Skeletons of stable maps II: superabundant geometries

1.1 Statement of results

All valued fields appearing in this paper will have equicharacteristic zero. Throughout, the symbol will be used to denote an abstract tropical curve.¹ We say that a parameterized tropical curve is realizable if there exists a smooth curve C over a nonarchimedean field and a map \([C\rightarrow \mathbb G_m^n]\) whose tropicalization is .

Our next result extends the reach of Speyer’s well-spacedness condition to the case of elliptic curves with good stable reduction, see Definition 3.2. Let be a tropical curve of genus 1 with a unique genus 1 vertex, so in particular the underlying graph of is a tree. Denote by the tropical curve obtained by first replacing the genus 1 vertex with a genus 0 vertex and then adding a self-loop of length 1 at v.

The work of Speyer shows that the well-spacedness condition is sufficient for realizability of tropical genus 1 curves. He also proves that this condition is also necessary, with the restriction that the curve is trivalent. The following result shows that outside the trivalent case, well-spacedness can be violated. This complements Baker, Payne, and Rabinoff’s generalization of Speyer’s condition [7, Theorem 6.9].

The new point of view taken in this paper is to attempt to understand the realizability locus inside the moduli space of all parameterized tropical curves as a global tropical geometric object. We do so by studying a fundamental object in logarithmic Gromov–Witten theory—the space of logarithmic prestable maps to the Artin fan. This is inspired by the insights of Abramovich–Wise, Gross–Siebert, and Ulirsch. By synthesizing these ideas, we are led to the following result.

Let X be a toric variety with fan \(\Delta \), and let \(\mathscr {L}^\circ _\Gamma (X)\) denote the moduli space of maps from smooth pointed genus g curves into X with fixed contact orders with the toric boundary along smooth marked points. In Sect. 2.2 a generalized extended cone complex \(T_\Gamma (\Delta )\) is constructed which parameterizes tropical stable maps with the analogous discrete data. The generalized cone complex \(T^\circ _\Gamma (\Delta )\) is the complement of the extended faces and parameterizes maps from tropical curves where all edge lengths are finite.

1.2 Further discussion

A number of experts have made the informal observation that the condition appearing in Speyer’s realizability theorem—that the minimum of a collection of numbers occurs at least twice—resembles the tropical variety of a tropical ideal. We view Theorem D as giving a simple and rigorous explanation for this phenomenon.

While the proof of our lifting theorems only relies on compactness of the realizable locus, the tropical structure is useful for applications to enumerative geometry. This is illustrated, for instance, by the results of Len and the author [20], in which the polyhedral structure of the realizability locus is used to derive multiplicities for tropical curve counting.

The above Theorem D also contributes to the study of tropical moduli spaces, which have received considerable interest in recent years. These results have aimed at an improved conceptual understanding of information contained in tropical moduli spaces, including applications to enumerative geometry [10, 11, 28] and to the geometry and topology of moduli spaces [9, 12, 13]. The novelty of the present paper is that the tropicalization of \(\mathscr {L}^\circ _\Gamma (X)\) is studied as the tropicalization of a map to a certain toroidal stack—the stack of prestable maps to the Artin fan. This is in sharp contrast to recent results on tropicalizations for moduli spaces, which have used toroidal structures on the spaces themselves.

The well-spacedness condition has inspired a great deal of research. However, to our knowledge, the results of Baker–Payne–Rabinoff [7, Theorem 6.9] are the only known nontrivial necessary conditions for realizability for nonmaximally degenerate tropical curves, although it may be possible to extract such conditions using the methods of [19]. By studying the limits of nonsuperabundant families of curves and applying Theorem A, one can obtain sufficient conditions for lifting tropical curves in nonmaximally degenerate situations, i.e., when vertices carry nonzero genus. Furthermore, Theorem B exhibits the first instance of a sufficient condition for the realizability for nonmaximally degenerate superabundant curves.

In addition to the work of Baker–Payne–Rabinoff mentioned above, Katz extracts a number of necessary conditions for realizability. These stem from interpreting the logarithmic tangent–obstruction complex for maps to toric varieties combinatorially for degenerate maps [19]. A similar approach is used by Cheung, Fantini, Park, and Ulirsch to prove that in a large range of cases, nonsuperabundance is a sufficient condition for realizability [15]. Note, however, that limits of nonsuperabundant tropical curves can often become superabundant. As a result, Theorem A extends the reach of these theorems as well.

An implicit goal of this paper is to demonstrate the usefulness of the perspective on tropicalization arising from logarithmic prestable maps to Artin fans arising from the work of Abramovich, Ulirsch, and Wise [5, 37, 38] and the insights of Gross and Siebert [17]. Indeed, accepting this technical input, the reader will note that the proofs of our realizability theorems follow from reductions to existing theorems in tropical geometry.

The first part of this project [28], which was a chapter in author’s doctoral dissertation, studies the tropicalization of the moduli space of logarithmic maps to toric varieties in genus 0, and it is in this sense that the present paper is a sequel. Superabundance never appears in the genus 0 setting, and the analogue of Theorem D can be used to derive a number of consequences concerning the geometry of the space of maps. We refer to loc. cit. for details. We also note that a similar polyhedrality result as above has been proved by Tony Yu in the context of nonarchimedean analytic Gromov–Witten theory using methods that are quite different from ours [40, 41].

1.3 Prerequisites

We assume familiarity with the fundamental concepts of Berkovich geometry and logarithmic structures. A rapid overview of the relevant concepts may be found in the preceding article [28, Section 2]. We refer the reader to two excellent recent surveys in this area by Abramovich, Chen, and their collaborators [3, 4].

2.1 Logarithmic stable maps

Our central object of study is the moduli space of genus g curves in a projective toric variety X in a fixed curve class, meeting each torus invariant divisor \(D_\rho \) at marked points with prescribed contact orders. We work with the compactification of this space of ramified curves, provided by the Abramovich–Chen–Gross–Siebert theory of logarithmic stable maps [2, 14, 17].

Remark 2.1

Minimality is a condition on the characteristic monoids of the logarithmic structure on the base of the family of maps. The reader may think about this concept as follows. A logarithmic stable map \([f:C\rightarrow X]\) is a stable map in the category of logarithmic schemes. Consequently, given a logarithmic scheme S, the moduli problem returns the collection of isomorphism classes of logarithmic stable maps over the base logarithmic scheme S. However, algebraic stacks are defined over the category of schemes rather than logarithmic schemes. A crucial and beautiful technical achievement of Abramovich, Chen, Gross, and Siebert was to identify an algebraic stack \(\mathscr {L}(X)\) that represented a logarithmic moduli problem—the moduli problem for minimal maps. Given a scheme \(\underline{S}\) without a chosen logarithmic structure and a map \(\underline{S}\rightarrow \mathscr {L}(X)\), there is a minimal logarithmic structure \(\mathscr {M}_S\) that can be placed on \(\underline{S}\), returning a diagram in the logarithmic category

which is a family of logarithmic stable maps. Families obtained by such logarithmic structures are referred to as minimal logarithmic stable maps. The algebraic stack \(\mathscr {L}(X)\) carries a universal logarithmic structure, and it is by pulling back this structure that we obtain the minimal logarithmic structure \(\mathscr {M}_S\) as above. One can thus interpret this moduli space \(\mathscr {L}(X)\) as a parameter space for minimal logarithmic stable maps and thus as a stack over the category of schemes. Given a different logarithmic structure \(\mathscr {M}_S'\) on \(\underline{S}\), the universal property of minimality ensures that any map \((\underline{S}, \mathscr {M}_S')\rightarrow \mathscr {L}(X)\) factors uniquely through a minimal family. We refer the reader to [17, Section 1] for an explicit description of these monoids and to [2, Section 2] for a conceptual discussion.

We package the discrete data by the symbol \(\Gamma = (g,n,\varvec{c})\). Let \(\mathscr {L}_\Gamma (X)\) denote the moduli space of minimal logarithmic maps carrying discrete data \(\Gamma \), and let \(\mathscr {L}^\circ _\Gamma (X)\) be the locus on which the logarithmic structure is trivial. The following result is established by Abramovich–Chen [2, Section 5] and Gross–Siebert [17, Corollary 4.2].

Note that in this paper, we will often work with logarithmic maps over valuation rings \({{\mathrm{Spec}}}(R)\) that are not discretely valued. The natural logarithmic structure in this case is coherent, but not fine. In this case, one may approximate R by sub-DVR’s and pass to a limit, see [5, Appendix A.1]. Alternatively, in Theorem A, one can pass to a subsequence of [0, 1) converging to 1, parameterizing tropical maps with rational edge lengths. This will eliminate the need to work with maps over nondiscrete valuation rings in the sequel.

2.2 Tropical stable maps

The purpose of this section is to construct a parameter space for tropical stable maps, which will serve as the target of our tropicalization map.

Definition 2.3

A prestable n-pointed tropical curve of genus g, denoted , is a finite graph and three additional data:

1. (1)

   To each edge of , a length \(\ell (e)\in {\mathbb R}_{\ge 0}\sqcup \{\infty \}\) such that if e is a marked leaf edge, \(\ell (e) = \infty \).

    
2. (2)

   To every vertex a genus \(g(v)\in {\mathbb Z}_{\ge 0}\).

    
3. (3)

   A labeling of a subset of the 1-valent genus 0 vertices of by the set \(\{p_1,\ldots , p_n\}\).

    

The genus of is defined to be

The tropical curve is made into a topological space as follows. Each edge e of finite length \(\ell (e)\) is identified with \([0,\ell (e)]\). Marked leaf edges are metrized as \([0,\infty ]\). Unmarked edges e with \(\ell (e) = \infty \) are topologized as two glued intervals \([0,\infty ]\sqcup _\infty [0,\infty ]\). We will refer to a curve whose lengths are finite away from the marked leaves as smooth.

Informally, is often thought of as being a metric space away from the infinite points and as a space with a “singular” metric when including the infinite points.

Recall that a morphism \(\phi : \Sigma _1\rightarrow \Sigma _2\) between polyhedral complexes is a map on the underlying point sets such that each polyhedron in \(\Sigma _1\) is mapped to a polyhedron in \(\Sigma _2\).

Definition 2.5

A tropical stable map from a smooth tropical curve is a continuous and proper morphism of polyhedral complexes

where is a smooth n-marked abstract tropical curve, such that the following conditions are satisfied.

1. (TSM1)

   For each edge , the direction of f(e) is an integral vector. Moreover, upon restriction to e, f has integral slope \(w_e\), taken with respect to this integral direction. This integral slope is referred to as the expansion factor of f along e.

    
2. (TSM2)

   The map f is balanced in the usual sense, i.e., at all points of the sum of the derivatives of f in each tangent direction is zero.

    
3. (TSM3)

   The map f is stable. That is, if has valence 2, then the image of \(\mathrm {Star}(v)\) is not contained in the relative interior of a single cone of \(\Delta \).

    

We will usually suppress the markings from the notation and simply write the map as .

The following definition indexes the “deformation class” of a tropical stable map and is obtained by dropping the data of the lengths of the edges of .

Definition 2.6

The combinatorial type of a tropical stable map is the data

1. (CT1)

   The finite graph model underlying .

    
2. (CT2)

   For each vertex , a cone \(\sigma _v\in \Delta \) containing the image of v.

    
3. (CT3)

   For each edge e, the slope \(w_e\) of f restricted to e and the primitive vector \(u_e\) in the direction of f(e).

    

The following proposition seems to be well-known to experts, and a formal proof in the \(g = 0\) case may be found in [25, Proposition 2.1] or [28, Proposition 3.2.1]. We give an outline of the argument in general.

Proof

Fix a combinatorial type and let V and E be the vertex and bounded edge sets of , respectively. To describe a particular tropical map [f] in this combinatorial type is to assign to each \(v_i\), a point \(f(v_i)\) in the associated cone \(\sigma _{v_i}\) and to each edge \(e_j\), a real edge length \(\ell _e\). In order that these assignments define a continuous and balanced piecewise-linear map to \(\Delta \), we simply need to force that for every edge e connecting to \(v_{e1}\) and \(v_{e2}\),

$$\begin{aligned} f(v_{e1})-f(v_{e2}) = \ell _ew_e. \end{aligned}$$

Here, in keeping with previous notation, \(w_e\) is the vector slope prescribed by the combinatorial type. In other words, the set of tropical curves is the subcone of \(\prod _{v\in V} \sigma _v \times \prod _{e\in E}{\mathbb R}_{\ge 0}\) given by

$$\begin{aligned} \sigma _\Theta = \left\{ ((f(v)_v,(\ell _e)_e)\in \prod _{v\in V} \sigma _v \times \prod _{e\in E}{\mathbb R}_{\ge 0}: \text {For all } e = v_{e1}v_{e2} \in E, v_{e1}-v_{e2} = \ell _ew_e\right\} . \end{aligned}$$

These relations cut out a subcone, and the result follows. The claimed finiteness follows from the boundedness of combinatorial types for logarithmic maps, as proved in [17, Section 3.1]. \(\square \)

Following [22, Proposition 2.14], the overvalence of a type \(\Theta \) is defined as

$$\begin{aligned} \mathrm {ov}(\Theta ) = \sum _{p:{{\mathrm{val}}}(p)\ge 4} {{\mathrm{val}}}(p)-3. \end{aligned}$$

The expected dimension of this cone of tropical maps is

where is the first Betti number of the underlying graph of . This dimension calculation explained, for instance, in [25, Section 1]. Just as in the algebraic case, it can be deduced from an analysis of the combinatorial tangent–obstruction complex, i.e., the “abundancy map” in [15, Definition 4.1]. See also [22, Sections 2.4–2.6]. The expected dimension above is a lower bound for the dimension of \(\sigma _\Theta \), but the actual dimension is often be larger. For instance, if cycles of are mapped into proper affine subspaces of \(|\Delta |\) of high codimension, the dimension of the actual deformation space will be larger than the above-expected dimension.

Following [1, Sections 2.3–2.4] there is a natural compactification of any cone \(\sigma \) with integral structure obtained as follows. Let \(S_\sigma \) be the monoid of integral linear functions on \(\sigma \) that are nonnegative, i.e., the dual monoid of \(\sigma \). The cone \(\sigma \) can be recovered as the space of monoid homomorphisms \({\text {Hom}}(S_\sigma ,{\mathbb R}_{\ge 0})\). The canonical compactification of \(\sigma \) is defined as

$$\begin{aligned} \overline{\sigma }:={\text {Hom}}(S_\sigma ,{\mathbb R}_{\ge 0}\sqcup \{\infty \}). \end{aligned}$$

Fixing a combinatorial type \(\Theta \), the faces of the extended cone \(\overline{\sigma }_\Theta \) parameterize tropical stable maps where is possibly singular and the vertices of map to the extended faces of \(\overline{\Delta }\). We will have no need to work with such maps directly, so we leave the precise formulation to an interested reader.

Definition 2.10

An isomorphism between two tropical stable maps and is an isometry of graphs, commuting with the vertex weights and with the map:

An isomorphism of a stable map with itself is said to be an automorphism. Similarly, an automorphism of the combinatorial type \(\Theta \) is an automorphism of the underlying finite graph preserving the edge directions, their expansion factors, vertex weights, and the cones associated with each vertex.

The topological space constructed above has the structure of a generalized extended cone complex in the sense of [1, Section 2]. By forming the union of the images of the ordinary (i.e., noncompact) cones \(\sigma _\Theta \) in \(T_\Gamma (\Delta )\), we obtain the moduli space \(T_\Gamma ^\circ (\Delta )\), parameterizing those maps with smooth source graph.

2.3 Prestable tropical maps

For our later study, it will be convenient to relax the stability condition above. A prestable tropical map to \(\Delta \) is a map as in Definition 2.5, but possibly violating the stability condition (TSM3). In other words, becomes a stable map after dropping finitely many 2-valent vertices from . There are infinitely many such destabilizations of any combinatorial type. However, there are no substantive changes to the construction above—prestable tropical curves of a fixed combinatorial type are still parameterized by a cone, though of arbitrarily high dimension. Gluing these cones together exactly as above, we obtain a generalized extended cone complex

$$\begin{aligned} T^{\mathrm {pre}}_\Gamma (\Delta ) : = \varinjlim _{\Theta : \mathrm {prestable}} (\overline{\sigma }_\Theta ,j_\Theta ). \end{aligned}$$

In analogy with toroidal embeddings that are locally of finite type, one might regard this cone complex as being locally of finite type.

This result follows from the fact that minimality of the map [f] can be characterized by placing constraints on the characteristic monoids of the base, in this case \({{\mathrm{Spec}}}(P\rightarrow {\mathbb C})\). As explained in [17, Section 1.6 & Remark 1.21], the map forces certain “minimal constraints” that every such base monoid has to satisfy. This implies that there is a natural injective map \({\text {Hom}}(P,{\mathbb N})\rightarrow T_\Theta ({\mathbb N})\). The universal property of minimality [17, Proposition 1.24] forces that, if P is minimal, the map is also surjective and thus an isomorphism.

The same result holds when X replaced with its Artin fan \(\mathscr {A}_X = [X/T]\).

2.4 Pointwise tropicalization for logarithmic stable maps

Any point \(x\in \mathscr {L}^\circ _\Gamma (X)^{\mathrm {an}}\) may be represented by a map

$$\begin{aligned} {{\mathrm{Spec}}}(K)\rightarrow \mathscr {L}^\circ _\Gamma (X), \end{aligned}$$

where K is a valued field extension of \({\mathbb C}\). After a ramified base change, the existence of the compactification \(\mathscr {L}_\Gamma (X)\) of \(\mathscr {L}^\circ _\Gamma (X)\) guarantees an extension to

$$\begin{aligned} {{\mathrm{Spec}}}(R)\rightarrow \mathscr {L}_\Gamma (X), \end{aligned}$$

where R is the valuation ring of K. By pulling back the universal curve and universal map we obtain a diagram

where the \(s_i\) are horizontal sections determining marked points on the geometric fibers. By [6, Section 1.4], this choice of model determines a retraction of the analytic generic fiber onto a tropical curve

with a canonical continuous section

Similarly, there is a deformation retraction

$$\begin{aligned} X^{\mathrm {an}}\rightarrow \overline{\Delta }, \end{aligned}$$

of the analytic toric variety onto its skeleton by [26, 33]. Composing the section map with the natural map

$$\begin{aligned} \mathscr {C}_\eta ^{\mathrm {an}}\rightarrow X^{\mathrm {an}}\rightarrow \overline{\Delta }, \end{aligned}$$

we obtain a map . By [28, Theorem 3.3.2], this map is seen to be a tropical stable map from a smooth tropical curve.

2.5 Tropicalization via the Artin fan

Given a scheme or stack Y defined over a valuation ring R, we will use \(Y^{\mathrm {an}}_\circ \) to denote Raynaud’s generic fiber functor applied to the formal completion of Y along the maximal ideal of R. See [37, 38, 40] for background on generic fibers of algebraic stacks. Note that if Y is proper, the generic fiber coincides with the Berkovich analytification and will drop the \(\circ \) in the subscript.

Theorem 2.12

([38, Theorem 1.1]) There is a canonical identification of extended cone complexes given by \(\mu _\Delta : |\mathscr {A}(\Delta )^{\mathrm {an}}_\circ |\rightarrow N(\Delta )\), making the diagram

commute.

We obtain the following as an immediate corollary.

Corollary 2.13

Let \(f: \mathscr {C}\rightarrow X\) be a logarithmic stable map with smooth generic fiber defined over \({{\mathrm{Spec}}}(R)\), where R is a valuation ring containing \({\mathbb C}\). Let be the skeleton of \(\mathscr {C}\). Then, the tropicalization of f coincides with the composite map

2.6 Maps to \(\mathscr {A}_X\)

The purpose of this subsection is to establish a “global” version of Corollary 2.13 above. In [5], Abramovich and Wise introduce a stack of prestable logarithmic morphisms to the Artin fan \(\mathscr {A}_X\) itself. Fixing the discrete data as before, we will use the following result of theirs.

The reader will note that the logarithmic smoothness of this stack is in sharp contrast to the geometry of \(\mathscr {L}_\Gamma (X)\), which satisfies Murphy’s law² in the sense of Vakil [39]. Indeed, this remarkable smoothness property was applied in [29] to show that every tropical stable map arose as the tropicalization of a family of stable maps to the Artin fan.

Any point of \(\mathscr {L}_\Gamma ^{\mathrm {pre}}(X)^\beth \) gives rise to a family of logarithmic prestable maps

over a valuation ring extending \({\mathbb C}\). Following the same procedure as in the previous section, passing to skeletons yields a map

$$\begin{aligned} \mathrm {trop}: \mathscr {L}_\Gamma ^{\mathrm {pre}}(\mathscr {A}_X)^\beth \rightarrow T^{\mathrm {pre}}_\Gamma (\Delta ). \end{aligned}$$

On the other hand, the stack \(\mathscr {L}_\Gamma ^{\mathrm {pre}}(X)\) is a toroidal (i.e., logarithmically smooth) stack in the lisse-étale topology. There exists a continuous deformation retraction of the analytic space onto an extended cone complex:

$$\begin{aligned} \varvec{p}_{\mathscr {L}}: \mathscr {L}_\Gamma ^{\mathrm {pre}}(\mathscr {A}_X)^\beth \rightarrow \overline{\Sigma }(\mathscr {L}_\Gamma ^{\mathrm {pre}}(\mathscr {A}_X)). \end{aligned}$$

Note that this deformation retraction has been constructed in various closely related settings in the literature, see [1, 33]. We briefly describe the necessary changes in the setting of Artin stacks. Let \(\mathcal X\) be a toroidal Artin stack. Let \(U\rightarrow \mathcal X\) be a smooth chart by a toroidal scheme without self-intersection. Consider the self-product

$$\begin{aligned} R = U\times _{\mathcal X} U\rightrightarrows U, \end{aligned}$$

and note that, since \(\mathcal X\) is toroidal, the arrows above are smooth morphisms. We wish to use the toric charts of U to obtain charts on R. This is done as follows. Given a smooth morphism \(R\rightarrow U\), after shrinking, the étale local structure theorem for smooth morphisms [32, Tag 039P] yields commutative diagram

In the diagram above, \(V_R\) and \(V_U\) are opens, and the rightward horizontal arrows are étale. Thus, we obtain toroidal charts for R. The crucial point is that the torus factors in the charts for R do not change the skeleton, i.e., the skeletons of \(U_\sigma ^\beth \) and \((U_\sigma \times \mathbb G_m^r)^\beth \) are canonically identified. The extended skeleton \(\overline{\Sigma }(\mathcal X)\) of \(\mathcal X^\beth \) can now be constructed via a colimit of the skeleton for the diagram \((R\rightrightarrows U)^\beth \). This is carried out using only cosmetic modifications to the arguments already present in the literature, see [1, 33, 35, 37]. The remaining details are left to an interested reader.

The main result of this section is the relationship between the generalized extended cone complexes \(T^{\mathrm {pre}}_\Gamma (\Delta )\) and \(\overline{\Sigma }(\mathscr {L}_\Gamma ^{\mathrm {pre}}(\mathscr {A}_X))\).

Theorem 2.15

There is a commutative diagram of continuous morphisms

where

$$\begin{aligned} \mathrm {trop}_\Sigma :\overline{\Sigma }(\mathscr {L}_\Gamma ^{\mathrm {pre}}(X))\rightarrow T^{\mathrm {pre}}_\Gamma (X) \end{aligned}$$

is a finite morphism of generalized extended cone complexes and becomes an isomorphism upon restriction to any cell of the source.

We derive the first part of Theorem D as a corollary.

As a consequence of the theorem, we rephrase the tropical lifting question as follows.

When does a tropical stable map lie in the image of the continuous map \(\mathrm {trop}\) above?

The next section takes advantage of the continuity of this map to establish the main applications.

3.1 Polyhedrality

Let \(p = [C\rightarrow X]\) be a minimal logarithmic stable map and consider the associated map \([C\rightarrow \mathscr {A}_X]\). Let U be a toric neighborhood of \([C\rightarrow \mathscr {A}_X]\) in \(\mathscr {L}_\Gamma ^{\mathrm {pre}}(\mathscr {A}_X)\). Let Z be the local model in the smooth topology of the moduli space \(\mathscr {L}_\Gamma (X)\) near p. After possibly shrinking Z, the points of the compact analytic space \(Z^\beth \) correspond to families over valuation rings of logarithmic stable maps whose special fiber, after composition with \(X\rightarrow \mathscr {A}_X\), lies in U.

3.2 Realizability of limits: Theorem A

Let for \({t\in [0,1)}\) be a continuously varying family of tropical stable maps. We have an induced moduli map

$$\begin{aligned}{}[0,1)\rightarrow T_\Gamma (\Delta ), \end{aligned}$$

whose image lies in the locus of realizable tropical curves. Since the realizable locus is a closed set, it follows that the limiting map also lies in this set, and the result follows immediately. \(\square \)

3.3 Well-spacedness for good reduction

We first recall the definition of Speyer’s well-spacedness condition. Note that any genus 1 abstract tropical curve has a unique cycle, possibly a single vertex of genus 1 or a self-loop at a vertex. A genus 1 tropical stable map is said to be superabundant if the image of the unique cycle L in \(|\Delta |\) is contained in a proper affine subspace.

3.4 Proof of Theorem B

Let be a tropical stable map of genus 1 such that there is a unique point satisfying \(g(p) = 1\). In other words, the underlying graph of is a tree. Let be the tropical stable map obtained by replacing the genus function with one such that \(g(p) = 0\), and attaching a self-loop at the vertex p, where \(t\in {\mathbb R}_+\) is the length of this self-loop. By hypothesis, the star of p in the modified map is realizable. By applying Speyer’s genus 1 realizability theorem [31, Theorem 3.4], we see that for each value of \(t>0\), is realizable. Letting \(t\rightarrow 0\) we obtain a continuous family of realizable tropical curves whose limit is f. Since the limiting map must be realizable by Theorem A, the result follows. \(\square \)

Remark 3.3

David Speyer has communicated to us a different approach to the proof of the above result using techniques akin to those in [31]. We record the argument here, in case it may be helpful to the reader. Let be a tropical stable map and the genus 1 vertex, and assume \(\mathrm {Star}(v)\rightarrow \Delta \) is realizable. Let \(a_1, a_2, ..., a_m\) be the directions of the edges pointing outward from v and write

$$\begin{aligned} a_i = (a_{i1}, a_{i2}, \ldots , a_{in}) \in {\mathbb Z}^n. \end{aligned}$$

Let K be a nonarchimedean field such that the vertices of map to points of \(\Delta \) that are rational over the value group. Let R be the valuation ring and k the residue field. Since the link is realizable, we may build an elliptic curve E over k with points \(z_1, z_2, ... z_m\in E(k)\) such that

$$\begin{aligned} \sum a_i z_i = 0 \in E^n. \end{aligned}$$

Choose a lifting of E to an elliptic curve \(\mathscr {E}\) over \({{\mathrm{Spec}}}(R)\) with reduction E over k. By the arguments of [31, Section 7,8], to lift the map we must lift the points \(\{z_i\}\) to the generic fiber of \(\mathscr {E}\) with prescribed relations, preserving \(\sum a_i z_i=0\). This can be done by mimicking the calculations in Speyer’s original argument, where the lifts of the points \(z_i\) lie in the fibers of the reduction map \(\mathscr {E}(R)\rightarrow E(k)\). By [30, Theorem 6.4], since we work in residue characteristic 0, the fibers of this map are isomorphic to the additive group \(R^\times \). The calculations are now identical to those in [31, Lemma 8.3] and the result follows.

3.5 Proof of Theorem C

For the reader’s benefit, a “proof-by-picture” is given in Fig. 1. Let be a genus 1 tropical curve whose cycle is contained in a hyperplane H. Assume that this map is well-spaced. Furthermore, assume that the minimum of the distances in Definition 3.2 is 0 and occurs exactly twice. It follows that there exist two vertices \(v_1\) and \(v_2\), belonging to the cycle L, where leaves H. Choose a path in L between \(v_1\) and \(v_2\) and any family of tropical curves for \({t\in [0,1)}\) such that the distance between \(v_1\) and \(v_2\) is \((1-t)\). Observe that for each value of \(t<1\), the tropical map is well-spaced and thus realizable by [31, Theorem 3.4]. However, in the limiting map, , there is a unique point on L at which the cycle L leaves the hyperplane H. It follows that the limiting map cannot be well-spaced. However, the map is realizable by Theorem A. \(\square \)