We continue with a pair (X, D) satisfying Assumptions 1.1 or 1.2 in the absolute and relative cases. We recall also we have fixed data:
-
(1)
A group of degree data \(H_2(X)\).
-
(2)
A saturated finitely generated monoid \(Q\subseteq H_2(X)\) such that \(Q\cap (-Q)=H_2(X)_{\mathrm {tors}}\) which contains the classes of all effective curves on X. We write the monomial maximal ideal of Q
$$\begin{aligned} {\mathfrak {m}}:=Q{\setminus } Q^{\times }. \end{aligned}$$
Throughout this section, we also fix a monoid ideal \(I\subseteq Q\) such that \(\sqrt{I}={\mathfrak {m}}\). Equivalently, we require \(Q{\setminus } I\) to be finite.
Recall of monomials on B
We have constructed \((B,{\mathscr {P}},\varphi )\) with \(\varphi \) a \(Q^{{{\text {gp}}}}_{{\mathbb {R}}}\)-valued MPL function as given in Construction 1.14. This choice of function then yields a local system \({\mathcal {P}}\) [39, Def. 1.15] on \(B_0=B{\setminus }\Delta \) fitting into an exact sequence
$$\begin{aligned} 0\longrightarrow \underline{Q}^{{{\text {gp}}}} \longrightarrow {\mathcal {P}}\longrightarrow \Lambda \longrightarrow 0, \end{aligned}$$
where \(\underline{Q}^{{{\text {gp}}}}\) denotes the constant sheaf with stalk \(Q^{{{\text {gp}}}}\) on \(B_0\).Footnote 7 We write the map \({\mathcal {P}}_x\rightarrow \Lambda _x\) as \(p\mapsto {\bar{p}}\). Further, for each \(x\in B_0\), [39, Def. 1.16] gives a submonoid \({\mathcal {P}}^+_x\subseteq {\mathcal {P}}_x\) of exponents of monomials defined at x.
For our purposes, rather than reviewing the definition of \({\mathcal {P}}\), it is easier to give explicit descriptions of the monoids \({\mathcal {P}}^+_x\) and the effects of parallel transport under these descriptions.
For \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\), \(x\in {\text {Int}}(\sigma )\), we have
$$\begin{aligned} {\mathcal {P}}^+_x= \Lambda _x \times Q. \end{aligned}$$
(3.1)
For \(\rho \in {\mathscr {P}}^{[n-1]}_{\partial }\), \(x\in {\text {Int}}(\rho )\), we have
$$\begin{aligned} {\mathcal {P}}^+_x=\Lambda _{\rho \sigma }\times Q \end{aligned}$$
(3.2)
where \(\Lambda _{\rho \sigma }\) is the monoid of integral tangent vectors contained in the tangent wedge \(T_{\rho }\sigma \) of \(\sigma \) along the face \(\rho \). If \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\) contains \(\rho \), parallel transport in the local system \({\mathcal {P}}\) from x to \(y\in {\text {Int}}(\sigma )\) induces the inclusion \({\mathcal {P}}^+_x\hookrightarrow {\mathcal {P}}^+_y\) given by \((\lambda ,q)\mapsto (\lambda ,q)\).
For \(\rho \in {\mathscr {P}}^{[n-1]}_{\mathrm {int}}\), \(x\in {\text {Int}}(\rho )\), we have
$$\begin{aligned} {\mathcal {P}}^+_x=(\Lambda _{\rho }\oplus {\mathbb {N}}Z_+\oplus {\mathbb {N}}Z_-\oplus Q)/ \langle Z_++Z_-=\kappa _{\rho }\rangle . \end{aligned}$$
(3.3)
This abstract description requires an ordering \(\sigma ,\sigma '\in {\mathscr {P}}_{{{\text {max}}}}\) of the maximal cells containing \(\rho \) and a choice of vector \(\xi \in \Lambda _x\) pointing into \(\sigma \) and representing a generator of \(\Lambda _{\sigma }/\Lambda _{\rho }\). Then for \(y\in {\text {Int}}(\sigma )\), \(y'\in {\text {Int}}(\sigma ')\), parallel transport in the local system \({\mathcal {P}}\) yields inclusions
$$\begin{aligned} \begin{aligned} {\mathfrak {t}}_{\rho \sigma }:{\mathcal {P}}_x^+\hookrightarrow {}&{\mathcal {P}}_y^+\\ (\lambda _{\rho },a Z_+,b Z_-,q)\mapsto {}&\big (\lambda _{\rho }+(a-b)\xi , q + b \kappa _{\rho }) \end{aligned} \end{aligned}$$
(3.4)
and
$$\begin{aligned} \begin{aligned} {\mathfrak {t}}_{\rho \sigma '}:{\mathcal {P}}_x^+\hookrightarrow {}&{\mathcal {P}}_{y'}^+\\ (\lambda _{\rho },a Z_+,b Z_-,q)\mapsto {}&\big (\lambda _{\rho }+(a-b)\xi , q + a \kappa _{\rho }) \end{aligned} \end{aligned}$$
(3.5)
respectively. See the discussion of [39, Sect. 2.2].
Given a choice of monoid ideal \(I\subseteq Q\), we also introduce the monoid ideal \(I_x\subseteq {\mathcal {P}}^+_x\) when \(x\in {\text {Int}}(\sigma )\), \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\) defined in the description (3.1) as
$$\begin{aligned} I_x:= \Lambda _x \times I. \end{aligned}$$
(3.6)
Notation 3.1
For \(x\in {\text {Int}}(\sigma )\), \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\), we will often write a monomial in \(\mathbbm {k}[{\mathcal {P}}^+_x]=\mathbbm {k}[Q][\Lambda _x]\) either as \(z^m\) for \(m\in {\mathcal {P}}^+_x\) or as \(t^qz^{{\bar{m}}}\) for \(({\bar{m}}, q)\in \Lambda _x\oplus Q\) via (3.1).
Similarly, if \(x\in {\text {Int}}(\rho )\) with \(\rho \in {\mathscr {P}}^{[n-1]}\), we have a canonically defined submonoid \(\Lambda _{\rho }\oplus Q\subseteq {\mathcal {P}}^+_x\) via (3.2) or (3.3), hence defining a subring \(\mathbbm {k}[Q][\Lambda _{\rho }]\subseteq \mathbbm {k}[{\mathcal {P}}^+_x]\). We again write monomials in this ring as \(t^qz^{{\bar{m}}}\) for \(q\in Q, {\bar{m}} \in \Lambda _{\rho }\).
We will frequently need to use parallel transport in \({\mathcal {P}}^+\) from a cell \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\) to a cell \(\sigma '\in {\mathscr {P}}_{{{\text {max}}}}\), either with \(\sigma =\sigma '\) or \(\sigma \cap \sigma '=\rho \in {\mathscr {P}}^{[n-1]}\). Take any \(x\in {\text {Int}}(\sigma )\), \(x'\in {\text {Int}}(\sigma ')\). If \(\sigma =\sigma '\), we may define
$$\begin{aligned} {\mathfrak {t}}_{\sigma ,\sigma '}:{\mathcal {P}}^+_x\rightarrow {\mathcal {P}}^+_{x'} \end{aligned}$$
to be given by parallel transport along a path contained in \({\text {Int}}(\sigma )\); under the representation (3.1), this map is the identity. On the other hand, if \(\sigma \cap \sigma '\in {\mathscr {P}}^{[n-1]}\), parallel transport along a path contained in \(\mathrm {Star}(\rho )\) gives a map
$$\begin{aligned} {\mathfrak {t}}_{\sigma ,\sigma '}:{\mathcal {P}}^+_x\rightarrow {\mathcal {P}}_{x'}. \end{aligned}$$
Noting that at the level of groups, \({\mathfrak {t}}_{\sigma ,\sigma '} ={\mathfrak {t}}_{\rho \sigma '}\circ {\mathfrak {t}}_{\rho \sigma }^{-1}\), it follows that if \(m\in {\mathcal {P}}^+_x\) and \({\bar{m}} \in T_\rho \sigma \), then \({\mathfrak {t}}_{\sigma ,\sigma '}(m)\in {\mathcal {P}}^+_{x'}\). Thus, either in this case or the case \(\sigma =\sigma '\), we may write \({\mathfrak {t}}_{\sigma ,\sigma '}(m)\in {\mathcal {P}}^+_{x'}\) whenever
$$\begin{aligned} m\in {\mathcal {P}}^+_x, \quad {\bar{m}} \in T_{\sigma \cap \sigma '}\sigma . \end{aligned}$$
If we let \(R'_I\) denote the subring of \((\mathbbm {k}[Q]/I)[\Lambda _{\sigma }]\) generated by monomials of the form \(t^q z^{{\bar{m}}}\) for \({\bar{m}}\in T_{\sigma \cap \sigma '}\sigma \), we obtain a ring homomorphism
$$\begin{aligned} {\mathfrak {t}}_{\sigma ,\sigma '}:R'_I \rightarrow (\mathbbm {k}[Q]/I)[\Lambda _{\sigma '}] \end{aligned}$$
(3.7)
The canonical wall structure
Recall of wall structures
We recall the notion of walls and wall structures from [39, Def. 2.11]:
Definition 3.2
A wall on \((B,{\mathscr {P}})\) is a codimension one rational polyhedral subset \({\mathfrak {p}}\not \subseteq \partial B\) of some \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\), along with an element
$$\begin{aligned} f_{{\mathfrak {p}}}=\sum _{m\in {\mathcal {P}}^+_x,{\bar{m}}\in \Lambda _{{\mathfrak {p}}}}c_m z^m\in \mathbbm {k}[{\mathcal {P}}^+_x], \end{aligned}$$
for \(x\in {\text {Int}}({\mathfrak {p}})\). Identifying \({\mathcal {P}}_y\) with \({\mathcal {P}}_x\) by parallel transport inside \(\sigma {\setminus }\Delta \), we require that \(m\in {\mathcal {P}}^+_y\) for all \(y\in {\mathfrak {p}}{\setminus }\Delta \) when \(c_m\not =0\). We further require that \(f_{{\mathfrak {p}}}\equiv 1\mod {\mathfrak {m}}\).
Definition 3.3
A wall structure \({\mathscr {S}}\) on \((B,{\mathscr {P}})\) is a finite set of walls.
Remark 3.4
The above definitions differ in a couple of ways from that of [39, Def. 2.11]. First, we are less permissive with wall functions, insisting that \(f_{{\mathfrak {p}}}\equiv 1\mod {\mathfrak {m}}\). Second, we are more permissive with the notion of wall structure. In [39], we insist walls form the codimension one cells of a rational polyhedral decomposition of B refining \({\mathscr {P}}\). This is imposed there to make it easier to describe gluing. However, a wall structure in the above more liberal sense is equivalent (in the sense of Definition 3.5 below) to one in the sense of [39, Def. 2.11], and we ignore this issue in this section and the next, only returning to the convention of [39] in Sect. 5.
Definition 3.5
For a wall structure \({\mathscr {S}}\), we define
$$\begin{aligned} |{\mathscr {S}}| := {}&\bigcup _{{\mathfrak {p}}\in {\mathscr {S}}} {\mathfrak {p}}\cup \bigcup _{\rho \in {\mathscr {P}}^{[n-1]}} \rho ,\\ {\text {Sing}}({\mathscr {S}}) := {}&\Delta \cup \bigcup _{{\mathfrak {p}}\in {\mathscr {S}}} \partial {\mathfrak {p}}\cup \bigcup _{{\mathfrak {p}},{\mathfrak {p}}'\in {\mathscr {S}}}({\mathfrak {p}}\cap {\mathfrak {p}}') \end{aligned}$$
where the last union is over all pairs of walls \({\mathfrak {p}},{\mathfrak {p}}'\) with \({\mathfrak {p}}\cap {\mathfrak {p}}'\) codimension at least two.
If \(x\in B{\setminus }{\text {Sing}}({\mathscr {S}})\), we define
$$\begin{aligned} f_x:= \prod _{x\in {\mathfrak {p}}\in {\mathscr {S}}} f_{{\mathfrak {p}}}. \end{aligned}$$
(3.8)
We say two wall structures are equivalent (modulo I) if \(f_x=f'_x\mod I\) for all \(x\in B{\setminus } ({\text {Sing}}({\mathscr {S}})\cup {\text {Sing}}({\mathscr {S}}'))\). Generally we omit mention of I if clear from context.
The construction
Definition 3.6
A wall type is a type \(\tau =(G,{\varvec{\sigma }},{{\mathbf {u}}})\) of tropical map to \(\Sigma (X)\) such that:
-
(1)
G is a genus zero graph with \(L(G)=\{L_{\mathrm {out}}\}\) with \({\varvec{\sigma }}(L_{\mathrm {out}})\in {\mathscr {P}}\) and \(u_{\tau }:=\mathbf{u}(L_{\mathrm {out}})\not =0\).
-
(2)
\(\tau \) is realizable and balanced.
-
(3)
Let \(h:\Gamma (G,\ell )\rightarrow \Sigma (X)\) be the corresponding universal family of tropical maps, and \(\tau _{\mathrm {out}}\in \Gamma (G,\ell )\) the cone corresponding to \(L_{\mathrm {out}}\). Then \(\dim \tau =n-2\) and \(\dim h(\tau _{\mathrm {out}})=n-1\). Further, \(h(\tau _{\mathrm {out}})\not \subseteq \partial B\).
A decorated wall type is a decorated type \({\varvec{\tau }}=(\tau ,\mathbf{A})\) with \(\tau \) a wall type.
Before using wall types to define the invariants we use in the canonical wall structure, we first make an observation in the relative case needed to show properness of the relevant moduli spaces. For most of the paper, we only work with the absolute moduli space \({\mathscr {M}}(X,\tau )\), but it turns out that in the relative case, we may also work with the relative moduli space \({\mathscr {M}}(X/S,\tau )\) when \(\tau \) is a wall type. To make this precise, we note that a realizable type \(\tau \) for a punctured map to X is a type for X/S if the universal tropical map over \(\tau \) to \(\Sigma (X)\) fits into a commutative diagram
Proposition 3.7
In the relative case, let \(\tau \) be a wall type for X, and let \(\beta \) be the class of punctured map determined by the data \(u_{\tau }\) and \(A\in Q{\setminus } I\) a non-zero curve class. Then \(\tau \) is a type of punctured map to X/S, and \({\mathscr {M}}(X,\beta )= {\mathscr {M}}(X/S,\beta )\).
Proof
We first show that the type \(\tau =(G,{\varvec{\sigma }},{\mathbf {u}})\) is in fact a type for X/S. Since G is connected, it is sufficient to show that for each \(E\in E(G)\cup L(G)\), \(\Sigma (g)_*(\mathbf{u}(E))=0\). Recall that as a wall type, \(\tau \) is realizable and balanced. Since \(\tau \) is realizable, we obtain a family of tropical maps \(h_s:G\rightarrow \Sigma (X)\), \(s\in {\text {Int}}(\tau )\). Composing with \(\Sigma (g)\) gives a tropical map \(\Sigma (g)\circ h_s:G\rightarrow \Sigma (S)={\mathbb {R}}_{\ge 0}\), which is balanced by Definition 2.2. But it is then immediate that this map must be constant, as any tropical map to \({\mathbb {R}}_{\ge 0}\) satisfying the balancing condition and with only one leg must be constant. Thus \(\tau \) is a type defined over S.
The equalities of moduli spaces now follow from [5, Prop. 5.11]. \(\square \)
Construction 3.8
Fix a wall type \(\tau \) and a non-zero curve class \(A\in Q{\setminus } I\). Let \(\beta \) be the class of punctured map determined by the data \(u_{\tau }\) and A. Then we obtain a reduced closed stratum \({\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\subseteq {\mathfrak {M}}({\mathcal {X}},\beta )\), and a moduli space \({\mathscr {M}}_{\tau }(X,\beta )\) along with a morphism
$$\begin{aligned} \varepsilon :{\mathscr {M}}_{\tau }(X,\beta )\rightarrow {\mathfrak {M}}_{\tau }({\mathcal {X}},\beta ). \end{aligned}$$
Lemma 3.9
\({\mathscr {M}}_{\tau }(X,\beta )\) is proper over \({\text {Spec}}\mathbbm {k}\) and carries a virtual fundamental class of dimension 0.
Proof
In the absolute case, \({\mathscr {M}}_{\tau }(X,\beta )\) is closed substack of \({\mathscr {M}}(X,\beta )\), which is proper over \({\text {Spec}}\mathbbm {k}\) by [5, Cor. 3.17].
We next consider the relative case. There is a morphism \({\mathcal {X}}\rightarrow {\mathcal {S}}= [{\mathbb {A}}^1/{\mathbb {G}}_m]\), induced by g, where \({\mathcal {S}}\) is the Artin fan of S. Write \({\mathcal {X}}_0:={\mathcal {X}}\times _{{\mathcal {S}}} [0/{\mathbb {G}}_m]\); this is a closed substack of \({\mathcal {X}}\). As \(\dim h(\tau _{\mathrm {out}})=n-1\) and \(h(\tau _{\mathrm {out}})\not \subseteq \partial B\), it follows that \(\Sigma (g):h(\tau _{\mathrm {out}}) \rightarrow \Sigma (S)={\mathbb {R}}_{\ge 0}\) is surjective by Proposition 1.15. By Proposition 3.7, \(\Sigma (g)\circ h_s\) is constant for each \(s\in \tau \), and hence for any vertex \(v\in V(G)\), \(\Sigma (g):h(\tau _v)\rightarrow {\mathbb {R}}_{\ge 0}\) is also surjective. From this it follows that any punctured map \(C^{\circ }\rightarrow {\mathcal {X}}\) in \({\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\) has image lying set-theoretically in \(|{\mathcal {X}}_0|\). Since \({\mathfrak {M}}_{\tau }({\mathcal {X}}, \beta )\) is reduced by construction, any map must thus factor through \({\mathcal {X}}_0\) log-scheme-theoretically, and hence any punctured map in \({\mathscr {M}}_{\tau }(X,\beta )\) must factor through \(X_0\) log-scheme-theoretically. Thus \({\mathscr {M}}_{\tau }(X,\beta )\) is a closed substack of \({\mathscr {M}}(X\times _S 0,\beta )\), which is again proper over \({\text {Spec}}\mathbbm {k}\).
We now calculate the virtual dimension. By [5, Prop. 3.28], if \({\bar{\tau }}\) is the global type induced by \(\tau \), then \({\mathfrak {M}}({\mathcal {X}}, {\bar{\tau }})\) is pure-dimensional and reduced of dimension
$$\begin{aligned} -3+|L(G)|-\dim \tau =-3+1-(n-2)=-n. \end{aligned}$$
The same is then true for \({\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\), as the forgetful map \({\mathfrak {M}}({\mathcal {X}},{\bar{\tau }})\rightarrow {\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\) is finite of generic degree \(|{\text {Aut}}({\bar{\tau }})|\). The virtual relative dimension of \({\mathscr {M}}_{\tau }(X,\beta )\) over \({\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\) at a punctured map \(f:C^{\circ }\rightarrow X\) over a geometric point is \(\chi (f^*\Theta _{X})=A\cdot c_1(\Theta _{X})+n\) by Riemann-Roch. Recall that \(c_1(\Theta _X)=-(K_X+D)\equiv _{{\mathbb {Q}}}-\sum _i a_i D_i\) by assumption. Further, for any generator \(D_i^*\) of \({\varvec{\sigma }}(L_{\mathrm {out}})\), it follows that \(a_i=0\) as \({\varvec{\sigma }}(L_{\mathrm {out}})\in {\mathscr {P}}\). Thus by [48, Cor. 1.14], \(A\cdot D_i=0\) whenever \(a_i\not =0\). Thus the total virtual dimension is 0 as claimed. \(\square \)
We now define
$$\begin{aligned} W_{\tau ,A}:=\deg [{\mathscr {M}}_{\tau }(X,\beta )]^{\mathrm {virt}}. \end{aligned}$$
(3.9)
In addition, \(h|_{\tau _{\mathrm {out}}}:\tau _{\mathrm {out}}\rightarrow \sigma \) induces a morphism
$$\begin{aligned} h_*: \Lambda _{\tau _{\mathrm {out}}}\rightarrow \Lambda _{\sigma }, \end{aligned}$$
and we define
$$\begin{aligned} k_{\tau }:=|{\text {coker}}(h_*)_{\mathrm {tors}}|= |\Lambda _{h(\tau _{\mathrm {out}})}/h_*(\Lambda _{\tau _{\mathrm {out}}})|. \end{aligned}$$
(3.10)
Finally, set
$$\begin{aligned} \boxed {{\mathfrak {p}}_{\tau ,A}:=\big (h(\tau _{\mathrm {out}}), \exp (k_{\tau } W_{\tau ,A} t^A z^{-u_{\tau }})\big ).} \end{aligned}$$
(3.11)
Here we view \(t^A z^{-u_{\tau }}\) as a monomial in \(\mathbbm {k}[{\mathcal {P}}^+_x]\) for \(x\in {\text {Int}}(h(\tau _{\mathrm {out}}))\) as in Notation 3.1. To view the exponential as a finite sum, note that \(\mathbbm {k}[Q][\Lambda _{h(\tau _{\mathrm {out}})}]\subseteq \mathbbm {k}[{\mathcal {P}}^+_x]\), and we may truncate the infinite sum by removing all monomials which are zero in \((\mathbbm {k}[Q]/I)[\Lambda _{h(\tau _{\mathrm {out}})}]\).
We then define:
Definition 3.10
$$\begin{aligned} {\mathscr {S}}_{{\mathrm {can}}}^{\mathrm {undec}}:=\{{\mathfrak {p}}_{\tau ,A}\,|\, \tau \hbox { an isomorphism class of wall type, }A\in Q{\setminus } I, W_{\tau ,A}\not =0\}. \end{aligned}$$
We note the superscript “\({\mathrm {undec}}\)” refers to the undecorated wall structure, in distinction with the decorated wall structure we will define in Construction 3.13, which will be equivalent to the above wall structure but which will be more useful in the proof of consistency.
Proposition 3.11
\({\mathscr {S}}^{\mathrm {undec}}_{{\mathrm {can}}}\) is a wall structure.
Proof
We need to verify that (1) \({\mathfrak {p}}_{\tau ,A}\) is always a wall and (2) \({\mathscr {S}}^{\mathrm {undec}}_{{\mathrm {can}}}\) is finite.
For the first item, fix \(\tau ,A\). Write \(u:=u_{\tau }\). It is obvious that \(h(\tau _{\mathrm {out}})\) is a rational polyhedral cone of codimension one by assumption. We need to check that the parallel transport of \((-u,A)\) to \({\mathcal {P}}_y\) lies in \({\mathcal {P}}^+_y\) for each point \(y\in {\mathfrak {p}}_{\tau ,A}{\setminus }\Delta \). Since \(\Delta \) is the union of all codimension two cones of \({\mathscr {P}}\), this is only an issue if \(y\in \rho \subseteq \sigma \) where \(\rho \) is codimension one and \({\mathfrak {p}}_{\tau ,A}\not \subseteq \rho \). In this case, \(\dim {\varvec{\sigma }}(L_{\mathrm {out}})=n\). Let \(v\in V(G)\) be the vertex adjacent to \(L_{\mathrm {out}}\), \(\tau _v\in \Gamma (G,\ell )\) the corresponding cone. We divide the analysis into three cases, depending on the relationship between y and \(h(\tau _v)\).
Case 1: \(y\not \in h(\tau _{v})\). From (2.1), necessarily \(-u\in T_{\rho }\sigma \). If \(y\in \partial B\), then it is immediate from (3.2) that \((-u,A)\in {\mathcal {P}}^+_y\). If instead \(y\not \in \partial B\), we may choose \(\xi \in \Lambda _y\) pointing into \(\sigma \) as in the description of \({\mathcal {P}}^+_y\) of (3.3), and then write \(-u=u'+a\xi \) for some \(a>0\) and \(u'\in \Lambda _{\rho }\). Thus by (3.4), \((-u,A)\) is identified with the element \((u',aZ_+,0,A)\) of \({\mathcal {P}}^+_y\).
Case 2: \(y \in h(\tau _v)\cap \partial B\). We are thus in the relative case. By Proposition 3.7, \(\beta \) must be defined over S, which in particular means that u is tangent to the fibres of \(g_{\mathrm {trop}}\), and hence u is tangent to \(\partial B\). However, by (3.2), it then follows that \((-u,A)\in {\mathcal {P}}^+_y\).
Case 3: \(y\in h(\tau _v){\setminus } \partial B\). Here we assume that \(W_{\tau ,A}\not =0\), (as otherwise such a wall would not be included in \({\mathscr {S}}^{\mathrm {undec}}_{{\mathrm {can}}}\)). Thus there is necessarily a punctured map \(f:C^{\circ }/W \rightarrow X\) over a geometric log point with type \(\tau '\) equipped with a contraction to \(\tau \). Thus we may mark f with \(\tau \), and this leads to a decorated type \({\varvec{\tau }}=(\tau ,\mathbf{A})\) with \(\mathbf{A}(v)\) given by the curve class of the map \(f_v:C^{\circ }_v\rightarrow X\), where \(C^{\circ }_v\) is the subcurve of \(C^{\circ }\) corresponding to \(v\in V(G)\), as in the proof of Lemma 2.1. In this case, u points into \(\sigma \), and in the notation \(\delta \) of Lemma 2.3, (2), \(\mathbf{A}(v)=d[X_{\rho }]\) with \(d \ge \delta (u)\). Thus the total curve class A satisfies \(A=\delta (u) [X_{\rho }]+A'\) for some \(A'\in Q\).
We may now use the description (3.3) along with (3.4) to test whether \((-u,A)\in {\mathcal {P}}^+_x\) lies in the image of \({\mathfrak {t}}_{\rho \sigma }:{\mathcal {P}}^+_y \rightarrow {\mathcal {P}}^+_x\). Choosing \(\xi \in \Lambda _y\) as before, we may now write \(u=u'+\delta (u)\xi \) for some \(u'\in \Lambda _{\rho }\). Thus \((-u,A)\) equals \({\mathfrak {t}}_{\rho \sigma }\) applied to
$$\begin{aligned} (-u',0,\delta (u) Z_-, A-\delta (u) \kappa _{\rho }) =(-u',0, \delta (u) Z_-, A')\in {\mathcal {P}}^+_y. \end{aligned}$$
We have now covered all possible cases for the location of y, and hence \({\mathfrak {p}}_{\tau ,A}\) is a wall.
To show \({\mathscr {S}}^{\mathrm {undec}}_{{\mathrm {can}}}\) is finite, we first observe that as \(Q{\setminus } I\) is finite by assumption on Q and I, there are only a finite number of choices for A. For determining \(\tau \), there are only a finite number of possibilities for \({\varvec{\sigma }}(L_{\mathrm {out}})\in {\mathscr {P}}\). Given a choice of A and wall type \(\tau \) with given \({\varvec{\sigma }}(L_{\mathrm {out}})\), \(W_{\tau ,A}\not =0\) implies that \({\mathscr {M}}_{\tau }(X,\beta )\) is non-empty, and then [48, Cor. 1.14] shows \(u_{\tau }\) is determined by the choice of A and \({\varvec{\sigma }}(L_{\mathrm {out}})\). If \(\beta \) is the punctured curve class determined by A and a given \(u_{\tau }\), then, since \({\mathscr {M}}(X,\beta )\) is finite type, there are only a finite number of types of tropical maps \(\tau \) appearing in the tropicalizations of curves in \({\mathscr {M}}(X,\beta )\), i.e., there are only a finite number of \(\tau \) such that \({\mathscr {M}}_{\tau }(X,\beta )\) is non-empty. \(\square \)
The main result of the paper, to be proved in Sect. 5, can then be stated:
Theorem 3.12
\({\mathscr {S}}^{\mathrm {undec}}_{{\mathrm {can}}}\) is a consistent wall structure in the sense of [39, Def. 3.9].
The above definition of the canonical wall structure is conceptually the simplest and most useful in practice (see [9] for some explicit examples). However, for the proofs of this paper, it is convenient to replace \({\mathscr {S}}^{\mathrm {undec}}_{{\mathrm {can}}}\) with an equivalent wall structure \({\mathscr {S}}_{{\mathrm {can}}}\) using decorated types as follows.
Construction 3.13
Fix a decorated wall type \({\varvec{\tau }}=(\tau ,\mathbf{A})\), and \(A=\sum _{v\in V(G)} \mathbf{A}(v)\) the total curve class. As \(\tau \) is realizable, we may view it equivalently as a global type. Hence we obtain a morphism of moduli spaces \(\varepsilon :{\mathscr {M}}(X,{\varvec{\tau }}) \rightarrow {\mathfrak {M}}({\mathcal {X}},{\varvec{\tau }})\). From the proof of Lemma 3.9, one sees that \({\mathfrak {M}}({\mathcal {X}},{\varvec{\tau }})\) is pure-dimensional and \([{\mathscr {M}}(X,{\varvec{\tau }})]^{\mathrm {virt}}\) is a zero dimensional cycle. Hence we may define
$$\begin{aligned} W_{{\varvec{\tau }}}:={\deg [{\mathscr {M}}(X,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}({\varvec{\tau }})|}. \end{aligned}$$
We may then define a wall
$$\begin{aligned} \boxed {{\mathfrak {p}}_{{\varvec{\tau }}}:= \big (h(\tau _{\mathrm {out}}), \exp (k_{\tau } W_{{\varvec{\tau }}}t^A z^{-u_{\tau }})\big )} \end{aligned}$$
(3.12)
and
$$\begin{aligned} {\mathscr {S}}_{{\mathrm {can}}}:=\left\{ {\mathfrak {p}}_{{\varvec{\tau }}}\,\Big |\, \begin{array}{c} {\varvec{\tau }}\hbox { an isomorphism class of decorated wall}\\ \hbox {type with total curve class lying in } Q{\setminus } I \end{array}\right\} . \end{aligned}$$
(3.13)
We note here we do not exclude walls with \(W_{{\varvec{\tau }}}=0\), so this wall structure may include an infinite set of trivial walls, i.e., with attached function 1. So technically this is not a wall structure, but if we remove all trivial walls, it becomes a finite set as in Proposition 3.11. However, for bookkeeping purposes, it will prove useful to include these trivial walls.
To see that this gives a wall structure equivalent (in the sense of Definition 3.5) to the previous definition, fix a wall type \(\tau \) and a curve class \(A\in Q{\setminus } I\) such that \(W_{\tau ,A}\not =0\); this data will give one wall in the earlier definition of \({\mathscr {S}}_{{\mathrm {can}}}\). With \(\beta =(\{u_{\tau }\},A)\) the associated class of punctured map, we have the canonical morphism \({\mathfrak {M}}({\mathcal {X}},\tau )\rightarrow {\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\), which is surjective and finite of degree \(|{\text {Aut}}(\tau )|\). Further, we have a Cartesian diagram in all categories [5, Prop. 5.19]
Here \(\mathbf{A}\) runs over all decorations of \(\tau \) with total curve class A. As j is a map of degree \(|{\text {Aut}}(\tau )|\) onto \({\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )\), we see that
$$\begin{aligned} W_{\tau ,A}&= {\deg \varepsilon ^![{\mathfrak {M}}_{\tau }({\mathcal {X}},\beta )]} = {\deg \varepsilon ^!j_*[{\mathfrak {M}}({\mathcal {X}},\tau )]\over |{\text {Aut}}(\tau )|}\\ \nonumber&= {\deg j'_*\varepsilon _{\tau }^![{\mathfrak {M}}({\mathcal {X}},\tau )]\over |{\text {Aut}}(\tau )|} = \sum _{{\varvec{\tau }}=(\tau ,\mathbf{A})} {\deg [{\mathscr {M}}(X,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}(\tau )|}. \end{aligned}$$
(3.14)
Here the third equality holds by push-pull of [65, Thm. 4.1], and the last summation runs over all choices of decorations \(\mathbf{A}\) of \(\tau \) with total curve class A. Note however that \({\text {Aut}}(\tau )\) acts on the set of all choices of decoration \(\mathbf{A}\), with orbits of this action giving isomorphism classes of decorated types \({\varvec{\tau }}\). The stabilizer of a given \({\varvec{\tau }}\) is the subgroup \({\text {Aut}}({\varvec{\tau }}) \subseteq {\text {Aut}}(\tau )\), and hence the orbit containing \({\varvec{\tau }}\) is of size \(|{\text {Aut}}(\tau )|/|{\text {Aut}}({\varvec{\tau }})|\). Thus the final expression of (3.14) can be now expressed as a sum over isomorphism classes of decorations \({\varvec{\tau }}\) of \(\tau \) with total class A:
$$\begin{aligned} W_{\tau ,A} =\sum _{{\varvec{\tau }}}{\deg [{\mathscr {M}}(X,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}(\tau )|} \cdot {|{\text {Aut}}(\tau )|\over |{\text {Aut}}({\varvec{\tau }})|} =\sum _{{\varvec{\tau }}}{\deg [{\mathscr {M}}(X,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}({\varvec{\tau }})|}. \end{aligned}$$
(3.15)
From this, we see the collection of walls in the new definition of \({\mathscr {S}}_{{\mathrm {can}}}\) coming from all \({\varvec{\tau }}\) with underlying type \(\tau \) and total curve class A is equivalent to the corresponding single wall in the original definition of \({\mathscr {S}}_{{\mathrm {can}}}\).
Example 3.14
The construction of \({\mathscr {S}}_{{\mathrm {can}}}\) agrees (up to equivalence) with the construction of the canonical scattering diagram of [37] when \(\dim X=2\) and \(K_X+D=0\), so that \(B=\Sigma (X)\). In other words, the construction given here is a generalization of the construction of [37].
To see this, we analyze decorated wall types \({\varvec{\tau }}\) in dimension two. Since we require \(\dim \tau =\dim X-2\), in fact such types will be rigid. Thus there is a unique tropical map \(h:G\rightarrow B\) of type \(\tau \), and all vertices of G are mapped to \(0\in B\); otherwise, rescaling B provides a non-trivial deformation of h. Further, we require \(\dim h(\tau _{\mathrm {out}})=1\). So \(h(\tau _{\mathrm {out}})\) is a ray in B with endpoint 0. Hence there is no choice but for \(u_{\tau }\in B({\mathbb {Z}}){\setminus } \{0\}\) and \(h(\tau _{\mathrm {out}})={\mathbb {R}}_{\ge 0} u_{\tau }\). Further, any edge of G contracted by h has length a free parameter of the tropical curve, and hence again by rigidity, there are no edges. Thus the only possibility for \({\varvec{\tau }}\) is that G has one vertex, with an attached curve class A, and one leg, \(L_{\mathrm {out}}\), with \(\mathbf{u}(L_{\mathrm {out}})=u_{\tau }\in B({\mathbb {Z}})\). Note further that \(k_{\tau }\) is then just the index of \(u_{\tau }\), i.e., the degree of divisibility of \(u_{\tau }\) in \(B({\mathbb {Z}})\). Of course \(|{\text {Aut}}({\varvec{\tau }})|=1\).
To make a comparison with the setup of [37], it is then convenient, for \(u \in B({\mathbb {Z}})\) primitive, to define
$$\begin{aligned} W_{A,u,k}:=W_{{\varvec{\tau }}}, \end{aligned}$$
where \({\varvec{\tau }}\) is the decorated wall type as described above with curve class A and \(u_{\tau }=ku\). In [37], only one wall for each ray \({\mathbb {R}}_{\ge 0}u\) of rational slope occurs, and hence it is then more useful to write the canonical wall structure in the equivalent form
$$\begin{aligned} {\mathscr {S}}'_{{\mathrm {can}}}:=\bigg \{\Big ({\mathbb {R}}_{\ge 0}u, \exp \Big (\sum _{A,k} k W_{A,u,k}t^Az^{-ku}\Big )\Big )\,\Big |\,u\in B({\mathbb {Z}}){\setminus } \{0\}\hbox { primitive} \bigg \}. \end{aligned}$$
Here the sum is over all positive integers k and all curve classes A. However, if we write \(u=aD_i^*+bD_j^*\) for some non-negative a, b with \(a+b>0\), then by the balancing condition as expressed in [48, Cor. 1.14], we may in fact sum over only those curve classes with \(A\cdot D_i=a\), \(A\cdot D_j=b\), \(A\cdot D_k=0\) for \(k\not =i,j\).
In [37], a wall function is associated to each ray \({\mathfrak {d}}={\mathbb {R}}_{\ge 0} u\) of rational slope using invariants defined as follows. Assuming \({\mathfrak {d}}\) does not coincide with a ray of \({\mathscr {P}}\), one performs a toric (log étale) blow-up \(\pi :{\widetilde{X}}\rightarrow X\) by refining \((B,{\mathscr {P}})\) along the given ray \({\mathfrak {d}}\). Otherwise, if \({\mathfrak {d}}\) coincides with a ray of \({\mathscr {P}}\), we may take \(\pi \) to be the identity. Let \({\widetilde{A}}\) be a curve class on \({\widetilde{X}}\) with the following properties, depending on whether or not \(\pi \) is the identity. If \(\pi \) is not the identity, then we require that \({\widetilde{A}}\) has trivial intersection number with all boundary components of \({\widetilde{X}}\) except for the exceptional divisor E of \(\pi \). If \(\pi \) is the identity, let E be the component of D corresponding to the ray \({\mathfrak {d}}\). We then require that \({\widetilde{A}}\) has zero intersection number with each component of D except for E. Then [37], following [40, Sect. 4], defines a number \(N_{{\widetilde{A}}}\). This number is defined as a relative Gromov–Witten invariant for the non-compact pair \((\widetilde{X}^{\circ },E^{\circ })\) where \({\widetilde{X}}^{\circ }\subseteq {\widetilde{X}}\) is obtained by removing the closure of \(\pi ^{-1}(D){\setminus } E\) from \({\widetilde{X}}\), and \(E^{\circ }=E\cap {\widetilde{X}}^{\circ }\). This relative invariant counts rational curves of class \({\widetilde{A}}\) with one marked point, with contact order \(k_{{\widetilde{A}}}:={\widetilde{A}}\cdot E\) with \(E^{\circ }\).
To compare \(N_{{\widetilde{A}}}\) with the type of number considered in this paper, note that the data of \({\widetilde{A}}\) and the contact order \(k_{{\widetilde{A}}}\) also specifies a type \({\widetilde{\beta }}\) of logarithmic map to the pair \(({\widetilde{X}},\pi ^{-1}(D))\), and we may compare the logarithmic Gromov–Witten invariant \(\deg [{\mathscr {M}}({\widetilde{X}},{\widetilde{\beta }})]^{\mathrm {virt}}\) with \(N_{\widetilde{A}}\). In fact, it follows from [7] that these two numbers will agree provided that every stable log map in the moduli space \({\mathscr {M}}({\widetilde{X}}, {\widetilde{\beta }})\) factors through \(\widetilde{X}^{\circ }\). However, it is an elementary exercise in tropical geometry to show that if given a stable log map \(f:C/W\rightarrow {\widetilde{X}}\) of type \({\widetilde{\beta }}\) with W a log point, any tropical map in the corresponding family of tropical maps has image \({\mathfrak {d}}\). This may be proved using an argument similar to the argument given in Lemma 2.5, but a similar tropical argument in two dimensions has already appeared in the proof of [17, Lem. 12], which may also be viewed as a tropical interpretation of [40, Prop. 4.2]. On the other hand, any stable log map of type \({\widetilde{\beta }}\) which does not factor through \({\widetilde{X}}^{\circ }\) will necessarily have a tropicalization whose image does not coincide with \({\mathfrak {d}}\), as follows immediately from the construction of tropicalization.
Finally, let \(A=\pi _*{\widetilde{A}}\). Note that in the case \(\pi \) is not the identity, the curve class \({\widetilde{A}}\) is uniquely determined by A and the constraint that \({\widetilde{A}}\) has zero intersection number with components of the strict transform of D. Then if u is the primitive generator of \({\mathfrak {d}}\cap B({\mathbb {Z}})\), we may interpret \(k_{{\widetilde{A}}}u\in B({\mathbb {Z}})\) as a contact order, and the class A and the contact order \(k_{{\widetilde{A}}}u\) determines a type of log map \(\beta \) to X. It then follows from the birational invariance of log Gromov–Witten theory of [8, Thm. 1.1.1] that \(\deg [{\mathscr {M}}(\widetilde{X},{\widetilde{\beta }})]^{\mathrm {virt}}=\deg [{\mathscr {M}}(X, \beta )]^{\mathrm {virt}}\). In particular, \(N_{{\widetilde{A}}}=W_{A,u,k_{{\widetilde{A}}}}\).
Conversely, given the data of A, u and k giving a wall type \(\tau \) with \({\mathscr {M}}(X,\tau )\) non-empty, we obtain via the above blow-up procedure a curve class \({\widetilde{A}}\). Thus we have a one-to-one correspondence between the set of data contributing to \({\mathscr {S}}'_{{\mathrm {can}}}\) and the data contributing to the canonical scattering diagram of [37]. Further, we have just observed an equality of the invariants. The equivalence of \({\mathscr {S}}'_{{\mathrm {can}}}\) with the canonical scattering diagram of [37] now follows by inspection of the definition of the canonical scattering diagram in [37].
Example 3.15
Returning to Examples 1.4 and 1.11, we first introduce notation for the relevant curve classes. Let \(e_i\) be the class of a fibre of \(\pi |_{E_i}:E_i \rightarrow Z_i\). Let f be the class of a curve of the form \(\{a\}\times {\mathbb {P}}^1 \times \{b\}\) disjoint from the centers \(Z_1,Z_2\).
One can show that the only decorated wall types \({\varvec{\tau }}\) with \(W_{{\varvec{\tau }}}\not =0\) are of the following 5 types. In each case, the underlying graph G of \({\varvec{\tau }}\) has one vertex v, no edges, and of course a unique leg \(L_{\mathrm {out}}\).
-
(1)
For some \(k>0\), we have \(u_{\tau }=k D_{2,0}^*\), \({\varvec{\sigma }}(L_{\mathrm {out}})= {\mathfrak {p}}_1:={\mathbb {R}}_{\ge 0} D_{2,0}^*+{\mathbb {R}}_{\ge 0} D_{1,0}^*\), and \(\mathbf{A}(v)= k e_1\).
-
(2)
For some \(k>0\), we have \(u_{\tau }=k D_{2,0}^* \), \({\varvec{\sigma }}(L_{\mathrm {out}})= {\mathfrak {p}}_2:={\mathbb {R}}_{\ge 0} D_{2,0}^*+{\mathbb {R}}_{\ge 0} D_{1,\infty }^*\), and \(\mathbf{A}(v)= k e_1\).
-
(3)
For some \(k>0\), we have \(u_{\tau }=k(E_2^*-D_{1,\infty }^*)\), \({\varvec{\sigma }}(L_{\mathrm {out}})= {\mathfrak {p}}_3:={\mathbb {R}}_{\ge } E_2^*+{\mathbb {R}}_{\ge 0} D_{1,\infty }^*\), and \(\mathbf{A}(v)= k(f-e_1-e_2)\).
-
(4)
For some \(k>0\), we have \(u_{\tau }=k D_{2,\infty }^*\), \({\varvec{\sigma }}(L_{\mathrm {out}})={\mathfrak {p}}_4:={\mathbb {R}}_{\ge 0} E_2^*+{\mathbb {R}}_{\ge 0}D_{2,\infty }^*\), and \(\mathbf{A}(v)= k(f-e_1)\).
-
(5)
For some \(k>0\), we have \(u_{\tau }=k D_{2,\infty }^*\), \({\varvec{\sigma }}(L_{\mathrm {out}})={\mathfrak {p}}_5:={\mathbb {R}}_{\ge 0} D_{1,0}^*+{\mathbb {R}}_{\ge 0}D_{2,\infty }^*\), and \(\mathbf{A}(v)= k(f-e_1)\).
In all cases, \(h(\tau _{\mathrm {out}})={\varvec{\sigma }}(L_{\mathrm {out}})\), so walls only have five possible supports. Further, in every case, \(k_{\tau }=k\) and \(W_{{\varvec{\tau }}}=(-1)^{k-1}/k^2\). After passing to an equivalent scattering diagram with only one wall with a given support, we see, for example, that we have a wall
$$\begin{aligned} \left( {\mathfrak {p}}_1, \exp \Big (\sum _{k>0} k {(-1)^{k-1}\over k^2} t^{k e_1} z^{-kD_{2,0}^*}\Big )\right) =\left( {\mathfrak {p}}_1, 1+t^{e_1}z^{-D_{2,0}^*}\right) . \end{aligned}$$
Similarly, we have four other walls:
$$\begin{aligned}&~~\left( {\mathfrak {p}}_2,1+t^{e_1} z^{-D_{2,0}^*}\right) , \left( {\mathfrak {p}}_3,1+t^{f-e_1-e_2} z^{D_{1,\infty }^*-E_2^*}\right) ,\\&\left( {\mathfrak {p}}_4,1+t^{f-e_1} z^{-D_{2,\infty }^*}\right) , \left( {\mathfrak {p}}_5,1+t^{f-e_1} z^{-D_{2,\infty }^*}\right) . \end{aligned}$$
Together, these walls cover the affine plane contained in B which is the union of all two-dimensional cones of \({\mathscr {P}}\) not containing \(D_{3,0}^*\) or \(D_{3,\infty }^*\). We omit a derivation of these results. Showing (1)–(5) are the only possibilities is not difficult using [48, Cor. 1.14] to encode balancing requirements for punctured maps. A direct calculation of \(W_{{\varvec{\tau }}}\) has not been carried out, but presumably these multiple cover calculations can be carried out as in the two-dimensional case of [40, Prop. 5.2]. Instead, the above formulas for the wall functions are proved in a more general context in [9]. In fact, the formulas for these wall functions follows from the consistency of \({\mathscr {S}}_{{\mathrm {can}}}\) proved in the present paper.
The walls \({\mathfrak {p}}_i\) for \(i\not =3\) arise from zero-dimensional strata in one-dimensional moduli spaces of ordinary (non-punctured) stable log maps. For example, the inclusion of every fibre of \(\pi |_{E_1}:E_1\rightarrow Z_1\) into X may be viewed as a stable log map. The walls \({\mathfrak {p}}_1\) and \({\mathfrak {p}}_2\) capture the curves in this family which are degenerate with respect to the log structure on X, i.e., fall into \(D_{1,0}\) or \(D_{1,\infty }\). For \(k>1\), we count multiple covers of these fibres.
The same holds for \({\mathfrak {p}}_4\) and \({\mathfrak {p}}_5\), with the curves in question being, in general, strict transforms of curves of the form \(\{a\}\times {\mathbb {P}}^1 \times \{1\}\subseteq \overline{X}\). Note as these curves intersect \(Z_1\) at one point, the class of the strict transform is indeed \(f-e_1\). However, as \(a\rightarrow \infty \), this curve degenerates to a union \(C=C_1\cup C_2\) of two irreducible components, of class \(f-e_1-e_2\) (the strict transform \(C_1\) of \(\{\infty \}\times {\mathbb {P}}^1\times \{1\}\)) and of class \(e_2\) (the curve \(C_2=\pi ^{-1}(\infty ,\infty ,1)\)). It is this degenerate curve and its multiple covers which contribute to \({\mathfrak {p}}_4\). Meanwhile the strict transform of \(\{0\}\times {\mathbb {P}}^1\times \{1\}\) and its multiple covers contribute to the wall \({\mathfrak {p}}_5\).
Finally, \({\mathfrak {p}}_3\) arises not from a family of curves with one marked point of positive contact order with the boundary, but from the punctured log map whose image is \(C_1\). This involves a negative contact order with the divisor \(D_{1,\infty }\), which contains \(C_1\). This curve is rigid, and if we did not include a wall for this curve, we would not get a consistent scattering diagram. This shows how it is essential, unlike in the case that \(\dim X=2\), to take into account punctured curves rather than just marked curves, as without \({\mathfrak {p}}_3\), \({\mathscr {S}}_{{\mathrm {can}}}\) would not be consistent.
Construction 3.16
(The torus action) As with the canonical wall structure in two dimensions in [37, Sect. 5], there is a natural torus action on the mirror family constructed using \({\mathscr {S}}_{{\mathrm {can}}}\) in all dimensions. We refer the reader to [39, Sect. 4.4] for the general setup for torus actions in the context of families of varieties built via wall structures, and do not review the notation here.
In this case, there is an action of a torus with character lattice
$$\begin{aligned} \Gamma :={\text {Div}}_D(X), \end{aligned}$$
i.e., the free abelian group generated by boundary divisors. In the language of [39, Sect. 4.4], the base ring A is taken here to be the ground field \(\mathbbm {k}\). To specify the torus action, we must specify the maps \(\delta _Q\) and \(\delta _B\) of the diagram [39, (4.8)]. Note that each irreducible component \(D_i\) of D defines an \({\mathbb {R}}\)-valued PL function on B; indeed, by construction \(B\subseteq {\text {Div}}_D(X)^*_{{\mathbb {R}}}\), and the linear functional \(D_i\) on this vector space restricts to a piecewise linear function on B. We continue to denote this PL function as \(D_i\). We may then define
$$\begin{aligned} \delta _Q:Q\longrightarrow \Gamma ,\quad \quad \delta _Q(A)= \sum _{i} (D_i\cdot A) D_i \end{aligned}$$
for \(A\in Q\), and
$$\begin{aligned} \delta _B:{\text {PL}}(B)^* \longrightarrow \Gamma ,\quad \quad \delta _B(\beta )= \sum _{i}\beta (D_i) D_i \end{aligned}$$
for \(\beta \in {\text {PL}}(B)^*\). Commutativity of [39, (4.8)] then follows by direct computation. Indeed, note that in our current context, \(Q_0\) is the free monoid with generators \(e_{\rho }\), \(e_{\rho }\in {\mathscr {P}}^{[n-1]}\), and \(h:Q_0\rightarrow Q\) is defined by \(h(e_{\rho })=[X_{\rho }]\), the kink of our choice of MPL function \(\varphi \) along \(\rho \). Then
$$\begin{aligned} \delta _Q(h(e_{\rho }))= \sum _i (D_i\cdot X_{\rho }) D_i. \end{aligned}$$
On the other hand, \(g:Q_0 \rightarrow {\text {PL}}(B)^*\) takes \(e_{\rho }\) to the linear functional on \({\text {PL}}(B)\) taking \(\psi \in {\text {PL}}(B)\) to the kink of \(\psi \) along \(\rho \). It is easy to check from toric geometry that the kink of the PL function induced by \(D_i\) along \(\rho \) is \(D_i \cdot X_{\rho }\) (noting that if \(D_i\) is not good, it induces the zero function on B). The claimed equality \(\delta _Q\circ h = \delta _B\circ g\) then follows.
Thus, by [39, Thm. 4.17], it follows there is an induced \({\text {Spec}}\mathbbm {k}[\Gamma ]\)-action on the flat family \(\check{\mathfrak {X}}\rightarrow {\text {Spec}}(\mathbbm {k}[Q]/I)\) constructed from the wall structure \({\mathscr {S}}_{{\mathrm {can}}}\) provided that \({\mathscr {S}}_{{\mathrm {can}}}\) is homogeneous in the sense of [39, Def. 4.16]. For this, it is enough to check that if \({\varvec{\tau }}\) is a decorated wall type with \(W_{{\varvec{\tau }}}\not =0\) and total curve class A, then \(\deg _{\Gamma } t^Az^{-u_{\tau }}=0\). Here \(\deg _{\Gamma } t^A = \delta _Q(A)=\sum _i (D_i\cdot A) D_i\). If we write \(u_{\tau }=\sum u_i D_i^*\) as a tangent vector to \({\varvec{\sigma }}(L_{\mathrm {out}})\) then \(\deg _{\Gamma }(z^{-u_{\tau }})\) is computed from [39, (4.9)] as \(-\sum _i u_i D_i\). It then follows from [48, Cor. 1.14] that necessarily \(u_i=D_i\cdot A\), and hence \(\deg _{\Gamma }t^Az^{-u_{\tau }}=0\) as desired.
The relative case
Construction 3.17
Suppose we are in the situation of Proposition 1.19. It is useful (see [9]) to compare the wall structures for (X, D) and \((X_s,D_s)\), which we do as follows.
Recall from that proposition that after extending the affine structure on B across the interior of cells \(\omega \in {\mathscr {P}}^{[n-2]}_{\partial }\), \(\partial B\) may be viewed as the polyhedral affine pseudomanifold corresponding to the pair \((X_s,D_s)\). As such, write \((\partial B)_0\) for the complement in \(\partial B\) of the union of cones of \({\mathscr {P}}^{[n-3]}_{\partial }\). Let \(\Lambda _{\partial B}\) denote the sheaf on \((\partial B)_0\) of integral tangent vectors. We may view \(\Lambda _{\partial B}\) as a subsheaf of \(\Lambda |_{\partial B}\), where the latter sheaf is now extended across the interiors of cells \(\omega \in {\mathscr {P}}^{[n-2]}_{\partial }\).
Further, Proposition 1.19, (3) then implies that if \({\mathcal {P}}_{\partial B}\) is defined using the MPL function \(\varphi |_{\partial B}\) on \(\partial B\), we obtain a natural inclusion \({\mathcal {P}}_{\partial B} \subseteq {\mathcal {P}}|_{\partial B}\). Again, the latter sheaf is viewed as extending across the interiors of \(\omega \in {\mathscr {P}}^{[n-2]}_{\partial }\). Note that \({\mathcal {P}}_{\partial B}\) consists of those sections m of \({\mathcal {P}}|_{\partial B}\) such that \({\bar{m}}\) is tangent to \(\partial B\).
On the other hand, we also have the MPL function \(\varphi _{(X_s,D_s)}\) taking values in \(H_2(X_s)\otimes _{{\mathbb {Z}}}{\mathbb {R}}\). This leads to a sheaf \({\mathcal {P}}_{(X_s,D_s)}\) on \((\partial B)_0\), and the map \(\iota :H_2(X_s)\rightarrow H_2(X)\) then induces a map
$$\begin{aligned} \iota _*:{\mathcal {P}}_{(X_s,D_s)} \rightarrow {\mathcal {P}}_{\partial B}. \end{aligned}$$
This allows us to define a map, for \(x\in (\partial B)_0\),
$$\begin{aligned} \iota _*:\mathbbm {k}[{\mathcal {P}}_{(X_s,D_s),x}^+]\rightarrow \mathbbm {k}[{\mathcal {P}}_{\partial B,x}^+], \end{aligned}$$
where \({\mathcal {P}}_{\partial B,x}^+:={\mathcal {P}}_{\partial B,x}\cap {\mathcal {P}}^+_x\). Finally, we may define
$$\begin{aligned} \iota ({\mathscr {S}}_{{\mathrm {can}},s}):= \big \{({\mathfrak {p}},\iota _*(f_{{\mathfrak {p}}}))\,|\, ({\mathfrak {p}},f_{{\mathfrak {p}}})\in {\mathscr {S}}_{(X_s,D_s)}\big \}. \end{aligned}$$
Like \({\mathscr {S}}_{{\mathrm {can}},s}\), this is a wall structure on \(\partial B\).
On the other hand, we may define the asymptotic wall structure of the wall structure \({\mathscr {S}}_{{\mathrm {can}}}\) on B by
$$\begin{aligned} {\mathscr {S}}^{\mathrm {as}}_{{\mathrm {can}}}:= \{({\mathfrak {p}}\cap \partial B,f_{{\mathfrak {p}}})\,|\, ({\mathfrak {p}}, f_{{\mathfrak {p}}})\in {\mathscr {S}}\hbox { with }\dim {\mathfrak {p}}\cap \partial B=n-2\}. \end{aligned}$$
(3.16)
We note that for each such wall, \(f_{{\mathfrak {p}}}\in \mathbbm {k}[{\mathcal {P}}^+_{\partial B,x}] \subseteq \mathbbm {k}[{\mathcal {P}}^+_x]\). Indeed, this follows from Proposition 3.7, which implies that for a wall type \(\tau \) for X, \(u_{\tau }\) is tangent to fibres of \(g_{\mathrm {trop}}\), and in particular tangent to \(\partial B\). Hence \({\mathscr {S}}^{\mathrm {as}}_{{\mathrm {can}}}\) may be viewed as a wall structure on \(\partial B\).
We then have:
Proposition 3.18
Suppose that all good divisors contained in \(g^{-1}(0)\) have multiplicity one in \(g^{-1}(0)\). Then the wall structures \(\iota ({\mathscr {S}}_{{\mathrm {can}},s})\) and \({\mathscr {S}}^{\mathrm {as}}_{{\mathrm {can}}}\) are equivalent.
Proof
Note \(\iota ({\mathscr {S}}_{{\mathrm {can}},s})\) is equal to the canonical wall structure for \((X_s,D_s)\) constructed using the curve class group \(H_2(X)\) rather than \(H_2(X_s)\). In the proof, we generally use \({\varvec{\tau }}'\) to denote a decorated wall type for (X, D) and \({\varvec{\tau }}\) for a decorated wall type for \((X_s,D_s)\), with curve class decoration \(\mathbf{A}\) taking values in \(H_2(X)\).
Given a decorated wall type \({\varvec{\tau }}'\), the corresponding cone \(\tau '\) comes with a structure map \(p:\tau '\rightarrow {\mathbb {R}}_{\ge 0}\) coming from the fact the type is defined over S. Note p is surjective as \(u_{\tau }\) is tangent to fibres of \(g_{\mathrm {trop}}\) by Proposition 3.7 and \(h_{\tau '}(\tau '_{\mathrm {out}})\not \subseteq \partial B\) by Definition 3.6, (3). Then \(p^{-1}(0)\) is a proper face of \(\tau '\), and hence corresponds to a contraction morphism \(\phi :{\varvec{\tau }}'\rightarrow {\varvec{\tau }}\) of decorated types. The type \(\tau \) is realizable provided that the length of \(L_{\mathrm {out}} \in L(G')\) isn’t zero on the face \(p^{-1}(0)\). Thus \(\tau \) realizable and \(\dim \tau =n-3\) is equivalent to \(\dim {\mathfrak {p}}_{{\varvec{\tau }}'}\cap \partial B = n-2\). From this it is immediate that \(\tau \) is a wall type for \((X_s,D_s)\) if and only if \(\dim {\mathfrak {p}}_{{\varvec{\tau }}'}\cap \partial B = n-2\).
Consider the sets
$$\begin{aligned} S':= {}&\{{\varvec{\tau }}'\,|\,{\varvec{\tau }}'\hbox { is a decorated wall type for }X \hbox { with }\dim {\mathfrak {p}}_{{\varvec{\tau }}'}\cap \partial B=n-2\},\\ S:= {}&\{{\varvec{\tau }}\,|\,{\varvec{\tau }}\hbox { is a decorated wall type for }X_s\}. \end{aligned}$$
The discussion of the previous paragraph has constructed a map \(\Phi :S'\rightarrow S\). To show the desired equivalence of wall structures, it is enough to show that for \({\varvec{\tau }}\in S\) we have
$$\begin{aligned} k_{\tau } W_{{\varvec{\tau }}} = \sum _{{\varvec{\tau }}'\in \Phi ^{-1}({\varvec{\tau }})} k_{\tau '} W_{{\varvec{\tau }}'}. \end{aligned}$$
(3.17)
To do so, we use the decomposition setup of [5, Thm. 5.21, Def. 5.22, Thm. 5.23]. Note that for \(0\not =s\in S\), s is a trivial log point, so in fact \({\mathscr {M}}(X_s,{\varvec{\tau }})={\mathscr {M}}(X_s/s,{\varvec{\tau }})\). Thus in particular we have
$$\begin{aligned} W_{{\varvec{\tau }}}= {\deg [{\mathscr {M}}(X_s/s,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}({\varvec{\tau }})|} = {\deg [{\mathscr {M}}(X_0/0,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}({\varvec{\tau }})|}, \end{aligned}$$
the first equality by definition and the second equality by [5, Thm. 5.23, (1)]. Further, by [5, Thm. 5.23, (2)], we have
$$\begin{aligned} \deg [{\mathscr {M}}(X_0/0,{\varvec{\tau }})]^{\mathrm {virt}} = \sum _{{\varvec{\tau }}'} {m_{{\varvec{\tau }}'} \over |{\text {Aut}}({\varvec{\tau }}'/{\varvec{\tau }})|} \deg [{\mathscr {M}}(X_0/0,{\varvec{\tau }}')]^{\mathrm {virt}}, \end{aligned}$$
(3.18)
where the sum is over codimension one degenerations \({\varvec{\tau }}'\) of \({\varvec{\tau }}\) in the sense of [5, Def. 5.22, (2)], and \(m_{{\varvec{\tau }}'}\) is the order of the cokernel of \(p_*:\Lambda _{\tau '}\rightarrow {\mathbb {Z}}\). It is immediate from that definition that if \({\mathscr {M}}(X_0/0,{\varvec{\tau }}')\not =\emptyset \) then such a \({\varvec{\tau }}'\) is a decorated wall type for X. (We need to assume non-emptiness to guarantee that \(\tau '\) is balanced.) It also follows from Proposition 3.7 and the proof of Lemma 3.9 that \({\mathscr {M}}(X_0/0,{\varvec{\tau }}')={\mathscr {M}}(X/S,{\varvec{\tau }}')={\mathscr {M}}(X,{\varvec{\tau }}')\). Thus we may rewrite (3.18) as
$$\begin{aligned} k_{\tau }{\deg [{\mathscr {M}}(X_0/0,{\varvec{\tau }})]^{\mathrm {virt}} \over |{\text {Aut}}({\varvec{\tau }})|} = \sum _{{\varvec{\tau }}'} {m_{{\varvec{\tau }}'} k_{\tau } \over |{\text {Aut}}({\varvec{\tau }}'/{\varvec{\tau }})||{\text {Aut}}({\varvec{\tau }})|} \deg [{\mathscr {M}}(X,{\varvec{\tau }}')]^{\mathrm {virt}}. \end{aligned}$$
Certainly, \(|{\text {Aut}}({\varvec{\tau }}'/{\varvec{\tau }})||{\text {Aut}}({\varvec{\tau }})|=|{\text {Aut}}({\varvec{\tau }}')|\). Further, \(\Lambda _{\tau '_{\mathrm {out}}}=\Lambda _{\tau _{\mathrm {out}}}\oplus {\mathbb {Z}}\) and \(p_*(\Lambda _{\tau '_{\mathrm {out}}})=m_{\tau '}{\mathbb {Z}}\). Next \(\Lambda _{{\varvec{\sigma }}'(L_{\mathrm {out}})}=\Lambda _{{\varvec{\sigma }}(L_{\mathrm {out}})}\oplus {\mathbb {Z}}\), with the induced map \(g_{\mathrm {trop},*}:\Lambda _{{\varvec{\sigma }}'(L_{\mathrm {out}})} \rightarrow {\mathbb {Z}}\) being surjective because of the assumption that \(g^{-1}(0)\) is reduced. Thus an elementary diagram chase gives a short exact sequence
$$\begin{aligned} 0\rightarrow {\text {coker}}(\Lambda _{\tau _{\mathrm {out}}}\rightarrow \Lambda _{{\varvec{\sigma }}(L_{\mathrm {out}})}) \rightarrow {\text {coker}}(\Lambda _{\tau '_{\mathrm {out}}}\rightarrow \Lambda _{{\varvec{\sigma }}'(L_{\mathrm {out}})}) \rightarrow {\mathbb {Z}}/m_{\tau '}{\mathbb {Z}}\rightarrow 0, \end{aligned}$$
so \(m_{\tau '}k_{\tau }= k_{\tau '}\), giving (3.17). \(\square \)
Logarithmic broken lines and theta functions
Wall types defined in the previous subsection allowed us to define the canonical wall structure. Consistency of a wall structure is defined partly via broken lines (reviewed in Definition 4.2). A key point of our proof of consistency is a correspondence result that associates counts of broken lines to certain punctured invariants. In this subsection, we shall define these punctured invariants. We proceed quite analogously with the notion of wall types.
Definition 3.19
A (non-trivial) broken line type is a type \(\tau =(G,{\varvec{\sigma }},{{\mathbf {u}}})\) of tropical map to \(\Sigma (X)\) such that:
-
(1)
G is a genus zero graph with \(L(G)=\{L_{\mathrm {in}},L_{\mathrm {out}}\}\) with \({\varvec{\sigma }}(L_{\mathrm {out}})\in {\mathscr {P}}\) and
$$\begin{aligned} u_{\tau }:=\mathbf{u}(L_{\mathrm {out}})\not =0, \quad p_{\tau }:=\mathbf{u}(L_{\mathrm {in}})\in {\varvec{\sigma }}(L_{\mathrm {in}}){\setminus }\{0\} \end{aligned}$$
(so that \(L_{\mathrm {in}}\) represents a marked rather than punctured point).
-
(2)
\(\tau \) is realizable and balanced.
-
(3)
Let \(h:\Gamma (G,\ell )\rightarrow \Sigma (X)\) be the corresponding universal family of tropical maps, and let \(\tau _{\mathrm {out}}\in \Gamma (G,\ell )\) be the cone corresponding to \(L_{\mathrm {out}}\). Then \(\dim \tau =n-1\) and \(\dim h(\tau _{\mathrm {out}})=n\).
We also consider the possibility of a trivial broken line type \(\tau \), which is not an actual type: the underlying graph G consists of just one leg \(L_{\mathrm {in}}=L_{\mathrm {out}}\) and no vertices, with the convention that \(u_{\tau }=-p_{\tau }\). Note this does not correspond to an actual punctured curve.Footnote 8
A decorated broken line type is a decorated type \({\varvec{\tau }}=(\tau ,{\mathbf {A}})\) with \(\tau \) a broken line type. In the trivial case, as there are no vertices, this does not involve any extra information, and in this case, the total curve class is taken to be 0.
A degenerate broken line type is a type \(\tau \) which satisfies conditions (1) and (2) above and instead of (3),
Lemma 3.20
Let \({\varvec{\tau }}\) be a non-trivial decorated broken line type. Then \({\mathscr {M}}(X,{\varvec{\tau }})\) is proper over \({\text {Spec}}\mathbbm {k}\) and carries a virtual fundamental class of dimension zero.
Proof
The argument is essentially identical to that of the proof of Lemma 3.9, and we leave it to the reader to make the necessary modifications, except for one point. In that proof, we appealed to Proposition 3.7 to argue that in the relative case, for each \(v\in V(G)\), we have \(\Sigma (g):h(\tau _v)\rightarrow {\mathbb {R}}_{\ge 0}\) surjective. Crucially, a broken line type is not in general a type of tropical map to X/S, so a different argument is necessary. However, note that by the assumption that \(\tau \) is a balanced type, for any \(s\in {\text {Int}}(\tau )\), \(\Sigma (g)\circ h_s:G\rightarrow {\mathbb {R}}_{\ge 0}\) is a balanced tropical map. Necessarily \(\Sigma (g)_*({\mathbf {u}}(L_{\mathrm {in}})) =\Sigma (g)(p_{\tau })\in {\mathbb {R}}_{\ge 0}\), and hence by balancing, \(\Sigma (g)_*(u_{\tau })\in {\mathbb {R}}_{\le 0}\). This implies, again by balancing, that if \(v_{\mathrm {in}}\), \(v_{\mathrm {out}}\) are the vertices adjacent to \(L_{\mathrm {in}}\), \(L_{\mathrm {out}}\) respectively, then \(a=\Sigma (g)\circ h_s(v_{\mathrm {out}}) \le \Sigma (g)\circ h_s(v_{\mathrm {in}})=b\), and for any other vertex \(v\in V(G)\), \(\Sigma (g)\circ h_s(v)\in [a,b]\). Further, since \(\dim h(\tau _{\mathrm {out}})=n\), necessarily \(h(\tau _{\mathrm {out}})\not \subseteq \partial B\), and thus \(\Sigma (g):h(\tau _{\mathrm {out}})\rightarrow {\mathbb {R}}_{\ge 0}\) is surjective, and so \(\Sigma (g):h(\tau _{v_{\mathrm {out}}})\rightarrow {\mathbb {R}}_{\ge 0}\) is also surjective. Putting this together, we see that \(\Sigma (g):h(\tau _v)\rightarrow {\mathbb {R}}_{\ge 0}\) is surjective for any vertex \(v\in V(G)\). This is sufficient to complete the argument of the proof of Proposition 3.9 for properness of the moduli spaces involved. \(\square \)
We now define, for \({\varvec{\tau }}\) a non-trivial decorated broken line type,
$$\begin{aligned} N_{{\varvec{\tau }}}:={\deg [{\mathscr {M}}(X,{\varvec{\tau }})]^{\mathrm {virt}}\over |{\text {Aut}}({\varvec{\tau }})|}. \end{aligned}$$
(3.19)
We also have a map \(h_*:\Lambda _{\tau _{\mathrm {out}}}\rightarrow \Lambda _{{\varvec{\sigma }}(L_{\mathrm {out}})}\), necessarily of finite index, and define
$$\begin{aligned} k_{\tau }:= |{\text {coker}}\, h_*|= |\Lambda _{{\varvec{\sigma }}(L_{\mathrm {out}})}/h_*(\Lambda _{\tau _{\mathrm {out}}})|. \end{aligned}$$
(3.20)
For a trivial decorated broken line type \({\varvec{\tau }}\), we set
$$\begin{aligned} N_{{\varvec{\tau }}} := 1, \quad k_{{\varvec{\tau }}}:=1. \end{aligned}$$
Definition 3.21
Fix \(p\in B({\mathbb {Z}}){\setminus } \{0\}\), \(\sigma \in {\mathscr {P}}_{{{\text {max}}}}\), \(x\in {\text {Int}}(\sigma )\) not contained in any rationally defined hyperplane in \(\sigma \). We then define
$$\begin{aligned} \vartheta ^{\log }_p(x)= \sum _{{\varvec{\tau }}} k_{\tau }N_{{\varvec{\tau }}} t^A z^{-u_{\tau }} \in \mathbbm {k}[{\mathcal {P}}^+_x]/I_x \end{aligned}$$
where the sum is over all isomorphism classes of decorated broken line types \({\varvec{\tau }}\) with \(p_{\tau }=p\), \(x\in h(\tau _{\mathrm {out}})\). Here A is the total curve class of \({\varvec{\tau }}\), and we require \(A\in Q{\setminus } I\).
Lemma 3.22
For \(p\in B({\mathbb {Z}}){\setminus } \{0\}\), the number of decorated broken line types \({\varvec{\tau }}\) with \(p_{\tau }=p\), total curve class in \(Q{\setminus } I\), and \(N_{{\varvec{\tau }}}\not =0\) is finite. In particular, the sum defining \(\vartheta ^{\log }_p(x)\) is finite.
Proof
As \(Q{\setminus } I\) is finite, there are only a finite number of total curve classes. Further, there are only a finite number of possibilities for \({\varvec{\sigma }}(L_{\mathrm {out}})\). For each curve class A and choice of \({\varvec{\sigma }}(L_{\mathrm {out}})\), it follows from [48, Cor. 1.14] that \(u_{\tau }\) is determined by p if \({\mathscr {M}}(X,{\varvec{\tau }})\) is non-empty. If \(\beta \) is the punctured curve class given by contact orders p and \(u_{\tau }\) and curve class A, then \({\mathscr {M}}(X,\beta )\) is of finite type, and there are only a finite number of decorated types \({\varvec{\tau }}\) appearing as the types of tropicalizations of punctured maps in \({\mathscr {M}}(X,\beta )\). Thus there are only a finite number of possible choices of \({\varvec{\tau }}\) with given total curve class A and \({\varvec{\sigma }}(L_{\mathrm {out}})\) satisfying \({\mathbf {u}}(L_{\mathrm {in}})=p\) and \({\mathbf {u}}(L_{\mathrm {out}})=u_{\tau }\). \(\square \)