the restriction on the distribution of $A$ to the subcurves given by the vertices.

For the following statement recall that $\mathscr {M}(X_0,{\boldsymbol{\tau }})$ is the stack defined in (2.26) of basic stable log maps over $b_0$ marked by the decorated type ${\boldsymbol{\tau }}$ and

\[ j_{\boldsymbol{\tau}}: \mathscr{M}(X_0,{\boldsymbol{\tau}}) \longrightarrow \mathscr{M}(X_0,\beta) \]

is the morphism forgetting the marking.

Lemma 3.10 Let $\tau =(G,\mathbf {g},\boldsymbol {\sigma },\mathbf {u})$ be the type of a tropical map to $\Sigma (X_0)$ and $\beta =(g,A,u_{p_1},\ldots ,u_{p_k})$. Then we have the decomposition

\[ \mathscr{M}_\tau(X_0,\beta) = \coprod_{\mathbf{A}}\mathscr{M}\big(X_0,{\boldsymbol{\tau}}\big), \]

where the sum is over all $\mathbf {A}: V(G)\to H_2^+(X_0)$ with $|\mathbf {A}|=A$ and ${\boldsymbol{\tau }}=(\tau ,\mathbf {A})$.

Before stating the main theorem, we note that $\mathscr {M}_\tau (X_0,\beta )$ inherits a perfect obstruction theoryFootnote ⁷ $\mathfrak {E}_\tau$ over $\mathfrak {M}(\mathcal {X}_0,\tau )$ from the perfect obstruction theory $\mathfrak {E}$ of $\mathscr {M}(X_0,\beta )$ over $\mathfrak {M}(\mathcal {X}_0,\beta ')$ by base change by $\iota _\tau : \mathfrak {M}(\mathcal {X}_0,\tau )\to \mathfrak {M}(\mathcal {X}_0,\beta ')$. Restricting to the open substacks $\mathscr {M}(X_0,{\boldsymbol{\tau }})\subset \mathscr {M}_\tau (X_0,\beta )$ in Lemma 3.10, we also have an obstruction theory $\mathfrak {E}_{\boldsymbol{\tau }}$ on $\mathscr {M}(X_0,{\boldsymbol{\tau }})$. If $\tau$ is rigid, $\mathfrak {M}(\mathcal {X}_0,\tau )$ is pure-dimensional of the same dimension as $\mathfrak {M}(\mathcal {X}_0,\beta ')$. Thus we have virtual fundamental classes

\[ [\mathscr{M}(X_0,\beta)]^{\operatorname{virt}},\quad [\mathscr{M}_\tau(X_0,\beta)]^{\operatorname{virt}},\quad [\mathscr{M}(X_0,{\boldsymbol{\tau}})]^{\operatorname{virt}} \]

on the moduli spaces $\mathscr {M}(X_0,\beta )$, $\mathscr {M}_\tau (X_0,\beta )$ and $\mathscr {M}(X_0,{\boldsymbol{\tau }})$.

Here is our main theorem, stated as Theorem 1.2 in the introduction.

Theorem 3.11 For any $\beta =(g,A,u_{p_1},\ldots ,u_{p_k})$ we have the equality

\[ [\mathscr{M}(X_0,\beta)]^{\operatorname{virt}} = \sum_{{\boldsymbol{\tau}}=(\tau,\mathbf{A})} \frac{m_\tau}{|\operatorname{Aut}(\boldsymbol{\tau})|}\, {j_{\boldsymbol{\tau}}}_*[\mathscr{M}(X_0,{\boldsymbol{\tau}})]^{\operatorname{virt}} \]

in the Chow group of the underlying stack $\underline{\mathscr {M}}(X_0,\beta )$ with coefficients in $\mathbb {Q}$. The sum is over all isomorphism classes of decorated types of rigid tropical maps ${\boldsymbol{\tau }}=(G,\mathbf {g},\boldsymbol {\sigma },\mathbf {u},\mathbf {A})=(\tau ,\mathbf {A})$ of total genus $|\mathbf {g}|=g$, total curve class $|\mathbf {A}|=A$ and $|L(G)|=k$.

Proof. By Corollary 3.8 and Lemma 3.9 we can write the fundamental class of $\mathfrak {M}(\mathcal {X}_0,\beta ')$ as

(3.4)\begin{equation} [\mathfrak{M}(\mathcal{X}_0,\beta')]= \sum_\tau \frac{m_\tau}{|\operatorname{Aut}(\tau)|}\, {\iota_\tau}_*[\mathfrak{M}(\mathcal{X}_0,\tau)]. \end{equation}

For each $\tau$, compatibility of virtual pull-back with push-forward [Reference ManolacheMan12, Theorem 4.1(3)] applied to the cartesian square

yields

\[ p_\mathfrak{E}^!{\iota_\tau}_*[\mathfrak{M}(\mathcal{X}_0,\tau)] = {j_\tau}_* q_{\mathfrak{E}_\tau}^! [\mathfrak{M}(\mathcal{X}_0,\tau)] = {j_\tau}_* [\mathscr{M}_\tau(X_0,\beta)]^{\operatorname{virt}}. \]

Moreover, from Lemma 3.10 and the definition of $\mathfrak {E}_{\boldsymbol{\tau }}$ by restriction of $\mathfrak {E}_\tau$, we have

\[ [\mathscr{M}_\tau(X_0,\beta)]^{\operatorname{virt}} = \sum_\mathbf{A} [\mathscr{M}(X_0,(\tau,\mathbf{A}))]^{\operatorname{virt}}. \]

Plugging the last two equalities into (3.4) now gives the desired result:

\begin{align*} [\mathscr{M}(X_0,\beta)]^{\operatorname{virt}} &= p_\mathfrak{E}^![\mathfrak{M}(\mathcal{X}_0,\beta')] = \sum_\tau \frac{m_\tau}{|\operatorname{Aut}(\tau)|}\, p_\mathfrak{E}^!{\iota_\tau}_*[\mathfrak{M}(\mathcal{X}_0,\tau)]\\ &= \sum_\tau \frac{m_\tau}{|\operatorname{Aut}(\tau)|}\, {j_\tau}_* [\mathscr{M}_\tau(X_0,\beta)]^{\operatorname{virt}}\\ &= \sum_{{\boldsymbol{\tau}}=(\tau,\mathbf{A})} \frac{m_\tau}{|\operatorname{Aut}(\boldsymbol{\tau})|}\, {j_{\boldsymbol{\tau}}}_*[\mathscr{M}(X_0,{\boldsymbol{\tau}})]^{\operatorname{virt}}. \end{align*}