Logarithmic Donaldson–Thomas theory

1.1 Cone complexes and their morphisms

We start with the building blocks of toric geometry. A polyhedral cone with integral structure $(\sigma ,M)$ is a topological space $\sigma $ equipped with a finitely generated abelian group M of continuous real-valued functions from $\sigma $ to $\mathbb R$ such that the evaluation

$$\begin{align*}\sigma\to\operatorname{Hom}(M,{\mathbb R}) \end{align*}$$

is a homeomorphism onto a strongly convex polyhedral cone. If this cone is rational with respect to the dual lattice of M, then we say that $(\sigma ,M)$ is rational. We define the lattice of integral points of $\sigma $ by taking the preimage of the dual lattice of M under the evaluation map. The set of elements of M that are nonnegative on $\sigma $ form a monoid $S_{\sigma }$ referred to as the dual monoid. The cone $\sigma $ is recovered as the space of monoid homomorphisms $\operatorname {Hom}(S_{\sigma },{\mathbb R}_{\geq 0})$ .

Henceforth, a rational polyhedral cone with integral structure will be referred to as a cone.

Definition 1.1.1 (Cone complexes).

A rational polyhedral cone complex is a topological space that is presented as a colimit of a partially ordered set of cones, where all arrows are given by isomorphisms onto proper faces. A morphism of cone complexes is a continuous map

$$\begin{align*}\Sigma'\to \Sigma \end{align*}$$

such that (i) the image of every cone in $\Sigma '$ is contained in a cone of $\Sigma $ and (ii) the restriction of the map to any cone of $\Sigma '$ is given by an integer linear map.

Cone complexes are nearly identical to the fans considered in toric geometry [Reference Fulton24]. The key differences are that they do not come equipped with a global embedding into a vector space, and two cones can intersect along a union of faces. However, we will typically restrict to the case where an intersection of two cones is a face of each, so the main thing to keep in mind is the lack of a global embedding. Concretely, the reader may keep in mind that toric fans of $\mathbb P^1$ and $\mathbb A^2\setminus \{(0,0)\}$ are isomorphic as cone complexes.

A cone complex is smooth if every cone is isomorphic to a standard orthant with its canonical integral structure. We record two combinatorial notions associated to morphisms between cone complexes.

The terminology is compatible with the identically named geometric notions, when applied to toric maps [Reference Abramovich and Karu7, Section 4 & 5].

1.2 Tropicalization for the target

A basic fact from the theory of logarithmic schemes is that every Zariski logarithmically smooth scheme gives rise to a cone complex. We take a moment to unpack this statement in concrete terms. Let $(X|D)$ be a smooth scheme of finite type over ${\mathbb C}$ equipped with a simple normal crossings divisor D. Assume that the intersections of irreducible components of D are always connected. The presence of D gives X the structure of a logarithmically smooth scheme. We let $X^{\circ }$ be the complement of the divisor, where the logarithmic structure is trivial.

We unwind the definition of the logarithmic structure in this case for the benefit of the reader. The components of D give rise to a distinguished class of functions in the structure sheaf. For an open set $U\subset X$ , we record the values of the logarithmic structure sheaf and the characteristic monoid sheaf:Footnote ¹

$$\begin{align*}M_X(U) = \left\{f\in{\mathcal O}_X(U): f \textrm{ is invertible on } U\setminus D\right\} \ \ \ \textrm{and} \ \ \overline M_X(U) = M_X(U)/{\mathcal O}_X^{\star}(U). \end{align*}$$

Locally at each point $x\in X$ , there are functions, canonical up to multiplication by a unit, cutting out the irreducible components of D passing through x. In particular, the stalk of the characteristic monoid sheaf at x is naturally identified with ${\mathbb N}^e$ , where e is the number the such components.

The dual cones of the stalks of the characteristic monoids of X give rise to a collection of cones, one for each point of X. The generization maps naturally give rise to gluing morphisms, and these cones form a finite type cone complex. We denote it by $\Sigma _X$ , refer to it as the tropicalization of X or the cone complex of X and regard it as a cone complex equipped with an integral structure.

In our case, the tropicalization has a more practical description. Given the pair $(X,D)$ , let d be the number of irreducible components of D and enumerate these components $D_1,\ldots , D_d$ . Each k-dimensional face of the cone ${\mathbb R}_{\geq 0}^d$ is spanned by rays $v_{i_i},\ldots , v_{i_k}$ . Call such a face relevant to $(X,D)$ if the corresponding intersection $D_{i_1},\ldots , D_{i_k}$ is nonempty. Then the tropicalization $\Sigma _X$ is the union of cones in ${\mathbb R}_{\geq 0}^d$ that are relevant to $(X,D)$ . In the case of a toric variety this construction recovers the fan, as an abstract cone complex. The construction extends to the case of logarithmically smooth schemes [Reference Abramovich, Caporaso and Payne1, Reference Ulirsch74]. Further details and generalizations may be found in the references [Reference Abramovich, Chen, Marcus, Ulirsch, Wise, Baker and Payne5, Reference Cavalieri, Chan, Ulirsch and Wise16, Reference Kato35, Reference Kempf, Knudsen, Mumford and Saint-Donat39].

1.3 Subdivisions

Let $(X|D)$ be a simple normal crossings pair with tropicalization $\Sigma _X$ . For simplicity, we assume that the nonempty intersections of components of D are connected. We will produce target expansions by using subdivisions of the tropicalization.

Definition 1.3.1. A subdivision is a cone complex $\Delta $ and a morphism of cone complexes

$$\begin{align*}\Delta\hookrightarrow \Sigma_X \end{align*}$$

that is injective on the support of $\Delta $ and further such that the integral points of the image of each cone $\tau \in \Delta $ are exactly the intersection of the integral points of $\Sigma _X$ with $\tau $ .

Note that this more flexible than the standard definition; the underlying map of sets need not be a bijection because we wish to have the flexibility to discard closed strata after blowing up. When the map on integral points is a bijection we call it a complete subdivision.

In toric geometry, subdivisions give rise to possibly nonproper birational models of X. The same is true for $(X|D)$ . The cone complex $\Sigma _X$ associated to D is smooth, and the intersection of any two cones is a face of each. If there are d rays in $\Sigma _X$ , we may embed it via

$$\begin{align*}\Sigma_X\hookrightarrow {\mathbb R}^{d}_{\geq 0} \end{align*}$$

by mapping each ray in $\Sigma _X$ isomorphically onto the positive ray on the corresponding axis. The positive orthant in this vector space is the fan associated to the toric variety $\mathbb A^d$ with dense torus ${\mathbb G}_m^d$ . The subdivision

$$\begin{align*}\Delta\hookrightarrow \Sigma_X\hookrightarrow {\mathbb R}^d_{\geq 0} \end{align*}$$

defines a noncomplete fan with associated toric variety ${\mathbb {A}}_{\Delta }$ , equipped with an ${\mathbb G}_m^d$ -equivariant birational map ${\mathbb {A}}_{\Delta }\to {\mathbb {A}}^d$ . By passing to quotients, we have a morphism of stacks

$$\begin{align*}[{\mathbb{A}}_{\Delta}/{{\mathbb G}}_m^d]\to[{\mathbb{A}}^d/{\mathbb G}_m^d]. \end{align*}$$

The presence of D gives rise to a tautological morphism $(X|D)\to [{\mathbb {A}}^d/{\mathbb G}_m^d]$ .

Definition 1.3.2. The birational model of X associated to the subdivision $\Delta \hookrightarrow \Sigma $ is given by

$$\begin{align*}X_{\Delta}:= X\times_{ [{\mathbb{A}}^d/{\mathbb G}_m^d]} [{\mathbb{A}}_{\Delta}/{\mathbb G}_m^d]. \end{align*}$$

1.4 Tropicalization for subschemes via valuations

In this section, we assume that X is proper, with the exception of Remark 1.5.5 and situations where it becomes active.

Locally at each point on X, the components of the divisor D provide a distinguished set of functions – a subset of coordinates – at that point. The tropicalization of a subscheme is the image of its ‘coordinatewise valuation’, in these coordinates.

Let $(K,\nu )$ be a rank $1$ valued field extending ${\mathbb C}$ , and let $\nu $ be the valuation. Consider a morphism

$$\begin{align*}\operatorname{\mathrm{Spec}} K \to X^{\circ}. \end{align*}$$

Since X is proper, this morphism extends to the valuation ring

$$\begin{align*}\operatorname{\mathrm{Spec}} R\to X. \end{align*}$$

Let x be the image of the closed point. The smallest closed stratum of D containing x is the intersection of a (possibly empty) subset of irreducible divisor components $D_{i_1},\ldots , D_{i_r}$ . The associated equations generate the stalk of the characteristic sheaf at the point x, which is abstractly isomorphic to ${\mathbb N}^r$ . For each element $\overline f \in {\mathbb N}^r$ , we may lift it to a function f on X in a neighborhood of the point x, pull back to $\operatorname {\mathrm {Spec}} R$ along the map above and compose with the valuation on R. The ratio of two such lifts is a unit, so this gives rise to a well-defined element

$$\begin{align*}[{\mathbb N}^r\to {\mathbb R}_{\geq 0}]\in \Sigma_X\subset \operatorname{Hom}({\mathbb N}^r,{\mathbb R}_{\geq 0}). \end{align*}$$

For any valued field K, we have a well-defined morphism

$$\begin{align*}\mathsf{trop}\colon X^{\circ}(K)\to \Sigma_X. \end{align*}$$

Let K be a valued field whose associated valuation map $K^\times \to {\mathbb R}$ is surjective. Let $Z^{\circ } \subset X^\circ $ be a subscheme. Let $\mathsf {trop}(Z^\circ )$ be the subset of $\Sigma _X$ obtained by restricting $\mathsf {trop}$ to $Z^\circ (K)$ ; it is independent of the choice of valued field K, and its and its functoriality properties are outlined in [Reference Gubler28, Reference Ulirsch75].Footnote ²

1.5 Properties of the tropicalization

The shapes of tropicalizations are governed by the Bieri–Groves theorem [Reference Bieri and Groves14, Reference Ulirsch74].

The set $\mathsf {trop}(Z^{\circ })$ has no distinguished polyhedral structure in general [Reference Maclagan and Sturmfels48, Example 3.5.4]. It is simply a set, and this set can be given the structure of a cone complex.

The role of tropicalization in degeneration and compactification problems has its origin in the following two theorems, proved by Tevelev for toric varieties and Ulirsch for logarithmic schemes; see [Reference Tevelev71, Theorem 1.2] and [Reference Ulirsch74, Theorem 1.2].

The first concerns the properness of closures of subschemes in partial compactifications of $X^{\circ }$ . Let $Z^{\circ }$ be a subscheme of $X^{\circ }$ and let $X'$ be a simple normal crossings compactification. Let $X"\subset X'$ be the complement of a union of closed strata of $X'$ and let $\Sigma _{X"}$ be the subfan of $\Sigma _{X'}$ obtained by deleting the corresponding union of open cones.

The second concerns transversality. We say the closure Z of $Z^{\circ }$ in $X'$ intersects strata in the expected dimension if

$$\begin{align*}\dim E\cap Z = \dim Z-\mathsf{codim}_{X'}E. \end{align*}$$

Remark 1.5.5. If X is not proper, there is no longer a tropicalization map defined on the set $X^{\circ }(K)$ because limits need not exist. However, the above relationship with the cone complex persists. For each valued field K with valuation ring R as above, let $X^\beth (K)$ be the subset $X^{\circ }(K)$ consisting of those K-points that extend to R-points. There is a morphism

$$\begin{align*}X^\beth(K)\to \Sigma_X \end{align*}$$

defined exactly as defined above. Similarly, given a subscheme $Z\subset X^{\circ }$ , we can define its tropicalization as the image of $Z^\beth (K)$ in $\Sigma _X$ ; see [Reference Ulirsch75, Section 5.2].

1.6 Tropicalization via compactifications

In the previous section, tropicalizations were seen to select partial compactifications of $X^{\circ }$ , in which the closure of a subvariety meets each stratum in the expected codimension. There is a partial converse.

Let $X'$ be a simple normal crossings compactification of $X^{\circ }$ . Let Z be a subscheme of $X'$ whose intersection with the strata of $X'$ have the expected codimension, with

$$\begin{align*}Z^{\circ} = X^{\circ}\cap Z. \end{align*}$$

The following perspective is due to Hacking–Keel–Tevelev, extended by Ulirsch and is sometimes referred to as geometric tropicalization; see [Reference Hacking, Keel and Tevelev29, Reference Ulirsch74].

Theorem 1.6.1. The tropicalization of $Z^{\circ }$ is equal to the union of cones $\sigma $ in the underlying set of $\Sigma _{X'}$ such Z nontrivially intersects the locally closed strata dual to $\sigma $ . That is, there is an equality of subsets of $\Sigma _{X'}$ given by

$$\begin{align*}\mathsf{trop}(Z^{\circ}) \ \ \ {=}\bigcup_{\sigma\in \Sigma_X: V(\sigma)\cap Z \neq \emptyset} \sigma\ \ \ \ \ \ \ \ \textrm{in} \ \ \ \ \ |\Sigma_X|, \end{align*}$$

where $V(\sigma )$ is the locally closed stratum of $X'$ corresponding to the cone $\sigma $ .

The definition of $\mathsf {trop}(Z^{\circ })$ via coordinatewise valuation described in the previous subsection depends on a choice of compactification X of $X^{\circ }$ , but if X is replaced with a blowup along a stratum, the set $\mathsf {trop}(Z^\circ )$ is unchanged. The result above can therefore be viewed as a computational tool; it describes the tropicalization using a single compactification in which Z is dimensionally transverse. We return to this in Section 7.

1.7 Tropicalization for a family of subschemes

The constructions extend to flat one-parameter families of subschemes. We will consider subschemes in X that are defined over a valued field K that extends the trivially valued ground field ${\mathbb C}$ .

In order to avoid foundational issues, we will assume that all subschemes are defined over the localization of a smooth algebraic curve of finite type. The assumption is made so we can appeal to the relationship between tropicalization via valuations and via logarithmic geometry, which has only received a definitive treatment under these hypotheses [Reference Ulirsch74]. The valued field will arise for us in the study of the valuative criterion for properness. The relevant moduli spaces will be shown to be of finite type, so this is a harmless assumption.

1.7.1 Tropicalization over a valued field

Let $(X|D)$ be a simple normal crossings compactification as above, with interior $X^{\circ }$ . Let K be a valued field extending ${\mathbb C}$ as above. Consider an algebraically transverse subscheme

$$\begin{align*}{\mathcal Z}_{\eta}\hookrightarrow X\times_{{\mathbb C}} K, \end{align*}$$

and let ${\mathcal Z}_{\eta }^{\circ }$ be the open subscheme contained in $X^{\circ }$ . By the algebraic transversality hypothesis, this open subscheme is dense. After passing from K to a valued extension L with real surjective valuation, we may once again consider the tropicalization map

$$\begin{align*}\mathsf{trop}\colon {\mathcal Z}^{\circ}_{\eta}(L)\to \Sigma_X. \end{align*}$$

The tropicalization is independent of the choice of L, provided the valuation is surjective onto the real numbers.Footnote ³

The basic structure result extends to this nontrivially valued setting with a small twist.

Theorem 1.7.1. The set $\mathsf {trop}({\mathcal Z}_{\eta }^{\circ })$ is the support of a rational polyhedral complex of $\Sigma _X$ . The topological dimension of $\mathsf {trop}({\mathcal Z}_{\eta }^{\circ })$ is bounded above by the algebraic dimension of ${\mathcal Z}^\circ $ . If $X^\circ $ is a closed subvariety of an algebraic torus, then the topological dimension of the tropicalization is equal to the algebraic dimension of ${\mathcal Z}^\circ $ .

The toric case of this result was established by Gubler without restriction on the base field [Reference Gubler28]. The statement for simple normal crossings targets is a consequence of [Reference Brandt and Ulirsch15, Theorem 5.1].

To emphasize the point, the tropicalization of a variety that is defined over a nontrivial valued extension of ${\mathbb C}$ is polyhedral but not necessarily conical. In other words, it can have bounded cells. The target X itself is still defined over ${\mathbb C}$ , not merely over K, so its tropicalization remains conical.

1.7.2 Families over a punctured curve

Let C be a smooth algebraic curve of finite type with a distinguished point $0\in C$ ; let $C^{\circ }$ be the complement of this point. The pair $(C,0)$ has a cone complex $\Sigma _C$ , canonically identified with the positive real line ${\mathbb R}_{\geq 0}$ .

Let $(X|D)$ be a pair as before and $X^{\circ }$ its interior. Consider a flat family of one-dimensional subschemes over $C^{\circ }$

$$\begin{align*}{\mathcal Z}^{\circ}\subset C^{\circ}\times X^{\circ}. \end{align*}$$

There are two tropicalizations associated to this family. For the first, we consider ${\mathcal Z}^{\circ }$ as a two-dimensional subscheme of $C^{\circ }\times X^{\circ }$ . The partial compactification $C\times X$ determines a tropicalization, as explained in Remark 1.5.5:

$$\begin{align*}\mathsf{trop}({\mathcal Z}^{\circ})\subset {\mathbb R}_{\geq 0} \times \Sigma_X. \end{align*}$$

For the second, note that the function field of C is equipped with a discrete valuation arising from order of vanishing at $0$ . Let K be an extension of this field with real surjective valuation, and note that there is a canonical inclusion of $\operatorname {\mathrm {Spec}} K$ to $C^{\circ }$ . Consider the tropicalization of the K-valued points of the base change

$$\begin{align*}{\mathcal Z}_{\eta} = {\mathcal Z}^{\circ}\times_{C^{\circ}} K\subset X(K), \end{align*}$$

where the morphism $\operatorname {\mathrm {Spec}} K \rightarrow C^{\circ }$ is determined by the fraction field of the local ring of C at $0$ . Denote the result by $\mathsf {trop}({\mathcal Z}_{\eta })$ – note that the subscript $\eta $ indicates that we are considering it as a generic fiber.

These two procedures are related. When considering the total space of ${\mathcal Z}^{\circ }$ as a surface over $C^{\circ }$ , there is a map

$$\begin{align*}\upsilon\colon \mathsf{trop}({\mathcal Z}^{\circ})\to {\mathbb R}_{\geq 0}. \end{align*}$$

The fiber $\upsilon ^{-1}(1)$ coincides with the second tropicalization $\mathsf {trop}({\mathcal Z}_{\eta })$ . This follows from a tracing of definitions for the tropicalization of fibers of maps of Berkovich spaces and the functoriality results in [Reference Ulirsch74].

1.8 Transversality for one-parameter families

The appropriate generalizations of the properness and transversality statements earlier in this section are as follows. Let R be the valuation ring of K, and equip $\operatorname {\mathrm {Spec}} R$ with the divisorial logarithmic structure at the closed point. The scheme $X\times \operatorname {\mathrm {Spec}} R$ is equipped with the simple normal crossings divisor given by $D\times \operatorname {\mathrm {Spec}} R\cup X\times 0$ . Once equipped with this divisor, it has tropicalization $\Sigma _X\times {\mathbb R}_{\geq 0}$ .

Let ${\mathcal Y}'\to X\times \operatorname {\mathrm {Spec}} R$ be a toroidal modification of the constant family, and let ${\mathcal Y}"\subset {\mathcal Y}'$ be the complement of a union of closed strata. In practice, ${\mathcal Y}"$ will be the complement of all closed strata of codimension $2$ in the special fiber. These have tropicalizations $\Sigma _{{\mathcal Y}'}$ and $\Sigma _{{\mathcal Y}"}$ that are, respectively, a complete subdivision and a subdivision of $\Sigma _X\times {\mathbb R}_{\geq 0}$ . There is a morphism of cone complexes by composition:

$$\begin{align*}\Sigma_{{\mathcal Y}"}\to {\mathbb R}_{\geq 0}, \end{align*}$$

and we denote the fiber over $1$ by $\Sigma _{{\mathcal Y}"}(1)$ . We view this as a polyhedral complex.

Let ${\mathcal Z}^{\circ }$ be a subscheme of $X^\circ \times \operatorname {\mathrm {Spec}} K$ . We examine the question of when the closure is proper.

Proof. This is well known to experts, but in the form stated, we have been unable to locate a suitable reference. We explain how it can be deduced from results that do appear in the literature.

First, we note that the closure of ${\mathcal Z}^{\circ }$ in the larger degeneration ${\mathcal Y}'$ is certainly proper over $\operatorname {\mathrm {Spec}} R$ because X is proper and ${\mathcal Y}'\to X\times \operatorname {\mathrm {Spec}} R$ is a proper and birational morphism. We now use the hypothesis on K. Specifically, since K is a localization of the function field of a smooth curve C with the valuation associated to a closed point $0$ with complement $C^\circ $ . Moreover, the given subscheme ${\mathcal Z}^\circ $ , as well as the degenerations ${\mathcal Y}'$ and ${\mathcal Y}"$ , can be assumed to arise via base change from corresponding families over C along the inclusion

$$\begin{align*}\operatorname{\mathrm{Spec}} R\to C \end{align*}$$

of the local ring at $0$ . Rather than overburdening the notation, we replace the families over $\operatorname {\mathrm {Spec}} R$ with the corresponding families over C. We now view C, $X\times C$ , ${\mathcal Y}'$ and ${\mathcal Y}"$ as logarithmic schemes over ${\mathbb C}$ . Note that, due to the potential nonproperness of C, Remark 1.5.5 is in effect. The reader is also advised to keep in mind the relationship between the two tropicalizations of ${\mathcal Z}^{\circ }$ described above.

We are now in a position to apply the Tevelev–Ulirsch lemma for logarithmic schemes as stated in [Reference Ulirsch74, Lemma 4.1]. From it, we deduce that the closure of ${\mathcal Z}^{\circ }$ in ${\mathcal Y}'$ coincides with the closure in ${\mathcal Y}"$ precisely under the hypotheses stated in the theorem. We conclude the result.

We keep the notation above and now deal with the corresponding transversality statement.

2.1 The plan

Given a simple normal crossings pair $(X|D)$ , we have recalled in the section above how a conical subdivision of its tropicalization $\Sigma _X$ determines an open subset in a proper birational modification of X. A conical subdivision of $\Sigma _X\times {\mathbb R}_{\geq 0}$ , by the same dictionary, will determine a expansion, a special type of degeneration, of X along its boundary D. It leads to a family defined over ${\mathbb {A}}^1$ .

A conical subdivision

$$\begin{align*}\widetilde{\Sigma_X\times{\mathbb R}_{\geq 0}}\to \Sigma_X\times{\mathbb R}_{\geq 0} \end{align*}$$

can be visualized as a polyhedral subdivision of $\Sigma _X$ : Take the height $1$ slice of the subdivision under the projection to ${\mathbb R}_{\geq 0}$ . The result is a union of polyhedra in $\Sigma _X$ glued along faces – a polyhedral complex. The cone over this polyhedral subdivision recovers the original conical subdivision.

As we have seen in the previous section, the tropicalization of a one-dimensional subscheme of X, as defined in the previous section, determines a polyhedral subdivision and therefore an expansion.

We capture the possible tropicalizations of subschemes by the notion of a $1$ -complex and use them to prove flat limit algorithms which will eventually establish the properness and separatedness of our yet-to-be-proposed moduli spaces. In order to build these moduli, we will form the parameter space for such $1$ -complexes and use logarithmic geometry to turn this into a parameterizing stack for expansions. The logarithmic Hilbert scheme will be built on top of this stack of expansions.

2.2 Graphical preliminaries

Let $\underline G$ be a finite graph, possibly disconnected but without loops or parallel edges. We enhance it with two additional pieces of data. The first is a finite set of rays, formally given by a finite set $R(\underline G)$ equipped with a map to the vertex set

$$\begin{align*}r\colon R(\underline G)\to V(\underline G). \end{align*}$$

The second is the metrization of the edge set, given by the edge length function

$$\begin{align*}\ell\colon E(\underline G)\to {\mathbb R}_{>0}. \end{align*}$$

These data give rise simultaneously to a metric space and a polyhedral complex, both enhancing the topological realization of $\underline G$ . The topological realization of $\underline G$ is a CW complex, and we endow an edge E with a metric by identifying it with an interval in ${\mathbb R}$ of length $\ell (E)$ . For each element $ h\in R(\underline G)$ , we glue on a copy of the metric space ${\mathbb R}_{\geq 0}$ to the point $r(h)$ . As we are now free to think of each edge or half edge as being either a polyhedron or a metric space, the result is a space G that is simultaneously a metric space and a polyhedral complex of dimension at most $1$ . In particular, it makes sense to talk about real-valued continuous piecewise affine functions on G.

Let $\Sigma $ be a smooth cone complex such that the intersection of any two cones is a face of each. Let $|\Sigma |$ be the associated topological space. Note that $\Sigma $ embeds canonically as a subcomplex of a standard orthant via $\Sigma \hookrightarrow {\mathbb R}_{\geq 0}^{\Sigma ^{(1)}}$ .

A piecewise affine map $F:G\to \Sigma $ is the data of a continuous map on the underlying topological spaces

$$\begin{align*}|G|\to \underline |\Sigma| \end{align*}$$

such that every face of G maps to a cone of $\Sigma $ and such that the map is integer affine upon restriction to each face.

We can measure the slope along an edge of G, well defined up to sign, as follows. Each noncontracted edge E maps to a cone $\sigma $ and thus maps onto a line $L_E$ in $\sigma ^{\mathsf {gp}}$ . We refer to the expansion factor of the induced map $F:E\to L_E$ as the slope of F along the edge E.

Definition 2.2.2. An embedded $1$ -complex in $\Sigma $ is a piecewise affine map, written,

$$\begin{align*}G\to \Sigma \end{align*}$$

that is injective on underlying topological spaces subject to the following conditions: (i) the slope along all edges E in G is equal to $1$ , and (ii) the image of each ray of G is parallel to a one-dimensional face of the cone containing it.

This latter condition will be satisfied for all the embedded $1$ -complexes that occur for us since it will be forced by algebraic transversality of subschemes in expansions. The reader can also drop condition, as these would correspond to components of the stack of expansions that are not relevant for our moduli spaces.

When it is clear from context that G has been embedded, we refer to it simply as a $1$ -complex and will use the notation G to refer to this embedded object.

2.3 Target geometry

Let $\Sigma _X$ be the cone complex of $(X|D)$ . An embedded $1$ -complex $G\subset \Sigma _X$ gives rise to a class of target geometries as follows. Place the embedded $1$ -complex G at height $1$ inside $\Sigma _X\times {\mathbb R}_{\geq 0}$ , and let $\mathsf C(G)$ be the cone over it.

By applying the construction in Section 1.3 and Remark 1.3.3 to the subdivision

$$\begin{align*}\mathsf C(G)\hookrightarrow \Sigma_X\times{\mathbb R}_{\geq 0}, \end{align*}$$

we obtain a target expansion

$$\begin{align*}{\mathcal Y}_G\to X. \end{align*}$$

It is typically not proper.

Every affine toric variety carries a canonical logarithmic structure. Consider a toric monoid P with associated toric variety $U_P$ and a closed point u in the closed torus orbit. By pulling back the toric logarithmic structure to u, we obtain the P-logarithmic point. It is denoted $\operatorname {\mathrm {Spec}} {\mathbb C}_P$ . The standard logarithmic point is $\operatorname {\mathrm {Spec}} {\mathbb C}_{\mathbb N}$ and is the pullback to $0$ of the toric logarithmic structure on ${\mathbb {A}}^1$ .

Definition 2.3.3 (Expansions and families).

A rough expansion of X over $\operatorname {\mathrm {Spec}} {\mathbb C}_{\mathbb N}$ is a logarithmic scheme ${\mathcal Y}\to X\times \operatorname {\mathrm {Spec}} {\mathbb C}_{\mathbb N}$ which is the fiber over $0$ in the modification of $(X|D)\times {\mathbb {A}}^1$ induced by a subdivision

$$\begin{align*}\Delta\hookrightarrow \Sigma_X\times{\mathbb R}_{\geq 0}. \end{align*}$$

A rough expansion is called an expansion if, in addition, the following two conditions are satisfied

1. (E1) The scheme ${\mathcal Y}$ is reduced.

2. (E2) The subdivision is given by the cone over an embedded $1$ -complex in $\Sigma _X$ . Equivalently, at every point of ${\mathcal Y}$ , the relative characteristic monoid of ${\mathcal Y}\to \operatorname {\mathrm {Spec}} {\mathbb C}_{\mathbb N}$ is free of rank at most $1$ .

If S is a fine and saturated logarithmic scheme, a morphism ${\mathcal Y}/S\to X\times S/S$ is called an expansion of X over S if all pullbacks $\operatorname {\mathrm {Spec}} {\mathbb C}_{\mathbb N}\to S$ are expansions and if the morphism $\mathcal {Y} \to S$ is flat and logarithmically smooth.

We make a few comments about the definition. The first concerns (E2). We deal exclusively with subschemes of dimension at most $1$ in this paper, and the transversality condition that we will impose on these subschemes will ensure that the meet the strata of their ambient spaces in a dimensionally transverse fashion. Once a rough expansion is constructed, the codimension $2$ strata become irrelevant, and by removing these, one is led to more efficient moduli spaces. The second concerns the definition of an expansion over general bases S. For experts in logarithmic geometry, we note that one can replace the condition ‘flat and logarithmically smooth’ with ‘logarithmically smooth and integral’. Similarly, the reducedness of the fibers can be replaced with ‘saturated’. Finally, we note that although the expansions themselves tend to be nonproper, we always have a map from ${\mathcal Y}$ to the original target X, which has been assumed to be projective throughout. Moreover, we will only ever consider sheaves on ${\mathcal Y}$ with proper support, so for example, the support always defines a cycle in X by pushforward.

A $1$ -complex and the corresponding geometric expansion are shown in Figure 2.

Figure 2 On the left, we sketch a $1$ -complex in the cone complex $\mathbb R_{\geq 0}^2$ . The cone complex $\Sigma _X$ is ${\mathbb R}_{\geq 0}^2$ drawn with a dashed arrow while the $1$ -complex is solid. On the right is a cartoon of the corresponding expansion. It is obtained by performing deformation to the normal cone of a codimension $2$ stratum and then passing to an open. The wavy lines indicate the divisors where the logarithmic structure is nontrivial. The holes indicate codimension $2$ strata that have been removed from the expansion. The red curve in the middle is a subscheme of the type that we will soon introduce.

2.4 Flat limit algorithms: existence

We let $X^{\circ }$ be the complement of D in X. Let C be a smooth curve with a distinguished point $0$ , whose complement is denoted $C^{\circ }$ .

We will refer to a family satisfying these properties as a dimensionally transverse family. A sketch picture of dimensional transversality can be seen on the right in Figure 2. Notice in particular that in the main component $Y_0$ of the expansion shown there, the subscheme does not intersect the logarithmic divisors. Correspondingly, the $1$ -complex on the left of the image does not include the dashed axes.

Proof. Let

$$\begin{align*}\operatorname{\mathrm{Spec}} {\mathbb C}[\![t]\!] \to C \end{align*}$$

be the spectrum of the completed local ring of C at $0$ . Pull back the subscheme family ${\mathcal Z}^{\circ }$ to the generic point of this valuation ring to obtain a subscheme ${\mathcal Z}^\circ _{\eta }$ of $X^\circ $ over ${\mathbb C}(\!(t)\!)$ . After passing to a valued field extension with value group ${\mathbb R}$ , apply the tropicalization map to obtain

$$\begin{align*}\mathsf{trop}({\mathcal Z}^{\circ})\subset \Sigma_X. \end{align*}$$

By the structure result for tropicalizations recorded in Theorem 1.5.1, this is the support of a polyhedral complex of dimension at most $1$ in $\Sigma $ . Choose a polyhedral structure on this set, and call it G. In practice, this amounts to expressing the tropicalization as a union of vertices, closed bounded edges and unbounded rays. This results in a polyhedral complex embedded in $\Sigma _X$ . Let $\mathsf C({G})$ be the cone over G in $\Sigma _X\times {\mathbb R}_{\geq 0}$ . We obtain an associated target family

$$\begin{align*}{\mathcal Y}_G\to C. \end{align*}$$

By construction, the tropicalization of ${\mathcal Z}^{\circ }$ when viewed as a single subscheme of the total space $X^{\circ }\times C^{\circ }$ is equal to $\mathsf C({G})$ . The theorems in Section 1.4 imply that the closure is proper. Since the base is a smooth curve, the closure is also flat over C.

We examine the strata intersections. Given a stratum W of ${\mathcal Y}_G$ dual to a cone $\sigma $ in $\mathsf {C}(G)$ , choose a point $v\in \sigma ^{\circ }$ . By definition, this is the image of a K-valued point of ${\mathcal Z}^\circ $ under the tropicalization map. Since the family of subschemes is proper, this extends to an R-valued point where R is the valuation ring of K. Since the tropicalization of this K-valued point lies in the interior of $\sigma $ , the closed point maps to W, so it follows that ${\mathcal Z}$ intersects all strata.

Finally, we show that we can obtain a degeneration with reduced special fiber after a base change. By using the toric dictionary, we observe that the special fiber of ${\mathcal Y}_G\to C$ is reduced if and only if the vertices of G are lattice points in $\Sigma _X$ . This can be engineered by using Kawamata’s cyclic covering trick, explained in [Reference Abramovich and Karu7]. The toroidal procedure is carried out by replacing the integral lattice in $\Sigma _C$ with a finite index sublattice, thereby ensuring that the fiber over the new primitive generator of $\Sigma _C$ is a dilation of G which has integral vertices. The main result of [Reference Abramovich and Karu7, Section 5] is that this produces the requisite ramified base change. The statements concerning strata intersections are unaffected by the base change, and we conclude the result.

2.5 Flat limit algorithms: uniqueness

The tropical limit algorithm in the previous section inherits a uniqueness property that should be thought of as close to a universal closedness result for the forthcoming moduli problem, where by ‘close to’ we mean that we will need to eventually strengthen the notion of transversality.

Let K be the discretely valued field associated to the valuation at $0$ in C, and let R be its valuation ring.

Proposition 2.5.1. Let ${\mathcal Z}_{\eta }^{\circ }$ be a flat family of subschemes of $X^{\circ }$ over $\operatorname {\mathrm {Spec}} K$ . Assume that the closure of ${\mathcal Z}_{\eta }^{\circ }$ in $X\times \operatorname {\mathrm {Spec}} K$ is dimensionally transverse. Then there exists a canonical triple $(R',{\mathcal Y}',{\mathcal Z}')$ comprising of a ramified base change $R\subset R'$ with fraction field $K'$ and expansion of X

$$\begin{align*}{\mathcal Y}'\to \operatorname{\mathrm{Spec}} R' \end{align*}$$

such that the closure ${\mathcal Z}'$ of ${\mathcal Z}_{\eta }^{\circ }\otimes _K K'$ in ${\mathcal Y}$ is dimensionally transverse. Moreover, the triple satisfies the following uniqueness property:

$(\star )$ For any other choice $(R",{\mathcal Y}",{\mathcal Z}")$ satisfying these requirements, there exists a unique toroidal birational morphism

$$\begin{align*}{\mathcal Y}"\to {\mathcal Y}^{\prime\prime}_1 \end{align*}$$

over $\operatorname {\mathrm {Spec}} R"$ with subscheme ${\mathcal Z}^{\prime \prime }_1$ and a ramified covering $\operatorname {\mathrm {Spec}} R"\to \operatorname {\mathrm {Spec}} R$ such that ${\mathcal Z}^{\prime \prime }_1\subset {\mathcal Y}^{\prime \prime }_1$ is obtained from ${\mathcal Z}'\subset {\mathcal Y}'$ by base change.

We deduce the uniqueness results from two simple combinatorial observations.

The term minimal here is used in the sense that any other polyhedral structure is obtained from the putative unique minimal one by subdividing along edges. Note that a morphism of polyhedral complexes is required to map vertices and edges of G to into cones of $\Sigma _X$ .

Passing to the cone over $\overline G$ , we obtain a uniqueness property concerning the dilations of $\overline G$ obtained by uniformly scaling the edge lengths.

Proof of Proposition

The uniqueness is a translation of the combinatorial observations above, as we now explain. Let ${\mathcal Z}_{\eta }^{\circ }$ be a subscheme as in the proposition. The algorithm for finding limits has the following steps. We calculate the tropicalization of ${\mathcal Z}_{\eta }^{\circ }$ and obtain a canonical $1$ -complex $\overline G\hookrightarrow \Sigma _X$ . We then pass to the cone over this complex $\mathsf C(\overline G)$ in $\Sigma _X\times {\mathbb R}_{\geq 0}$ . Finally, we find the minimum positive integer b in ${\mathbb R}_{\geq 0}$ over which the fiber in $\mathsf C(\overline G)$ has integral vertices. We perform the order b cyclic ramified base change to obtain the requisite triple $(R',{\mathcal Y}',{\mathcal Z}')$ .

Now, consider another set $(R",{\mathcal Y}",{\mathcal Z}")$ satisfying these properties. Let $\mathsf C(G")\to {\mathbb R}_{\geq 0}$ be the fan associated to this expansion. Tropicalization is invariant under taking valued field extensions; since the subscheme ${\mathcal Z}"$ nontrivially intersects all the strata of the expansion, we can conclude using Theorem 1.6.1 that the support of the tropicalization is equal to $\mathsf C(G"),$ where $G"$ is the height $1$ slice of the cone. If we replace $G"$ with its minimal polyhedral structure $\overline G"$ , we obtain a refinement of cone complexes $\mathsf C(G")\to \mathsf C(\overline G")$ . The latter cone complex gives rise to an expansion of X in which the closure of ${\mathcal Z}_{\eta }^{\circ }$ is still dimensionally transverse.

For the base change, by using the preceding lemma we let $\overline G'$ be the minimal dilation of $\overline G$ whose vertices are all integral. The cones over $\overline G'$ and $\overline G"$ coincide, so $\overline G"$ must be a dilation of $\overline G'$ . It follows that the morphism

$$\begin{align*}\mathsf C(\overline G")\to {\mathbb R}_{\geq 0} \end{align*}$$

is obtained from

$$\begin{align*}\mathsf C(\overline G')\to {\mathbb R}_{\geq 0} \end{align*}$$

by passing to a finite index sublattice in the integral structure of the base. The result follows. $\Box $

Before proceeding to algebraic transversality, we offer two examples of the procedure above. We begin with a zero-dimensional example.

Example 2.5.4. Let X be the ${\mathbb {A}}^2$ equipped with its toric logarithmic structure. Consider the subscheme ${\mathcal Z}$ given by the following union of two reduced points defined over the field ${\mathbb C}(\!(t)\!)$ :

$$\begin{align*}{\mathcal Z} = \{(t^2,t^3)\} \cup \{(t^4,t^5)\}\subset {\mathbb{A}}^2. \end{align*}$$

The flat limit is a nonreduced point supported at the origin in ${\mathbb {A}}^2$ . The tropicalization of ${\mathcal Z}$ consists of two points

$$\begin{align*}\mathsf{trop}({\mathcal Z}) = \{(2,3)\}\cup \{(4,5)\} \subset {\mathbb R}^2_{\geq 0}. \end{align*}$$

According to the algorithm above, associated expansion is obtained by performing a toric modification of ${\mathbb {A}}^2\times {\mathbb {A}}^1$ , adding rays passing through points $(2,3,1)$ and $(4,5,1)$ in the fan ${\mathbb R}^2_{\geq 0}\times {\mathbb R}_{\geq 0}$ . The resulting expansion consists of two components, both isomorphic to ${\mathbb G}_m^2$ . The new flat limit in the expanded family is the union of two reduced points, with one in each of these two components.

We now consider a one-dimensional example.

Example 2.5.5. Let X be $\mathbb {P}^2$ equipped with its toric logarithmic structure and homogeneous coordinates X, Y and Z. Consider the subscheme ${\mathcal Z}$ given by a line defined over the field ${\mathbb C}(\!(t)\!)$ :

$$\begin{align*}{\mathcal Z} = \mathbb V(t^{-3}X+t^{-2}Y+Z) \subset \mathbb P^2. \end{align*}$$

The tropicalization is easily computed by hand as the subset of ${\mathbb R}^2$ obtained by the union of $3$ rays $\rho _1,\rho _2,\rho _3$ at the point $(-3,-2)$ . The ray $\rho _1$ is parallel to the negative x-axis, the ray $\rho _2$ is parallel to the negative y-axis and $\rho _3$ is parallel to the first-quadrant diagonal. It is the translate of the $1$ -skeleton of the fan of $\mathbb {P}^2$ to the point $(-2,-3)$ . The associated $1$ -complex G has two vertices: one vertex at the point $(-3,-2)$ which is trivalent and we call $V_1$ and another at the point $(-1,0)$ where the set $|G|$ intersects the $1$ -skeleton of the fan of $\mathbb {P}^2$ , which we call $V_2$ . See Figure 3

Figure 3 The $1$ -complex depicted in solid arrows with the fan of $\mathbb P^2$ in dashed arrows.

The associated expansion consists of two components corresponding to the two vertices above. The component $X_1$ corresponding to $V_1$ is a copy of $\mathbb P^2$ minus its three fixed points. The component $X_2$ corresponding to $V_2$ is a copy of $\mathbb P^1\times {\mathbb G}_m$ . A straightforward calculation shows that the limiting subscheme is a line in $X_1$ union a fiber in $X_2$ .

2.6 Algebraic transversality from dimensional transversality

We have constructed limits of transverse subschemes of X inside expansions of X along D that are dimensionally transverse to the strata of the expansion. We require a stronger form of transversality.

Definition 2.6.1 (Algebraic transversality).

Let ${\mathcal Y}$ be an expansion of X, and let ${\mathcal Z}\subset {\mathcal Y}$ be a subscheme with ideal sheaf $\mathcal I_{{\mathcal Z}}$ . Then ${\mathcal Z}$ is said to be algebraically transverse if it is dimensionally transverse and for every closed divisor stratum $\mathcal S\subset {\mathcal Y}$ , the induced map

$$\begin{align*}\mathcal I_{{\mathcal Z}}\otimes_{{\mathcal O}_{{\mathcal Y}}} {\mathcal O}_{\mathcal S}\to {\mathcal O}_{{\mathcal Y}}\otimes_{{\mathcal O}_{{\mathcal Y}}} {\mathcal O}_{\mathcal S} \end{align*}$$

is injective.

We call this algebraic transversality because it depends on more than just the dimension of the intersection with the strata, which is only set theoretic. On the other hand, it is weaker than genuine geometric transversality because it allows the intersection of the subscheme with a divisor to be nonreduced.Footnote ⁵

Let us give a few different ways to think about this condition. We maintain the notation above.

A more conceptual characterization comes from the proof of the proposition above. Recall that $\Sigma _X$ is a union of faces in the standard orthant fan ${\mathbb R}_{\geq 0}^d$ . Let ${\mathsf A}_{\Sigma }$ be the associated union of orbits in the Artin stack $[{\mathbb {A}}^d/{\mathbb G}_m^d]$ . The expansion ${\mathcal Y}\to X$ is pulled back from an associated expansion ${\mathsf A}_{{\mathcal Y}}$ . We stress that this is a gluing together of Artin fans along divisors.

Via the proposition above, algebraic transversality is equivalent to the logarithmic flatness of ${\mathcal Z}$ , when equipped with the pullback logarithmic structure from the expansion, together with the condition all strata have nonempty intersection with ${\mathcal Z}$ . The perspective coming from logarithmic flatness play a key role in Kennedy-Hunt’s work on the general logarithmic Hilbert scheme [Reference Kennedy-Hunt41].

Finally, we note that by basic properties of tensor products, algebraic transversality is equivalent to the condition that the higher Tor functors of ${\mathcal O}_Z$ with ${\mathcal O}_S$ vanish. This is how transversality is stated by Li–Wu, who refer to the condition as normality to the divisor; see [Reference Li and Wu47].

Proposition 2.6.5. Let ${\mathcal Z}_{\eta }^{\circ }$ be a flat family of subschemes of $X^\circ $ over $\operatorname {\mathrm {Spec}} K$ whose closure in X is algebraically transverse. Then there exists a canonical triple $(R',{\mathcal Y}',{\mathcal Z}')$ comprised of a ramified base change $R\subset R'$ with fraction field $K'$ and expansion of X

$$\begin{align*}{\mathcal Y}'\to \operatorname{\mathrm{Spec}} R' \end{align*}$$

such that the closure ${\mathcal Z}'$ of ${\mathcal Z}_{\eta }^{\circ }\otimes _K K'$ in ${\mathcal Y}$ is algebraically transverse, satisfying the following uniqueness property:

$(\star )$ For any other choice $(R",{\mathcal Y}",{\mathcal Z}")$ satisfying these requirements, there exists a unique toroidal birational morphism

$$\begin{align*}{\mathcal Y}"\to {\mathcal Y}^{\prime\prime}_1 \end{align*}$$

over $\operatorname {\mathrm {Spec}} R"$ with subscheme ${\mathcal Z}^{\prime \prime }_1$ , and a ramified covering $\operatorname {\mathrm {Spec}} R"\to \operatorname {\mathrm {Spec}} R$ such that ${\mathcal Z}^{\prime \prime }_1\subset {\mathcal Y}^{\prime \prime }_1$ is obtained from ${\mathcal Z}'\subset {\mathcal Y}'$ by base change.

2.6.2 A proof via Gröbner theory

We take a different approach, inspired by Gröbner theory considerations. The approach has a slightly more constructive nature.

The strategy is to reduce to a theorem of Tevelev on subvarieties of toric varieties [Reference Tevelev71]. Specifically, if X is a toric variety with dense torus T, Tevelev proves that, for any subscheme $Z\hookrightarrow X$ , there exists a toric blowup $X'\to X$ , with $X'$ smooth such that the strict transform $Z'\hookrightarrow X'$ is what he calls tropical [Reference Tevelev71, Theorem 1.2]. This means that the induced map $Z\to [X'/T]$ is flat. When this theorem is applied to a flat family of subschemes in X of relative dimension $1$ , over ${\mathbb {A}}^1$ , the output is precisely the algebraically transverse family that we seek.

We will reduce our more general situation to the toric one. The reduction is essentially straightforward; once dimensional transversality has been achieved, the additional birational modifications required to produce an algebraically transverse family can be analyzed affine locally. The details follow.

We will use the following lemma in the course of our proof, which allows us to take an algebraically transverse limit and contract unnecessary blowups.

Lemma 2.6.6 (Contraction lemma).

Let ${\mathcal Y}\to \operatorname {\mathrm {Spec}} R$ be a expansion of X whose special fiber includes two irreducible components $Y_1$ and $Y_2$ that meet transversely along a smooth divisor D. Let ${\mathcal Y}'\to {\mathcal Y}$ be the blowup at D with exceptional E. Assume that the special fiber of ${\mathcal Y}'$ is reduced .

Suppose ${\mathcal Z}'\hookrightarrow {\mathcal Y}'$ is a flat and proper family of algebraically transverse subschemes of dimension $1$ , with generic fiber ${\mathcal Z}^{\prime }_{\eta }$ , and such that ${\mathcal Z}'\cap E$ is the pullback of a subscheme along $E\to D$ .

Let ${\mathcal Z}$ be the closure of the generic fiber ${\mathcal Z}_{\eta }$ in ${\mathcal Y}$ . Then ${\mathcal Z}$ is a flat family of algebraically transverse subschemes.

Proof. Throughout the proof, we add the subscript $0$ to indicate special fibers. Suppose ${\mathcal Z}$ were not algebraically transverse. By the characterization of algebraic transversality, this means that the special fiber ${\mathcal Z}_0\hookrightarrow {\mathcal Y}_0$ has embedded points contained in D. Since ${\mathcal Z}$ is the closure of its generic fiber, it is flat over $\operatorname {\mathrm {Spec}} R$ . It follows that the special fibers of ${\mathcal Z}^{\prime }_0$ and ${\mathcal Z}_0$ have the same holomorphic Euler characteristic. However, this now gives a numerical contradiction. Since ${\mathcal Z}^{\prime }_0$ is algebraically transverse, its holomorphic Euler characteristic is given by

$$\begin{align*}\chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0) = \chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap Y_1)+\chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap Y_2)+\chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap E)- \chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap E\cap Y_1)-\chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap E\cap Y_2). \end{align*}$$

But we also have

$$\begin{align*}\chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap E\cap Y_i) = \chi_{\mathsf{hol}}({\mathcal Z}^{\prime}_0\cap E) \end{align*}$$

since ${\mathcal Z}^{\prime }_0\cap E$ is a $\mathbb P^1$ -bundle over ${\mathcal Z}^{\prime }_0\cap E\cap Y_i$ .

Now, the intersection of ${\mathcal Z}'$ with $(Y_1\cup Y_2)\setminus E$ maps isomorphically onto its image in the complement of D in ${\mathcal Y}_0$ . On the one hand, the closure of this image cannot contain the embedded points above and is a closed subscheme, properly contained in ${\mathcal Z}_0$ . On the other hand, the Euler characteristic of this closure is equal to that of ${\mathcal Z}^{\prime }_0$ above. Since the embedded points contribute positively to the holomorphic Euler characteristic of ${\mathcal Z}_0$ , we arrive at a contradiction.

We move on to the main proof. We first identify the subdivision that is required to produce an algebraically transverse limit and then prove that it has the properties required. We require the notion of an initial degeneration from tropical geometry. In the notation of the proposition, let ${\mathcal Z}_{\eta }^{\circ }$ be a flat family of subschemes of $X^{\circ }$ over $\operatorname {\mathrm {Spec}} K$ . Then we have

$$\begin{align*}\mathsf{trop}({\mathcal Z}_{\eta})\subset \Sigma_X \end{align*}$$

which is the support of a one-dimensional polyhedral complex embedded in $\Sigma _X$ . Given a rational point w in $\Sigma _X$ , we consider a nonproper degeneration of X as follows. Consider the linear map

$$\begin{align*}{\mathbb R}_{\geq 0}\to \Sigma_X\times{\mathbb R}_{\geq 0} \end{align*}$$

mapping isomorphically onto the ray joining the origin to $(w,1)$ . This determines an associated flat degeneration

$$\begin{align*}{\mathcal Y}_w\to \operatorname{\mathrm{Spec}}(R) \end{align*}$$

whose general fiber is the interior $X^{\circ }$ of X and whose special fiber is a torus torsor over a locally closed stratum of X and, precisely, the locally closed stratum corresponding to the cone that contains w in its interior. We denote the special fiber by $X_w$ . We note, in particular, that, up to isomorphism, the scheme $X_w$ is constant when w varies in the interior of each cone of $\Sigma _X$ .

Definition 2.6.7. The initial degeneration of ${\mathcal Z}_{\eta }$ at a point w of $\Sigma _X$ is the special fiber of the closure of ${\mathcal Z}_{\eta }\cap X^{\circ }$ in ${\mathcal Y}_w$ , viewed as a subscheme of $X_w$ . It will be denoted $\mathsf {in}_w({\mathcal Z}_{\eta })$ .

Recall from the previous section that $\mathsf {trop}(Z_{\eta })$ has a minimal polyhedral structure, and this determines a degeneration ${\mathcal Y}_{\Gamma }$ . Let w be a point on an edge e of this minimal polyhedral structure. Then $X_w$ is a ${\mathbb G}_m$ -torsor over a codimension $1$ stratum $X_e$ in the special fiber of the associated degeneration. We will say that the initial degeneration $\mathsf {in}_w({\mathcal Z}_{\eta })$ is tubular if it is the schematic preimage of a zero-dimensional subscheme under the projection $X_w\to X_e$ .

The following basic finiteness theorem holds.

Proposition 2.6.8. The subset of points w in $\mathsf {trop}({\mathcal Z}_{\eta })$ with the property that $\mathsf {in}_w({\mathcal Z}_{\eta })$ is not tubular is finite.

Proof. In the notation of the preceding paragraph, if we are given a finite set of points in the set $\mathsf {trop}(Z_{\eta })$ , there is a unique coarsest polyhedral structure on the tropicalization that includes these points as vertices. It of course refines the minimal polyhedral structure on $\mathsf {trop}(Z_{\eta })$ . We refer to the vertices of this new polyhedral structure that do not appear in the minimal one as extra vertices. They should be viewed as lying in the interior of an edge in the minimal structure.

By the paragraph above and the dictionary explained in the background section, each such choice of finite set produces a model of X over the DVR. We assume it has reduced special fiber by performing an appropriate base change. The closure of $Z_{\eta }$ in this model is a flat family of subschemes.

Let w be an extra vertex. The component it is dual to is a $\mathbb P^1$ -bundle. Since the Hilbert polynomial of the subscheme is constant in the family, for all but finitely many choices of w in $\mathsf {trop}(Z_{\eta })$ , the curve class of the subscheme in this component is a fiber class. Similarly, since the holomorphic Euler characteristic is fixed, we find that $\mathsf {in}_w({\mathcal Z}_{\eta })$ must be tubular for all but finitely many choices of w.

We are now in a position to construct an algebraically transverse limit.

Construction 2.6.9 (The Gröbner limit).

Let $\mathscr G$ denote the unique minimal polyhedral structure on $\mathsf {trop}({\mathcal Z}_{\eta })$ whose vertices include all points w in $\mathsf {trop}({\mathcal Z}_{\eta })$ whose initial degeneration is not tubular. Consider the associated rough expansion $\mathcal V_{\mathscr G}\to \operatorname {\mathrm {Spec}}(R)$ . Perform a ramified base change $R\subset R'$ of order equal to the least common multiple of the denominators appearing in the coordinates of the vertices of $\mathscr G$ . Let ${\mathcal Y}_{\mathscr G}\to \operatorname {\mathrm {Spec}}(R')$ be the associated expansion. Let ${\mathcal Z}\to \operatorname {\mathrm {Spec}}(R')$ be the new flat limit, obtained as the closure of ${\mathcal Z}_{\eta }$ after extension of scalars.

We now show that this Gröbner limit is algebraically transverse.

Lemma 2.6.10. The Gröbner limit family ${\mathcal Z}\to \operatorname {\mathrm {Spec}}(R')$ is a family of algebraically transverse subschemes of ${\mathcal Y}_{\mathscr G}$ .

Proof. Consider the Gröbner family of subschemes ${\mathcal Z}\hookrightarrow {\mathcal Y}_{\mathscr G}$ over $\operatorname {\mathrm {Spec}}(R')$ as above. The family is certainly dimensionally transverse, as we have already shown, and we assume that the generic fiber is algebraically transverse. Algebraic transversality is therefore equivalent to the condition that there are no embedded points on the double locus or the divisorial strata in the special fiber. We treat the case of the double locus; the other case is essentially identical.

The condition may be checked Zariski locally on the subscheme ${\mathcal Z}$ and depends only on the map from ${\mathcal Z}$ to the Artin fan of the target ${\mathcal Y}_{\mathscr G}$ . By shrinking to a neighborhood U around the double divisor and denoting the subscheme by ${\mathcal Z}_U$ , each divisor may be assumed to be principal. Choose generators for the coordinate ring of U that include the equations for these principal divisors, to obtain a maps

$$\begin{align*}{\mathcal Z}_U\to \mathbb A^n_R\to \mathbb A^2_R, \end{align*}$$

where the second arrow is given by projection onto the first two coordinates. Note that with respect to the logarithmic structure on $\mathbb A^2_R$ the tropicalization of ${\mathcal Z}_U$ is a single edge, and we henceforth replace ${\mathscr G}$ to be this edge. Moreover, the logarithmic structure induced on ${\mathcal Z}_U$ by pullback from ${\mathcal Y}_{\mathscr G}$ is the pullback of the logarithmic structure coming from the coordinate axes of $\mathbb A^2$ . Our task is check flatness of the induced map

$$\begin{align*}{\mathcal Z}_U\to \left[{\mathbb{A}}^2/{\mathbb G}_m^2\right]. \end{align*}$$

We now reduce to the toric case. Choose $n-2$ generic hyperplanes and add them to the logarithmic structure of $\mathbb A^n$ above; this logarithmic structure is now toric. This gives rise to a factorization of the above map as

(1) $$ \begin{align} {\mathcal Z}_U\to \left[{\mathbb{A}}^n/{\mathbb G}_m^n\right]\to \left[{\mathbb{A}}^2/{\mathbb G}_m^2\right]. \end{align} $$

The second map is certainly flat, so we need only check flatness for the first.

With the subscheme now embedded in a toric variety, with its toric logarithmic structure, we may now appeal to the methods of Gröbner theory and tropical geometry, in the sense of [Reference Gubler28, Section 12] or [Reference Maclagan and Sturmfels48, Section 6.4]. Since the affine hyperplanes are generic, the closures of the initial degenerations of ${\mathcal Z}_U$ are constant on the faces of the tropicalization of ${\mathcal Z}_U$ . In order to see this, we note that the tropicalization of ${\mathcal Z}_U$ in the new embedding is obtained from ${\mathscr G}$ by adding unbounded rays to each of its two vertices. Note that, since the affine hyperplanes are generic, the initial degenerations at the points of these unbounded rays are tubular and are constant along these new rays.

We apply the result of [Reference Gubler28, Theorem 12.3]. This constructs (nonuniquely) a Gröbner complex ${\mathscr G}'$ on the underlying set of ${\mathscr G}$ , which is an explicit polyhedral structure on the tropicalization of ${\mathcal Z}_U$ . The associated degeneration ${\mathcal Y}_{{\mathscr G}'}$ of the generic fiber ${\mathcal Y}_{{\mathscr G},\eta }$ has the property that (i) the initial degenerations are constant along faces, and (ii) the strict transform of ${\mathcal Z}_U$ in the associated degeneration is algebraically transverse.Footnote ⁶

The degenerations ${\mathcal Y}_{{\mathscr G}'}$ and ${\mathcal Y}_{\mathscr G}$ can only differ by further subdivision of edges in the tropicalization; call these new $2$ -valent vertices extra vertices. By algebraic transversality of the closure of ${\mathcal Z}_U$ in ${\mathcal Y}_{{\mathscr G}'}$ , if we restrict the subscheme to the components corresponding to the extra vertices, the resulting subscheme is pulled back along the blowup ${\mathcal Y}_{{\mathscr G}'}$ and ${\mathcal Y}_{\mathscr G}$ . We now apply Lemma 2.6.10 to erase all the extra vertices and see that the closure in ${\mathcal Y}_{\mathscr G}$ is already algebraically transverse. This is equivalent to the flatness of the first arrow in Equation (1), and the proof of the lemma is complete.

We now establish the uniqueness property that we need.

Lemma 2.6.11. The Gröbner limit family ${\mathcal Z}\to \operatorname {\mathrm {Spec}}(R')$ satisfies the uniqueness property $(\star )$ of Proposition 2.6.5.

Proof. Consider another family ${\mathcal Z}'\to \operatorname {\mathrm {Spec}}(R")$ that is algebraically transverse obtained from a polyhedral structure on $\mathsf {trop}(Z_{\eta })$ . The first claim is that if a point w on $\mathsf {trop}(Z_{\eta })$ has nontubular initial degeneration, then it must appear in the vertex set of any algebraically transverse family. To see this, first observe that if ${\mathcal Z}'$ is an algebraically transversely family of subschemes in ${\mathcal Y}$ , then if we blowup ${\mathcal Y}$ further, then the strict and total transforms of ${\mathcal Z}'$ must coincide. However, the strict transform is precisely the flat limit of ${\mathcal Z}_{\eta }$ in the new family.

With this in mind, we consider a polyhedral structure $\Lambda $ for which there is a point w not contained in the vertex set of such that the initial degeneration at w is not tubular. Form a new polyhedral structure $\Lambda '$ by introducing this point w as part of the vertex set. Examine the total transform of ${\mathcal Z}$ under the blowup ${\mathcal Y}_{\Lambda '}\to {\mathcal Y}_{\Lambda }$ restricted to the locally closed stratum corresponding to this vertex w. By algebraic transversality, it must coincide with the initial degeneration at w, but the total transform is necessarily tubular since it is the preimage of the intersection of ${\mathcal Z}$ with a double divisor. We arrive at a contradiction.

The reducedness condition for an expansion ${\mathcal Y}_{\Lambda '}$ is that every vertex of $\Lambda '$ must be integral. There is a unique minimal set of vertices – precisely those with nontubular initial degenerations – that must be included in every algebraically transverse model. It follows that the order of every base change for the Gröbner limit divides the order of the base change for every other algebraically transverse model. The uniqueness statement is a consequence.

Remark 2.6.12. We have the following important consequence of the algorithm for producing algebraically transverse limits. If $G \hookrightarrow \Sigma _X$ denotes the $1$ -complex associated to the minimal algebraically transverse limit, G may involve refining the minimal polyhedral structure $\overline G$ by adding bivalent vertices to subdivide edges. However, if we look at the irreducible components $Y_i$ corresponding to these bivalent vertices, the embedded subscheme $Z_i \subset Y_i$ is stable, that is, not fixed by the natural $\mathbb {G}_m$ -action on $Y_i$ .

With the results of this section as motivation, we define a subscheme of X relative to D.

Definition 2.6.13 (Relative subschemes).

Let S be a logarithmic scheme. A subscheme of X relative to D over S is an expansion ${\mathcal Y}/S$ of X over S together with a flat family of algebraically transverse subschemes ${\mathcal Z}\subset {\mathcal Y}$ .

Proposition 2.6.5 suggests that algebraically transverse subschemes will satisfy the valuative criterion for properness. Our goal is now to construct moduli for these objects – first for the expansions themselves which happens in the next section and then for the algebraically transverse subschemes which happens in the one after.

3.1 Artin fans and cone stacks

The stack $\mathsf {Exp}(X|D)$ is similar in nature to Olsson’s stack $\mathsf {LOG}$ of logarithmic structures and are instances of the Artin fans that are extracted from $\mathsf {LOG}$ ; see [Reference Abramovich and Wise9, Reference Olsson62].

An Artin cone is a global toric quotient stack:

$$\begin{align*}\mathsf{A}_P = \operatorname{\mathrm{Spec}} {\mathbb C}[P]/\operatorname{\mathrm{Spec}} {\mathbb C}[P^{\mathsf{gp}}], \end{align*}$$

where P is a toric monoid. These stacks are endowed with a logarithmic structure in the smooth topology from the toric logarithmic structure on $\operatorname {\mathrm {Spec}} {\mathbb C}[P]$ . Observe that $\mathsf {A}_P$ recovers P from its logarithmic structure, so these data naturally determine each other.

In practice, this means that we are interested in algebraic stacks that arise by gluing stacks of the form ${\mathsf A}_P$ along open substacks corresponding to torus invariant subvarieties of $\operatorname {\mathrm {Spec}} {\mathbb C}[P]$ . Note that we do not impose the faithful monodromy condition on Artin fans, but our constructions are explicit rather than conceptual, and in practice all of our Artin fans will be monodromy free.

Under mild assumptions, every logarithmic structure on a scheme arises from a morphism to an Artin fan [Reference Abramovich, Chen, Marcus, Ulirsch, Wise, Baker and Payne5, Proposition 3.2.1]. In practice, it is convenient pass to an equivalent but explicitly combinatorial $2$ -category, for which we follow [Reference Cavalieri, Chan, Ulirsch and Wise16, Sections 2.1–2.2].

A cone complex is obtained when the further assumption is placed that there is at most one morphism between any two cones. A cone stack can be defined similarly, as a fibered category over the category of rational polyhedral cones satisfying natural conditions. We will not need to work with cone stacks in a serious way, so we refer to the reader to [Reference Cavalieri, Chan, Ulirsch and Wise16, Section 2] for a definition. Their results allow us to systematically work with Artin fans using combinatorics. In fact, any cone space can be subdivided into a cone complex, so even cone spaces can be avoided for our purposes. However, it can be convenient to allow them to construct minimal choices of stacks of target expansions; for example, the stack of expansions in [Reference Li and Wu47] arises from a cone space.

Under this equivalence, an Artin cone $\mathsf {A}_P$ is carried to the cone $\operatorname {Hom}(P,{\mathbb R}_{\geq 0})$ . If $\Sigma $ is a fan embedded in a vector space and $W_{\Sigma }$ is the resulting toric variety with dense torus ${\mathbb G}_{\Sigma }$ , the equivalence carries the global quotient Artin fan $[W_{\Sigma }/{\mathbb G}_{\Sigma }]$ to the cone complex $\Sigma $ .

3.2 Embedded complexes

Section 2 constructed expansions of X along D that accommodate limits of transverse subschemes. The construction was presented over valuation rings with logarithmic structure. In order to globalize this, we construct moduli for the tropical data over higher-dimensional cones and then glue these cones to form a cone space. By the categorical equivalence in Theorem 3.1.3, we will have produced $\mathsf {Exp}(X|D)$ as an Artin stack.

Recall that $\Sigma $ is a cone complex with smooth cones such that the intersection of any two cones is a face of each, and we view it as a subcomplex of the standard orthant by the embedding

$$\begin{align*}\Sigma\hookrightarrow {\mathbb R}_{\geq 0}^{\Sigma^{(1)}}. \end{align*}$$

We will make use of the Euclidean geometry of this ambient vector space.

In Section 2.2 we defined an abstract $1$ -complex to be a possibly disconnected metric graph G together with polyhedral complex structure. An embedded $1$ -complex in $\Sigma $ , or $1$ -complex for short, was an injective piecewise affine map of polyhedral complexes

$$\begin{align*}G\hookrightarrow \Sigma \end{align*}$$

given by piecewise affine functions of slope $1$ . Let $|{\mathsf {T}(\Sigma )}|$ be the set of isomorphism classes of $1$ -complexes G equipped with an embedding

$$\begin{align*}G\hookrightarrow \Sigma. \end{align*}$$

For each embedded $1$ -complex, we can dilate the embedded complex by any positive real scalar, giving rise to an copy of ${\mathbb R}_{\geq 0}$ in $|{\mathsf {T}(\Sigma )}|$ . We will endow this set with the structure of a cone space, whose rays will be a subset of the rays obtained in this fashion.

3.3 Construction of cones of embedded graphs

By the conventions in this text, a graph refers to a possibly disconnected finite graph without loop or parallel edges, together with a finite collection of rays placed at the vertices. As is standard in Gromov–Witten theory, we visualize the rays as ‘legs’.

In order to put a cone space structure on $|{\mathsf {T}(\Sigma )}|$ , we first study spaces of maps from graphs to $\Sigma $ and then identify maps with the same image. For now, we suppress discussion of the integral structure and fix the correct integral lattice afterwards.

We define a linear path in G to be a path (consisting of edges and/or rays) where every edge/ray has the same direction vector. Let $P(G)$ denote the set of linear paths in G.

An isomorphism of combinatorial $1$ -complexes is an isomorphism of the underlying graphs that is compatible with the labels. Fix a representative for each isomorphism class. We consider embeddings of a combinatorial $1$ -complex into $\Sigma $ . Let $|\Sigma |$ denote the support of the cone complex. Define the set $\mathbb X_G$ to be the set of functions

$$\begin{align*}f: V(\underline G)\to |\Sigma| \end{align*}$$

such that

1. (1) For each $v\in V(\underline G)$ , the image $f(v)$ lies in $\sigma _v$ .

2. (2) For each $e\in E(\underline G)$ with adjacent vertices v and W, the line segment between $f(v)$ and $f(w)$ has direction vector equal to the one labeling e.

Each such function $f\in \mathbb X_G$ determines a polyhedral complex in $\underline \Sigma $ . It can be constructed as follows. Take the collection of points obtained as images of vertices of $\underline G$ under f. Given an edge e between v and w introduce a segment between $f(v)$ and $f(w)$ , whose edge direction is necessarily the one given by e. For each ray, glue an unbounded ray with the corresponding base point and edge direction dictated by G. This topological space has a minimal polyhedral complex structure whose vertices are of two types: (1) images of vertices of $\underline G$ under f and (2) points where edges or rays intersect.

We refer to this embedded complex as the image,Footnote ⁷ and denote it by $\mathsf {im}(f)$ . We restrict to those G such that there exists at least one function $f\in \mathbb X_G$ such that $\mathsf {im}(f)$ is isomorphic to the complex $|\underline G|$ , the latter with its canonical polyhedral structure.

As mentioned above, each point $f\in \mathbb X_G$ determines an embedded $1$ -complex $\mathsf {im}(f)$ . Our primary interest is in embedded complexes, so we would like to replace the space $\mathbb X_G$ with the corresponding set of images. However, the topological type of the image is not constant even for points in the relative interior of a cone in $\mathbb X_G$ . A key example is provided by two skew edges in a three-dimensional cone that intersect in their interiors. We record the example explicitly to illustrate the behavior.

Example 3.3.3. Consider the family of embedded $1$ -complexes in $\mathbb R^3_{\geq 0}$ parameterized by a cone $\sigma $ which we identify with ${\mathbb R}_{\geq 0}^2$ and furnish with coordinates $(s,t)$ . At a point $(s,t)$ , the associated $1$ -complex is a union of two components $\rho _1(s,t)$ and $\rho _2(s,t)$ . At the point $(s,t)$ , the ray $\rho _1(s,t)$ is the unbounded ray connecting $(0,0,s)$ and $(1,1,s)$ . At the point $(s,t)$ , the segment $\rho _2(s,t)$ is the line segment that connects $(t,0,0)$ and $(0,t,t)$ . The two segments intersect along a codimension $1$ subspace in the interior of the base cone $\sigma $ . The total spaces of $\rho _1(s,t)$ and $\rho _2(s,t)$ , taken indivisually, form two-dimensional cones in the four-dimensional space ${\mathbb R}^3_{\geq 0}\times \sigma $ . However, their union is not a cone complex, as their intersection is not a face of either. For each fixed value of $(s,t)$ , there is a minimal polyhedral structure on the union $\rho _1(s,t)\cup \rho _2(s,t)$ . The combinatorial type, and in particular the face poset, of this fiber can change in the interior of the cone $\sigma $ . The figure below displays a slice of this picture; see Figure 4

Figure 4 The figure shows a family of embedded $1$ -complexes in $\mathbb R^3$ over a one-dimensional base. The $1$ -complex on the left corresponds to a generic point, but at $t = 0$ the two skew edges cross forming a new vertex. As a consequence, the total space has two two-dimensional cones meeting at a point in their interior. In order to obtain a total space that, after taking cones, is a cone complex, vertices have to be added to $e_1$ and $e_2$ .

The topological changing in the interior of a cone, as in the above example, can be thought of as a nonflatness of the family of $1$ -complexes. The following definition is meant to capture this phenomenon of varying topological type. The topological type can also change due to specialization, for example, a vertex moving from a cone to a face. But it is the changes of topological type in the interior that is the exotic phenomenon, as compared to earlier moduli problems [Reference Abramovich, Caporaso and Payne1, Reference Cavalieri, Chan, Ulirsch and Wise16, Reference Gross and Siebert27].

Definition 3.3.4. Let G and H be combinatorial $1$ -complexes in $\Sigma _X$ . A surjection from G to H is given by the a pair of maps

$$ \begin{align*} \tau: V(\underline G)&\ \ \to\ \ V(\underline H)\\ \upsilon: E(\underline G)\sqcup R(G)&\ \ \to\ \ P(\underline H) \end{align*} $$

subject to the following conditions for:

1. (1) If v and w are the vertices adjacent to e, then $\tau (v)$ and $\tau (w)$ are the endpoints of the path $\upsilon (E)$ .

2. (2) Let e be an edge of $\underline G$ . Orient the edges of $\nu (e)$ to form a directed path and choose the orientation of E compatibly, that is, if e is directed from endpoint v to endpoint w then $\nu (e)$ is oriented from $\tau (v)$ to $\tau (w)$ . Then,

   If e is an edge of $\underline G$ , then the edge directions for each edge in $\upsilon (e)$ coincide with that of e.

3. (3) For each vertex of $\underline G$ , the cone $\sigma _{\tau (v)}$ is equal to a face of $\sigma _v$ .

4. (4) Each edge or ray in H is contained in a path which is in the image of $\upsilon $ .

Each point $f\in \mathbb X_G$ determines a surjection of G, as follows. Given a point $f\in \mathbb X_G$ , we consider graph associated to the image $\mathsf {im}(f)$ , considered with its natural structure as a combinatorial $1$ -complex. A basic finiteness result holds.

Proof. We may view the cone $\mathbb X_G$ as a cone of tropical maps with type G to $\Sigma $ in the sense of [Reference Gross and Siebert27, Remark 1.21]. It comes equipped with a moduli diagram

The fibers of the vertical map are the metrizations of G given by endowing each edge e with length equal to the distance in $\Sigma $ between the images of its endpoints. Since the horizontal map is linear on cones, the image of $\mathbf G$ in $\Sigma \times \mathbb X_G$ can be given the structure of a cone complex. Choose any such cone complex structure $\mathsf {im}(\mathbf G)$ . After subdividing this image further and replacing $\mathbb X_G$ by a subdivision, the induced map

$$\begin{align*}\widetilde {\mathbf G}\to \widetilde {\mathbb X}_G \end{align*}$$

is flat in the combinatorial sense – every cone of the source surjects onto a cone of the image. This follows from [Reference Abramovich and Karu7, Section 4]. After this subdivision, the combinatorial type of the $1$ -complex in the fibers in the relative interior of any cell is constant. It follows that there are only finitely many surjections that appear from taking image, as claimed.

Let G be as above, and let H be a surjection of G associated to a point in $\mathbb X_G$ . Note that we will only consider surjections associated to points of cones $\mathbb X_G$ . Let $\mathbb X_H$ be the cone associated to the combinatorial $1$ -complex H. There is an associated inclusion

$$\begin{align*}\mathbb X_H\hookrightarrow \mathbb X_G. \end{align*}$$

Specifically, a point of $\mathbb X_H$ in particular determines a function on $V(\underline G)$ by using the map on vertex sets $V(\underline G)\to V(\underline H)$ .

Lemma 3.3.6. Let $G\to H$ and $G\to K$ be two surjections of G obtained from points of $\mathbb X_G$ by the image construction. Consider a point

$$\begin{align*}f\in \mathbb X_H\cap \mathbb X_K\subset \mathbb X_G. \end{align*}$$

The surjection $G\to J$ determined by f is a common surjection of H and K.

Fix a combinatorial $1$ -complex G and let $\mathsf {Aut}_G$ be the automorphism group. There is a natural action of $\mathsf {Aut}_G$ on the cone $\mathbb X_G$ . Our goal will be construct an automorphism-equivariant subdivision of $\mathbb X_G$ such that the subsets $\mathbb X_H$ for surjections H of G become subcomplexes. The following lemma will be of use to us.

Proof. Let $N_{\mathbb R}$ denote the associated vector space of the cone $\mathbf C$ . We consider $\mathbf F$ and $\mathbf C$ as subsets of this vector space. The existence of equivariant completions for toric varieties guarantees that there exists a complete fan $\Delta $ in $N_{\mathbb R}$ that contains $\mathbf F$ as a subcomplex [Reference Ewald and Ishida23]. Consider the common refinement of $\Delta $ and  $\mathbf C$ :

$$\begin{align*}\mathbf C' = \{\sigma_1\cap\sigma_2: \sigma_1\in \Delta \ \ \sigma_2\in \mathbf C\}. \end{align*}$$

This is a cone complex structure on the intersection of the supports of the two fans, that is on $|\mathbf C|$ ; see [Reference Maclagan and Sturmfels48, Section 2.3]. It clearly contains $\mathbf F$ as a subcomplex. The subdivision is not $\Gamma $ equivariant. We fix this by averaging over the group. Given $\gamma \in \Gamma $ , the set of translates of cones in $\mathbf C'$ form a cone complex with support $|\mathbf C|$ . The common refinement over all group elements gives rise to the subdivision $\widetilde {\mathbf C}$ as claimed in the lemma. Since $\mathbf F$ is $\Gamma $ -stable, this last common refinement step does not subdivide it. The final statement on smoothness follows from the fact that toric resolution of singularities for any fan can be performed without subdividing smooth cones [Reference Cox, Little and Schenck20, Theorem 11.1.9] and the fact that these subdivisions can be made $\Gamma $ equivariant [Reference Abramovich and Wang8, Section 2].

Note that this procedure typically involves choices and is not canonical. Our application is to the cones associated to combinatorial one complexes.

Lemma 3.3.8. For each combinatorial $1$ -complex G, there exists an $\mathsf {Aut}_G$ -equivariant subdivision $\mathbb Y_G\to \mathbb X_G$ such that for each surjection $G\to H$ induced by a point of $\mathbb X_G$ , the induced map

$$\begin{align*}\mathbb Y_H\hookrightarrow \mathbb Y_G \end{align*}$$

is an inclusion of a subcomplex. If $\mathbb Y_H$ is smooth, then $\mathbb Y_G$ can be chosen to be smooth.