Introduction↩︎

In this paper, we use the combinatorics of degenerations of Hurwitz covers to study the tropical realizability problem for maps from tropical curves. We establish new realizability criteria for curves of genus two and provide new proofs of several known realizability theorems using these techniques.

0.1 The problem↩︎

The tropical inverse problem is a fundamental question in tropical geometry: when does a synthetically defined tropical object arise from algebraic geometry? A key instance concerns the realizability of parameterized tropical curves, see [1]–[6]. Fix an abstract tropical curve \(\Gamma\) of genus \(g\) and a balanced map \[F\colon \Gamma \to \mathbb{R}^r.\] We say that \(F\) is realizable if there exists a smooth curve \(\mathcal{C}\) of genus \(g\), defined over a non-archimedean field, and a rational map \[\varphi\colon \mathcal{C} \dashrightarrow \mathbb{G}_m^r,\] such that the tropicalization of \(\varphi\) equals \(F\). The relevant basic notions of tropical geometry are recalled in Section 1. In this paper, we focus on the case where all vertex genera are equal to \(0\).

0.2 State of the art↩︎

To contextualize our results, we explain how the complexity of the realizability problem grows with the genus \(g\) and the target dimension \(r\). When \(g = 0\), for any value of \(r\), all balanced tropical maps are realizable [6]. More generally, an explicit combinatorial condition called non-superabundance is sufficient for realizability for all values of \(g\) and \(r\) [2]. The (non-)superabundance condition depends only on the combinatorics of the map: the underlying graph and the directions of edges under the map.

However, superabundant maps are expected to be generic objects, and understanding their realizability lies at the heart of the tropical inverse problem. Superabundant phenomena appear for each value of \((g,r)\) starting at \((1,1)\). For a superabundant configuration, additional conditions on the edge lengths of the tropical curve are typically required to guarantee realizability. The basic goal is to identify these additional realizability conditions.

Once a superabundant configuration appears, it “propagates” to all higher genera and target dimensions: smaller genus graphs appear as subgraphs of larger genus graphs, and maps to lower-dimensional targets appear as projections of maps to higher-dimensional ones. For each value of \((g,r)\), new phenomena arise that are not inherited from lower genus or lower dimension, see [7]. Realizability conditions can also be propagated into higher genus and higher dimensions via the methods of [8], which we further develop here.

The first “atomic” superabundance phenomenon occurs at \((1,1)\), and its propagation explains all genus-one superabundance. Realizability in this case is characterized by a well-known condition discovered by Speyer, called well-spacedness, see [9]–[11]. Speyer’s work has implications for the enumerative geometry of elliptic curves [10], [12], and the higher-genus propagation of well-spacedness has applications in Brill–Noether theory [8].

In genus two and higher, prior to this paper, there had been no example of a condition guaranteeing the realizability of superabundant tropical curves, except those arising from genus one by propagation. A reason for this lack of progress is that existing proofs in genus one rely on features of elliptic curves—such as non-archimedean uniformization or the classification of elliptic singularities—that do not have straightforward generalizations to higher genus [10], [11].

0.3 Main results↩︎

The contributions of this paper are as follows:

(i) Qualitatively new conditions guaranteeing the realizability of balanced functions \(\Gamma\to{\mathbb{R}}\) on tropical curves of genus \(2\) with contracted genus two subgraphs.

(ii) A new proof that a balanced function \(\Gamma\to{\mathbb{R}}\) on a tropical curve of genus \(1\) is realizable if and only if it is well-spaced.

(iii) A new proof that non-superabundant functions \(\Gamma\to{\mathbb{R}}\) are realizable in any genus.

In Section 5, we develop a new dimensional induction that allows us to deduce partial results about maps to \(\mathbb{R}^r\) using the natural projections to \({\mathbb{R}}\). Consequently, results (ii) and (iii) imply the full well-spacedness result in all dimensions for genus one [10], [11], as well as the corresponding result for non-superabundant maps in all dimensions [2]. This approach also realizes many superabundant maps to \({\mathbb{R}}^r\) in genus two.

The geometric input is that the Hurwitz space of covers of \(\mathbb{P}^1\) admits two natural moduli-theoretic compactifications: the space of logarithmic (or relative) stable maps [13], [14] and the space of admissible covers [15], [16]. We analyze the combinatorics of the natural comparison map in key base cases, and deduce the main results by combining these local analyses with global moduli arguments.

0.3.1 Genus two↩︎

We work in characteristic \(0\) for simplicity. An abstract tropical curve \(\Gamma\) of type \(\Theta\) is a genus two abstract tropical curve that contains a subgraph homeomorphic to a \(\Theta\) graph. Such a subgraph is necessarily unique; we call it the core and denote it by \(\Theta\). The three edges of \(\Theta\) will be called the edges of the \(\Theta\).

Consider a piecewise linear function \(F\colon \Gamma \to {\mathbb{R}}\) that is constant of value \(0\) on the core. In the connected component of \(F^{-1}(0)\) containing \(\Theta\), the points that are incident to non-contracted edges in \(\Gamma\) are called critical points. There is a unique path from each critical point to \(\Theta\), which we call a critical path. The intersection of a critical path with \(\Theta\) is called its location.

Let \(F\colon \Gamma\to{\mathbb{R}}\) be a balanced function on an abstract tropical curve of \(\Theta\) type, and assume that \(F\) contracts the core. If

(i) for each edge \(E\) of \(\Theta\), there exists a critical path of minimal length whose location is on \(E\), and

(ii) the lengths of all other critical paths are sufficiently large compared to the minimal length,

then \(F\) is realizable.

In Figure 1, we show a balanced function of this kind. Except for the non-constructive nature of (ii), the result above is exactly parallel to the \(r = 1\) case of the well-spacedness condition.

Our next result is qualitatively different from the genus \(1\) case. Observe that the \(\Theta\) graph admits a unique involution whose quotient is a tree, given by flipping each edge. Pairs of points that are exchanged by the involution are conjugate.

Let \(F\colon \Gamma\to{\mathbb{R}}\) be a balanced function on an abstract tropical curve of \(\Theta\) type, and assume that \(F\) contracts the core. If

(i) there exists an edge of \(\Theta\) and two critical paths of minimal length whose locations are conjugate, and

(ii) the lengths of all other critical paths are sufficiently large compared to the minimal length,

then \(F\) is realizable.

In Figure 2, we depict an example of a balanced function of this kind. In Proposition 22 we show an analogous result where the Weierstrass points, i.e. fixed points of the hyperelliptic involution, play a crucial role.

The moduli of maps from \(\Theta\) curves to \({\mathbb{R}}\) that contract the core has excess dimension \(2\) compared to the expected dimension, and so one expects realizability to be a codimension \(2\) condition. In these two theorems, we see that the two conditions can arise from both the locations of critical paths and their lengths.

In the main text, we include additional such results, including when the core is a dumbbell rather than \(\Theta\). We focus on the combinatorics of the \(\Theta\) here, largely for expository reasons.

0.3.2 Hurwitz modifiability and combinatorial rederivations↩︎

Let \(F\colon \Gamma\to {\mathbb{R}}\) be a balanced function. It is known by work of Amini–Baker–Brugallé–Rabinoff that the realizability of \(F\) is equivalent to the existence of, what we dub here, an \(H\)-modification [17], [18]. In Section 1, we re-explain this using the geometry of relative maps and admissible covers. Either way, the upshot is that realizability is equivalent to the existence of a commutative diagram \[\begin{tikzcd} \hat{\Gamma}\arrow[swap]{d}{\hat{F}}\arrow{r} &\Gamma\arrow{d}{F}\\ \hat{T}\arrow{r} & {\mathbb{R}}, \end{tikzcd}\] where the horizontal arrows are tree attachments, the map \(\hat{F}\) is finite and preserves balanced functions under pullback, and at every vertex, the local Hurwitz problem has a solution.

The \(H\)-modifiability of a balanced function might seem like a basic tool for studying realizability, but both the graph-theoretic aspects and the Hurwitz problem are of high combinatorial complexity, and few general techniques apply to either. One motivation for this paper was to investigate whether \(H\)-modifiability can be used in practice. The answer appears to be positive.

We now examine what these techniques imply for the existing body of work on tropical realizability [2], [3], [6], [11]. The methods used in these results are very different from \(H\)-modifiability. Approaching the problem from the perspective of \(H\)-modifiability, however, leads naturally to nontrivial combinatorial statements. For example: (i) every balanced function on a trivalent tropical curve that does not contract a cycle is \(H\)-modifiable, and (ii) well-spacedness for a balanced function on a trivalent genus-\(1\) tropical curve is equivalent to \(H\)-modifiability.

Our second main contribution is to provide direct proofs of these combinatorial statements.

Let \(\Gamma\) be an abstract, trivalent tropical curve, and let \(F\colon \Gamma \to {\mathbb{R}}\) be a balanced map. Assume that \(F\) does not contract a cycle. Then \(F\) is \(H\)-modifiable. As a consequence, such maps are realizable.

This result reproduces a theorem of Cheung–Fantini–Park–Ulirsch [2] specialized to target dimension \(1\), and, with the dimensional reduction technique developed in Section 5, provides a new proof of their results.

We now turn to genus \(1\). Speyer’s “well-spacedness” condition [11] gives a combinatorial criterion for realizability. In the trivalent case, it asserts that a genus-\(1\) tropical curve \(\Gamma\) with a balanced map \(F\colon \Gamma \to {\mathbb{R}}\) that contracts the core is realizable if and only if there are at least two critical paths of minimal length.

Let \(\Gamma\) be an abstract, trivalent, genus-\(1\) tropical curve, and let \(F\colon \Gamma \to {\mathbb{R}}\) be a balanced map that contracts a cycle. Then \(F\) is \(H\)-modifiable if and only if \(F\) is well-spaced. Consequently, such a map is realizable if and only if it is well-spaced.

We discuss the non-trivalent case in the main text. The result provides an independent proof of Speyer’s realizability theorem [11] for balanced functions. The bootstrapping technique of Section 5 implies the higher dimensional case from the above statement.

0.4 Outlook and techniques↩︎

Our approach to realizability is more elementary than earlier ones, which rely on heavier technical machinery such as Tate’s non-archimedean uniformization for elliptic curves and the residue theory of elliptic singularities. These techniques have not yet been extended to higher genera, but our results hint at the subtleties involved in this. The cost of our method is high combinatorial complexity: finding appropriate tropical modifications and verifying the relevant Hurwitz conditions is challenging in practice, and much of the paper is devoted to these combinatorial constructions. Once such techniques are developed, it seems potentially fruitful to combine them with other viewpoints.

One promising direction is the connection to the singularity theory of curves. Our methods are particularly effective for producing examples, whereas the geometry of Gorenstein curves used in [10] seems better suited for proving statements once they have been conjectured. We expect that progress in genus \(2\) and beyond will involve revisiting the genus-\(2\) results using Abel–Jacobi maps of Gorenstein curves of genus \(2\), following Battistella–Carocci [19].¹

We use some general algebro-geometric techniques – essentially “pure thought” arguments – and we mention them here in case they may be useful. First, while we use results on modifiability [18], we explain this as a comparison between two moduli spaces of maps from curves. The basic object for us is a map \(C\to\mathbb{P}^1\) over a non-archimedean field. We view this as a point in the Hurwitz space and produce tropical pictures from the combinatorics of limits in two different compact moduli spaces: the admissible cover space [15], [16] and the logarithmic stable map space [13], [14]: \[\mathsf{AdmCovers}\supset\mathsf{HurwCovers}\subset\mathsf{LogStMaps}.\] The tropical realization problem is a type of smoothing problem for degenerate maps. But \(\mathsf{AdmCovers}\) is logarithmically smooth, so this problem always has a solution. Therefore, a smoothable point of \(\mathsf{LogStMaps}\) must be “promotable” to a point of \(\mathsf{AdmCovers}\), and this is the modifiability condition. It suggests a general picture: the interacting combinatorics of different semistable reduction theorems create constraints on tropical realizability.

Second, as mentioned above, we explain how to deduce results about realizing maps to higher dimensional tori from maps to lower dimensional tori. This is based on a lifting argument for tropical intersections. In genus \(0\), \(1\), and in the non-superabundant case, this is sufficient to understand the problem completely. In higher genus, this is not the case, as shown by Koyama’s recent results [7]. The target dimension \(2\), genus \(2\) situation in complete generality requires new ideas.

And third, the non-constructive aspect of our genus \(2\) results – the fact that lengths have to be “very long” – seems to provide a useful conclusion that is relatively cheap to prove. It uses the structure of compactified tropicalizations to deduce the existence of points in the interior of the moduli space by constructing points at infinity in the combinatorial picture.

We conclude with a word about applications. The present results in genus \(2\) can be leveraged, using the genus induction of [8], to prove regeneration theorems for linear series on tropical curves, and we believe this should have applications in Brill–Noether theory. In particular, one can hope to use this to understand the codimension of the Brill–Noether locus when the Brill–Noether number is negative [8], [20].

We expect other applications, for example, to the study of fixed complex structure enumerative geometry in genus \(2\), following [12], [21]. In a different direction, Hicks has shown that the realizability of tropical curves has a surprising link, via mirror symmetry, to the construction of Floer theoretically unobstructed Lagrangian submanifolds in the torus [22]. For this link to Lagrangian geometry, it is important that the tropical curves are embedded, or at least immersed. Interestingly, this makes the dimensional reduction in Section 5 something on the algebraic side that might not be available on the mirror.

Acknowledgements↩︎

The question pursued here has its origins in an uncompleted project of R. Cavalieri, H. Markwig, and the third author dating back to 2014, concerning the tropicalization of the double ramification cycle [23]. We are grateful for many years of conversations with them on these and related topics. Our thanks also go to D. Jensen for encouraging us to return to the question, as well as to L. Battistella, F. Carocci, and J. Wise for inspiring discussions on realizability and Gorenstein singularities. Finally, we are very grateful to K. Christ, E. Katz, J. Hicks, S. Koyama, S. Payne, I. Tyomkin, and D. Speyer for their helpful conversations. The manuscript was improved thanks to the input of an anonymous referee.

ML was supported by the Philippa Fawcett program. ML and CKN conducted the research here under the Cambridge SRIM program in 2023. DR was supported by the EPSRC New Investigator Grant EP/V051830/1 and the EPSRC Frontier Research Grant EP/Y037162/1.

1 Tropical covers and modifications↩︎

We recall the basics of tropical curves and maps between them, following [17], [18], [23].

1.1 Tropical curves↩︎

A tropical curve is a connected, finite metric graph with unbounded rays and a genus labeling at its vertices. Precisely, it is an enhancement of an abstract connected graph \(G\) by

(i) a length function \(\ell\colon E(G)\to{\mathbb{Q}}_{>0}\),

(ii) a finite set of rays attached to the vertices by \(m\colon [n]\to V(G)\), and

(iii) a genus function \(\hat{g}\colon V(G)\to \mathbb{Z}_{\geq 0}\).

Given this data, we form a non-compact metric space as follows. Identify the edge \(e\) with an interval of length \(\ell(e)\) to form a metric graph \(\Gamma_0\). We form \(\Gamma\) by \[\Gamma = \left(\Gamma_0\sqcup \coprod_{i\in[n]}{\mathbb{R}}_{\geq 0}\right)/\sim,\] where the equivalence relation identifies each \(0\) in the disjoint union with the point corresponding to the vertex \(m(i)\). The unbounded rays are referred to as rays.

Of course, one can also study disconnected tropical curves. The connected case is better for most of our exposition, so when disconnected curves arise, we explicitly use the phrase “possibly disconnected”. We typically work in the case where \(\hat{g}\) is everywhere equal to \(0\). This is sometimes referred to as an explicit tropical curve. The only exception to this is in Section 1.4, which discusses tropicalization via semistable reduction. The exposition of this is more natural with an allowance for vertex genera.

Note that it is sometimes natural to allow the edge lengths to take real values, rather than only rational values. However, the space of realizable tropical maps, whose analysis is our main goal, is a rational polyhedral subcomplex of the moduli space of all tropical maps [9]. Realizability results in this more general context, therefore, follow from the results that we discuss here.

A metric graph \(\Gamma\) comes with a distinguished set of piecewise linear functions with integral slopes, or piecewise linear functions for short. These are continuous functions \[\Gamma\to{\mathbb{R}}\] that restrict to linear functions with integer slopes on the edges and rays.

Given two metric graphs \(\Gamma'\) and \(\Gamma\), we study piecewise linear maps between them. These are continuous maps \(F\colon\Gamma'\to \Gamma\) that induce a graph map, with the further condition that, when restricted to every edge or ray of \(\Gamma'\), the map \(F\) is linear with an integer slope.

For ease of terminology, given a vertex \(v\) on \(\Gamma\), we refer to flags consisting of a vertex \(v\) and an edge or ray incident to it as the tangent directions at \(v\). The “trivial” flag \((v,v)\) is formally viewed as the \(0\)-tangent direction. Given a vertex \(v\) and an edge or ray \(e\) to which it is incident, we refer to the flag \((v,e)\) as the “direction associated with \(e\)”. We visualize these as germs of edges or rays starting at \(v\), moving in the direction of one of the flags.

Given a piecewise linear map \(F\colon \Gamma'\to \Gamma\), every tangent direction based at \(v'\) in \(\Gamma'\) maps to a tangent direction based at \(F(v')\). It is the \(0\) direction when the edge or ray that determines the tangent direction is contracted to \(F(v')\).

A piecewise linear function has a well-defined integral slope along a tangent direction, by convention directed away from \(v\). Given a map \(\Gamma'\to\Gamma\), we can measure the slope along any tangent direction in \(\Gamma'\). Indeed, a tangent direction of \(\Gamma'\) maps to a tangent direction of \(\Gamma\), locally as an integer linear map. If \((v',e')\) is a tangent direction on \(\Gamma'\), and if \(e'\) maps to the vertex \(v\), the slope is equal to \(0\).

1.2 Balanced functions and harmonic maps↩︎

Piecewise linear functions and piecewise linear maps will arise for us as tropicalizations of meromorphic functions on Riemann surfaces and maps between Riemann surfaces, respectively. However, not all such maps are relevant to geometry. The balancing condition identifies a subset of such functions.

Balanced functions are the tropical analogues of rational functions on algebraic curves. The corresponding notion for maps between tropical curves is recorded next.

In other words, given a map \(\Gamma' \to \Gamma\), the local degree at a vertex \(v'\) of \(\Gamma'\) can be measured after choosing a tangent direction at its image \(v = F(v')\) in \(\Gamma\). This measurement is given by the sum of the slopes along all directions in \(\Gamma'\) based at \(v'\) that map to the chosen direction in \(\Gamma\). The purity condition asserts that this choice of direction in the target does not affect the local degree. In practice, this is how the condition is used. The harmonic condition asserts that purity holds at every vertex of the domain \(\Gamma'\).

The harmonic condition for a map \(F\colon \Gamma' \to \Gamma\) can also be described more conceptually though less practically in terms of balanced functions. Specifically, a piecewise linear map is harmonic if and only if, for any open set \(U \subset \Gamma\), the pullback of any balanced function \(U \to \mathbb{R}\) is again a balanced function \(F^{-1}(U) \to \mathbb{R}\). In particular, compositions of harmonic maps are harmonic. See [17] for further details.

1.3 Mildness conditions↩︎

We introduce two conditions on harmonic maps that allow realizability to be analyzed in a straightforward manner. The discussion here assumes that all vertex genera are equal to \(0\). The contents of this section do not require a connectivity hypothesis on the tropical curves.

Given a finite harmonic map of tropical curves, each vertex of the domain determines potential discrete data for a map between Riemann spheres. We record this as a construction below.

Given a finite harmonic map \(F\colon \Gamma'\to \Gamma\), the set \(\mathsf R(F,v')\) may be empty — for example, the conditions on ramification provided by Riemann–Hurwitz might obstruct the existence of such a map. Even if the Riemann–Hurwitz constraints are satisfied, the set of maps may still be empty.

In the next subsection, we introduce tropicalization for maps between curves and the associated inverse problem. Locally realizable finite harmonic maps will turn out always to be realizable.

1.4 Tropicalization for families of curves, functions, and covers↩︎

We discuss tropicalization for maps between curves. The notion has been treated before: using Berkovich spaces, by Amini, Baker, Brugallé, and Rabinoff [17], [18], as well as using admissible covers and logarithmic mapping spaces in [9], [23]. All these papers prove much more than we need; however, they require additional technical machinery. We provide an elementary presentation tailored to our applications.

1.4.1 First for curves↩︎

Let \(K\) be a discretely valued field with valuation ring \(R\) and residue field \(\mathbb{C}\). Let \(t\in R\) be a uniformizing parameter and let \(\mathsf{val}\) denote the valuation on \(K^\times\).

Consider a smooth marked curve \((C,p_1,\ldots,p_r)\) of genus \(g\) over \(K\). We will assume that \(2g(C)-2+r\) is positive. By the stable reduction theorem, or equivalently by the weak valuative criterion of properness for \(\overline{{\mathcal{M}}}\vphantom{{\mathcal{M}}}_{g,r}\), after possibly replacing \(K\) with a finite field extension \(K\subset K'\), there exists a unique stable model. Specifically, this is a family: \[(\mathcal{C},p_1,\ldots,p_r)\to \operatorname{Spec}R',\] The special fiber \(\mathcal{C}_0\) of this family is a stable, nodal, genus \(g\) curve. The map \(\operatorname{Spec}R'\to \operatorname{Spec}R\) is a finite surjective map, possibly ramified over its closed point.

The total space \(\mathcal{C}\) is a toroidal surface, with the boundary given by the special fiber, together with the closure of the \(r\) sections. This means that the total space is smooth away from the nodes of \(\mathcal{C}_0\). At the nodes, the space is allowed to have singularities of \(A_{n-1}\)-type, given locally by \[xy = \alpha\cdot s^{n},\] where \(\alpha\) is a unit in \(R'\), and \(s\) is a uniformizer in \(R'\).

By convention, we extend the valuation \(\mathsf{val}\colon K^\times\to {\mathbb{Z}}\subset {\mathbb{R}}\) to \(K'\), in such a way that the valuation of the parameter \(t\) is still \(1\), and the valuation of \(s\), the new uniformizer in \(R'\), lies in \(\mathbb{Q}\). In fact, it will be equal to \(\frac{1}{m}\) where \(m\) is the order of the field extension \(K\subset K'\).

The tropicalization of \((C,p_1,\ldots,p_n)\) is a tropical curve denoted \(\Gamma(C)\), \(\Gamma(\mathcal{C})\), or simply \(\Gamma\) when the context is clear, determined as follows:

• The abstract graph is the dual graph of \(\mathcal{C}_0\), with genus decoration at the vertices.

• The marking function \([n]\to V(G)\) sends \(i\) to the vertex whose associated component contains \(p_i\).

• Let \(e\) be an edge with associated node \(q_e\). If \(q_e\) is an \(A_{k-1}\) singularity in \(\mathcal{C}\), then the length of the edge is \(k\).

The cone over this tropical curve \(\Gamma\) is a generalized cone complex \(\Sigma(\mathcal{C})\), see [24]. The valuation on \(R'\) gives rise to a map \[\Sigma(\mathcal{C})\to {\mathbb{R}}_{\geq 0}.\] Recall that the valuation of \(s\) is an element of \(\mathbb{Q}_{\geq 0}\). The fiber of the map above over the point \(\mathsf{val}(s)\) is canonically identified with \(\Gamma\), and enhances it with the structure of a metric graph with rational edge lengths.

1.4.2 ...then with a function...↩︎

We add a meromorphic function to the picture. Let \((C,p_1,\ldots,p_r)\) be a curve over a discretely valued field \(K\). Consider a rational map \[\begin{tikzcd} C \arrow{d}\arrow[dashed]{r}{\varphi} &\mathbb{G}_m\\ \operatorname{Spec}K, & \end{tikzcd}\] whose indeterminacy is contained in the set of marked points \(\{p_1,\ldots,p_r\}\). The added structure of the map to \(\mathbb{G}_m\) enhances the tropical curve \(\Gamma\) with a balanced function \[F\colon \Gamma\to{\mathbb{R}}.\] The slope of \(F\) along the ray corresponding to \(p_i\) equals the order of the zero or pole of \(\varphi\) at the point \(p_i\) on \(C\).

The map \(F\) can be constructed using the theory of harmonic functions on Berkovich spaces [1]. Alternatively, we can use the semistable reduction theorem for maps to the pair \((\mathbb{P}^1|0+\infty)\), as worked out by J. Li [14]. The output is that, after possibly replacing \(K\) with a field extension, there is a diagram \[\begin{tikzcd} \mathcal{C}\arrow{dr}\arrow{rr}{\psi} & & \mathcal{P}\arrow{dl}\\ &\mathop{\mathrm{Spec}}R',& \end{tikzcd}\] completing the map \(\varphi\) above, and further: (i) where \(\mathcal{C}\) is a semistable family of genus \(g\) curves with general fiber \(C\), (ii) the space \(\mathcal{P}\) is a semistable family of \(2\)-pointed genus \(0\) curves, with special fiber a chain of \(\mathbb{P}^1\)’s, and (iii) the preimages of nodes/markings of \(\mathcal{P}_0\) are nodes/markings of \(\mathcal{C}_0\).

The conditions (i)–(iii) are exactly the conditions that guarantee that \(\mathcal{C}_0\to\mathcal{P}_0\) is a relative stable map in the sense of Li.² The surfaces \(\mathcal{C}\) and \(\mathcal{P}\) are again toroidal surfaces with at worst \(A_k\)-singularities. The toroidal structure on \(\mathcal{P}\) is given by the special fiber, together with two sections, corresponding to the closure of \(0\) and \(\infty\) in \(\mathbb{P}^1_K\). By construction, the preimage of the toroidal boundary of \(\mathcal{P}\) is contained in that of \(\mathcal{C}\), so there is an induced map of cone complexes \(\Sigma(\mathcal{C})\to \Sigma(\mathcal{P})\). However, \(\operatorname{Spec}R\) is also a toroidal pair with boundary given by the closed point and the cone complex \({\mathbb{R}}_{\geq 0}\). Since the arrows in the triangle above have the property that the preimage of the boundary is contained in the toroidal boundary, the maps are toroidal, and by functoriality results for tropicalizations [24], [25], we obtain a triangle of cone complexes: \[\begin{tikzcd} \Sigma(\mathcal{C})\arrow{rr}\arrow{dr} & & \Sigma(\mathcal{P})\arrow{dl}\\ &{\mathbb{R}}_{\geq 0}.& \end{tikzcd}\] Again, the fiber of \(\mathcal{P}\) over the valuation of the uniformizer in \(R'\) is a canonically polyhedral decomposition of \({\mathbb{R}}\). The fiber produces a map \[F\colon \Gamma'\to {\mathbb{R}}.\] We note that it is possible that, when viewed as the fiber over \(\Sigma(\mathcal{C})\), the graph \(\Gamma'\) contains vertices that are \(2\)-valent and whose corresponding components have \(0\) genus. However, by balancing, it is straightforward to check that \(F\) descends to a piecewise linear function after coarsening \(\Gamma'\) to remove these \(2\)-valent vertices from the graph structure, i.e., “erasing” the vertex. This is precisely the graph \(\Gamma\) defined by the stable limit in the previous discussion.

We call the resulting balanced function \[\Gamma\to {\mathbb{R}}\] the tropicalization of the data \((C,p_1,\ldots,p_r,\varphi)\).

1.4.3 ...and finally for covers↩︎

In the discussion above, we obtain from the meromorphic function on a curve \((C,\varphi)\) a map \[C\to \mathbb{P}^1\] of pairs – where \(C\) is marked at the roots and poles of \(\varphi\) and \(\mathbb{P}^1\) is marked at \(0\) and \(\infty\). The resulting semistable reduction theorem for maps of pairs [14] gave rise to the tropical curve and balanced function.

Going a step further, we can in addition mark all the branch points of \(\varphi\) on \(\mathbb{P}^1\), and then mark all the preimages of these points in \(C\). We denote the marked points on \(\mathbb{P}^1\) by \(b_1,\ldots, b_m\) and the points on the domain \(p_{i1},\ldots,p_{ik_i}\), where \(\varphi\) sends \(p_{im}\) to \(b_i\) for all \(m\). We now view \(\varphi\colon C\to\mathbb{P}^1\) as a map of pairs, with all ramification and branch points (and possibly more) marked.

Now suppose \(\varphi\colon C\to \mathbb{P}^1\) is as above, but defined over a field \(K\) equipped with a discrete valuation, inducing a valuation subring \(R\). We may again apply semistable reduction for maps of pairs. The results of Li apply again in this setting, but in fact, once all ramification is marked, this follows from earlier work. Specifically, we may use the semistable reduction theorem for admissible Hurwitz covers of nodal curves, due to Harris–Mumford [16]. To summarize the output, possibly after replacing \(K\) with a finite extension, we have a map \[\begin{tikzcd} \mathcal{C}\arrow{dr}\arrow{rr}{\psi} & & \mathcal{P}\arrow{dl}\\ &\mathop{\mathrm{Spec}}R,& \end{tikzcd}\] where \(\mathcal{C}\) is a semistable model for \(C\) and \(\mathcal{P}\) is a semistable model for \(\mathbb{P}^1\). The special fiber of \(\mathcal{P}\) is a tree, but not necessarily a chain, of smooth rational curves.

Proceeding as in the previous case, we obtain a triangle of cone complexes \[\begin{tikzcd} \Sigma(\mathcal{C})\arrow{rr}\arrow{dr} & & \Sigma(\mathcal{P})\arrow{dl}\\ &{\mathbb{R}}_{\geq 0}.& \end{tikzcd}\] If we pass to the fibers over \(1\), we obtain a map \[F\colon \Gamma\to T\] where \(\Gamma\) is a tropical curve and \(T\) is a tropical curve whose underlying graph is a tree.

In all these constructions, we ultimately obtain a tropical curve with possibly nonzero genus decorations at the vertices. But we flag again that in our applications, which concern reversing this process, we will start with tropical curves whose vertex decorations are everywhere equal to \(0\).

1.5 Hurwitz modifiability↩︎

In this section, we explain the main technical theorem on which our results and examples rest. We begin with the strategy.

1.5.1 Basic strategy↩︎

Let \(F\colon \Gamma\to{\mathbb{R}}\) be a balanced function. Our goal is to determine if it arises via the procedure described in Section 1.4.2. In a moment, we will explain a procedure that modifies harmonic maps of graphs.

The idea is that these modifications will make \(F\) closer to being finite harmonic. If this is achieved completely, and if one can independently verify local realizability of Definition 5, then one can appeal to the deformation theory of admissible covers to establish realizability.

Conversely, if we have realizability of \(F\colon \Gamma\to{\mathbb{R}}\), the semistable reduction theorem that establishes the properness of the space of admissible covers will guarantee that such modifications to locally realizable finite harmonic maps exist. We emphasize that the result itself is known [17], [18], see also [26]; however, we believe the presentation here may be valuable.

1.5.2 Modifications and \(H\)-modifiability↩︎

We introduce the key combinatorial move in this paper.

Note that modifications are local on \(\Gamma\), so they apply equally well in the disconnected setting.

Modifications are our key tool in analyzing realizability, and this relies on the following:

2 Non-superabundant functions↩︎

We prove that every balanced function \(\Gamma \to {\mathbb{R}}\) is H-modifiable, provided it contracts no cycles. In particular, this implies realizability in genus \(0\). The section includes technical constructions that are needed in later sections.

2.1 Basic structures↩︎

We begin with preparatory results about realizable or \(H\)-modifiable balanced functions.

The theorem above will be used to reduce realizability statements to nearly trivalent cases.

The next result shows that modifications cannot make things “worse” for H-modifiability. Consequently, in many cases, we can assume that the harmonic maps we are interested in do not contract any rays.

We recall that the star of a vertex \(v\) in a metric graph \(\Gamma\) consists of \(v\), all rays adjacent to \(v\), and the interiors of all edges whose closures contain \(v\). These stars may contain non-compact edges; the notions of harmonicity and local realizability extend verbatim to this setting.

2.2 Realizability without contracted cycles↩︎

We now work towards realizability for balanced functions with tree domains, and more generally, balanced functions that do not contract any cycles.

The following proposition serves as technical input to glue \(H\)-modifications of graphs along contracted trees.

  **Construction of the $H$-modification:** For each $i$, let $B_i$
  be the point on $e_i$ at distance $a$ from $A_i$, and let
  $\mathcal{B}$ be the set of these points. For each distinct
  $P \in \mathcal{B}$, attach $d_P$ rays
  $\ell_{P,1},\dots,\ell_{P,d_P}$ to the domain at $P$. This
  produces a modification $T'$ of the domain $T$.

  Add a ray $\ell$ to the target graph at $0 \in \mathbb{R}$, and
  place a vertex $N$ on $\ell$ at distance $a$ from $0$. This yields
  a modification $\Sigma'$ of the target. Map each newly added ray
  in the domain to $\ell$ with slope $1$, starting at $N$.

  **Applying the inductive hypothesis:** Remove the half-open
  segments $[A_i,B_i)$ from $T$, giving a disjoint union of trees
  $S_1,\dots,S_m$, each containing strictly fewer edges than $T_1$.
  For each $j$, form $S'_j$ by attaching the stars of all vertices
  in $\mathcal{B} \cap S_j$ to $S_j$.

  Viewing the interior of $\ell$ as a copy of $\mathbb{R}$, each
  $S'_j$ induces a map $h_j \colon S_j \to \mathbb{R}$ that
  contracts $S_j$ to $N$. By the inductive hypothesis, each $h_j$
  admits an $H$-modification satisfying the required properties.
  Applying the gluing construction of
  Lemma [3](#lemma:glueing){reference-type="ref"
  reference="lemma:glueing"}, we obtain an $H$-modification of the
  original map $T \to \mathbb{R}$ with the desired properties.

We now have the ingredients to show that maps without contracted cycles are \(H\)-modifiable.

In particular, this implies the following:

3 Genus one results↩︎

We consider balanced functions on tropical curves of genus \(1\), focusing on the case where the cycle is contracted. Our goal is to discover well-spacedness by studying modifiability.

3.1 The well-spacedness condition↩︎

We recall the statement of well-spacedness. Following [10], the condition here differs slightly from Speyer’s definition outside the trivalent case.

In this section, let \(\Gamma\) denote a tropical curve of genus \(1\). It has a unique cycle, and the vertices on this cycle are called cycle vertices. Every point of \(\Gamma\) lies on a unique path leading to the cycle. Suppose \[F\colon \Gamma \to T\] is a harmonic map that is constant on the cycle. A point \(P \in \Gamma\) is a critical point if \(F\) is constant along the path from \(P\) to the cycle but non-constant on some edge incident to \(P\). Intuitively, a critical point is the first point along a path from the cycle at which \(F\) begins to vary.

In what follows, the path from a critical point to the cycle of \(\Gamma\) will be called a critical path.

If \(\Gamma\) is trivalent and \(V\) is a critical point of a balanced map, then \(n(V)\) must be \(2\). Thus, when \(\Gamma\) is trivalent, the condition says that there are at least two critical paths of minimal length, which is the condition stated in [11]. The flag number will only be important outside the trivalent case. Our goal is the following restatement of Theorem [thm:32well-spacedness].

3.2 Necessity of the well-spacedness condition↩︎

We will often study directed paths, which we will denote by \(w\), and will typically drop the adjective “directed” but specify the direction explicitly when needed. A path consisting of \(k\) edges has finitely many vertices \(P_1,\ldots,P_{k+1}\), and we will sometimes use the notation³ \[w = \overline{P_1,\ldots,P_{k+1}}.\] Given such a path, we say \(P_1\) is the starting vertex of \(w\), \(P_{k+1}\) is the terminal vertex, and any other vertices are the interior vertices.

We now introduce two important notions required for the analysis of modifiability.

The next lemma uses these notions to show that harmonicity allows us to “copy” paths, eventually producing the multiple critical paths that appear in the well-spacedness condition.

We now record a technical consequence of the lemma.

We now prove necessity of the well-spacedness condition.

3.3 Preparation disguised as illustrative examples↩︎

We present several examples illustrating realizability and well-spacedness via modifications. These examples are intended to aid the reader’s intuition, but they also serve as preparatory base cases for the proof of sufficiency of well-spacedness in the next section.

3.3.1 A single ray attached to the cycle↩︎

As a first example, we explain how realizability may be obstructed. We begin with the following lemma.

We can now give the example in the form of the following:

If instead the critical point had three outgoing non-constant directions, realizability holds.

3.3.2 Two paths from the cycle↩︎

We next consider a slightly more complicated situation: a superabundant tropical curve that is well-spaced due to the relative positions of two critical paths.

Next, we consider an example where there are two critical paths with a shared edge.

3.4 The full realizability criterion↩︎

With the cases above done by hand, we now explain how to inductively deduce the general case.

We begin with a basic reduction.

In what follows, we assume the conditions of the lemma hold. We denote the minimal critical path by \(A_1B_1\), where \(B_1\) lies on the cycle and \(A_1\) is the critical point.

The structure of our proof of the theorem above is to induct on the number of non-minimal critical trees, with the base cases being the ones above. In the induction, a few finer properties of the modification are relevant. Before proceeding further, we record a combinatorial trick that arises frequently in practice, and is relevant for the proofs and definitions that follow.

Remark 18 (Long images and slope adjustments). Consider \(F\colon \Gamma\to{\mathbb{R}}\) as in Figure 7. If we add a new critical path starting at a new point of the cycle, we have the following:

Before this new attachment, the standard \(H\)-modification adds a ray based at \(O\) in the target; it then maps \(A_1B\) and \(A_2B\) onto this ray with slope \(1\). Balancing forces the outgoing slope at \(B\) to be \(2\), and we map \(BC\) with slope \(2\), the two edges of the loop with slope \(1\), and then modify by adding two rays at a point of the loop.

We now contemplate adding \(A_3D\). It can happen that \(|A_3D|\) is bigger than \(|BC|+|A_1B|\), but smaller than \(|A_1B|+2|BC|\). Now \(A_3\) and \(A_1\) map to the same point on the target, and since the slope of the modification \(A_3D\) may be forced to be \(1\), this can make it impossible to extend the standard modification of Figure 7 to accommodate the new critical tree.

The solution is as follows. Choose a point \(E\) on \(BC\) such that \(|A_1B|+2|BE|+|EC|=|A_3 D|\)length of \(A_1B\), plus twice the length of \(BE\), plus the length of \(EC\) is equal to the length of \(A_3D\) . Map \(BE\) with slope \(2\) as before, but map \(EC\) with slope \(1\). This creates an issue with harmonicity at \(E\), but we will come back to fix this later. The image of \(C\) is a point of the new ray added at \(O\) in the target. Add another infinite ray at this image, so there are now two rays based at the image of \(C\); call them \(r\) and \(r'\). Now, map the clockwise arc onto \(r\) by dividing the arc at its midpoint, map the first half onto \(r\), and map the second half back towards the image of \(C\). The midpoint of this arc is a turning point. By adding further rays at this turning point, we can ensure harmonicity. Do the same for the counter-clockwise arc, mapping it to \(r'\).

Finally, to fix the issue at the point \(E\), we attach a tree to \(E\). Precisely, if we delete the image of \(E\) from the target, there are two connected components. Let \(T\) be the (closure of the) component containing the image of the cycle. Attach a copy of \(T\) to the domain based at \(E\) and map it down isomorphically (with slope \(1\)). This produces an \(H\)-modification.

The remark above shows that the lengths of images of paths in modifications plays a role in the analysis, and also records the “slope adjustment trick”, which we will use later.

We are now ready to proceed with the rest of the proof.

The examples/base cases discussed above are each cycle decomposing.

In the process of adding non-minimal critical paths, it is more straightforward to do so if the path is “very long” – we make this precise below. Recall that \(A_1B_1\) is always a minimal critical path in our notation.

The next lemma explains how to attach “very long” critical paths, while preserving the cycle decomposing nature of the modification.

Given a balanced function \(F\colon \Gamma\to {\mathbb{R}}\) that contracts the cycle, if we delete the cycle, there is a forest of contracted edges that connects the cycle to the critical points. We refer to the connected components as critical trees. It is possible to have a well-spaced map \(F\) with a unique critical tree. There are two essential ways in which this can happen: first, when there are two critical points \(A_1\) and \(A_2\) whose associated critical paths share an edge. We call this a “\(Y\)-shape”, as the critical points are the top endpoints of the \(Y\) and the bottom of the \(Y\) attaches to the cycle. The other case we isolate is the “\(3\)-flag” case – there is a single critical point, but there are three flags based at this point where the map is non-constant.

The next lemma enables us to take the \(Y\)-shaped picture and connect it to a tree via a contracted edge that attaches to the this unique critical tree.

The analogous lemma holds, with essentially the same proof, for the \(3\)-flag case, rather than the \(Y\)-shaped case.

We come to the final statement.

4 Genus two results↩︎

In this section, we prove Theorems [thm:32genus-2-A] and [thm:32genus-2-B], along with related results. We begin by establishing the existence of \(H\)-modifications in several base cases via explicit constructions. We then use moduli-theoretic considerations to explain why “sufficiently long” trees may be grafted onto these constructions, thereby obtaining the general results.

4.1 Explicit constructions↩︎

Any abstract tropical curve of genus \(2\) can be stabilized by pruning trees and removing bivalent vertices. In the trivalent case, the resulting graph is necessarily either a \(\Theta\)-graph or a dumbbell graph. The complexity in the two cases is analogous, so we focus primarily on the \(\Theta\) case and include a representative result for the dumbbell case at the end of the section.

We study balanced functions that contract a genus \(2\) subgraph. A dimension count suggests that realizability should impose two conditions on the moduli of the domain curve. We exhibit several distinct ways in which these two conditions may arise.

We begin with the genus \(2\) analogue of Proposition 15. Here, the codimension-\(2\) condition on moduli appears as a direct generalization of Speyer’s condition.

Based on this construction, one might guess that “three edge lengths equal” is a perfect generalization to the genus \(2\) case. But the next proposition shows that the picture is more subtle.

A tropical curve of genus \(2\) is equipped with a unique involution whose quotient is a tree – the “tropical hyperelliptic involution”. The fixed points of this involution are Weierstrass points and points that are swapped by the involution are called conjugate. For a \(\Theta\) graph, the Weierstrass points are the midpoints of each of the three edges, and the conjugate points are pairs of points whose center of mass is the midpoint.

Realizability is a closed condition, the proposition also holds when both attaching points are a single Weierstrass point. However, at the Weierstrass points, new phenomena appear.

We expect that there are analogues of the theorems above in the other combinatorial type of genus \(2\) curve – the dumbbell graph. These are building blocks of the now-famous “chain of cycles”, appearing in [27] and later developments in Brill–Noether theory. We include the following representative construction for the dumbbell.

4.2 Appending long trees↩︎

Let \(F\colon \Gamma \to {\mathbb{R}}\) be a balanced function on a tropical curve. A subgraph \(\Gamma_0 \subset \Gamma\) is called a frame if \(\Gamma \smallsetminus\Gamma_0\) is a forest and each connected component of this forest is attached to the frame by an \(F\)-contracted edge. These contracted edges will be called connecting edges. A frame is realizable if it is realizable as a tropical map.

Our goal is to prove the following.

In the situations of interest, the frames will be the realizable tropical maps constructed in Section 4.1. However, we present the argument in a way that allows any independently constructed realizable tropical maps to serve as frames.

Together with the constructions of Section 4.1, this theorem immediately implies Theorems [thm:32genus-2-A] and [thm:32genus-2-B].

4.2.1 Preparations: tropicalization for the double ramification locus↩︎

We deduce the theorem from formal properties of tropical compactifications [25], [28] and extended tropicalizations [24], applied to the double ramification locus in \(\mathcal{M}_{g,n}\).

Fix a vector \(A = (a_i) \in \mathbb{Z}^n\) with vanishing sum. The double ramification locus \[{\sf D}_g(A) \subset \mathcal{M}_{g,n}\] parameterizes pointed curves \([C,p_1,\ldots,p_n]\) such that \(\mathcal{O}_C\!\left(\sum a_i p_i\right)\) is trivial.

Let \(K\) be a non-archimedean valued field. By results of Abramovich–Caporaso–Payne⁴, there is a natural tropicalization map \[\mathsf{trop}\colon \mathcal{M}_{g,n}(K)\to \mathcal{M}_{g,n}^{\sf trop},\] from the \(K\)-points of the moduli space of smooth pointed curves to its tropicalization. This map sends a curve over \(K\) to the metrized dual graph of its stable model, as described in Section 1.4.

Composing with the inclusion \[{\sf D}_g(A)(K)\hookrightarrow \mathcal{M}_{g,n}(K),\] we obtain a map whose domain parameterizes exactly those curves that admit a map to \(\mathbb{P}^1\) with ramification orders prescribed by \(A\) at the marked points. Such a map, when it exists, is unique up to scalar.

The image of this map is precisely the locus of tropical curves that admit a balanced function with slope \(a_i\) along the \(i^{\rm th}\) ray. Again, such tropical maps are unique up to translation when they exist. Thus, this image is exactly the realizable locus of interest.

Since this construction works for any non-archimedean valued field, we choose \(K\) whose valuation is surjective onto \(\mathbb{R}\). The image of \({\sf D}_g(A)\) under \(\mathsf{trop}\) is then called its tropicalization. The structure theory of tropicalizations implies the following [24], [25].

Denote the tropicalization by \({\sf D}_g^{\sf trop}(A)\) and choose any cone complex structure on it. By toroidal geometry, this subcomplex of \(\mathcal{M}_{g,n}^{\sf trop}\) determines an open subset of a toroidal modification of of the moduli space of curves \(\overline{{\mathcal{M}}}\vphantom{{\mathcal{M}}}_{g,A}\). By the basic theorems of tropical compactifications [25], [28], we have the following.

We denote the closure by \(\overline{\sf D}_g(A)\).

We consider compactifications on the tropical side. Every cone complex \(\Sigma\) admits a canonical compactification [24]. Every cone \(\sigma\) can be expressed as \(\operatorname{Hom}(P,{\mathbb{R}}_{\geq 0})\) where \(P\) is a finitely generated monoid. By replacing \({\mathbb{R}}_{\geq 0}\) with \({\mathbb{R}}_{\geq 0}\cup\{\infty\}\) with the obvious monoid structure, one compactifies the cone. Gluing the compactified cones together, we obtain a compactification \[\Sigma\hookrightarrow\overline{\Sigma}.\] The complement of \(\Sigma\) decomposes into a disjoint union of cone complexes, called the infinite faces. See [24] for further details.

Now, just as \(K\)-points of \({\sf D}_g(A)\) map onto the cone complex \({\sf D}_g^{\sf trop}(A)\), the \(K\)-points of the compactification space \(\overline{\sf D}_g(A)\) map onto a compactification \(\overline{\sf D}_g^{\sf trop}(A)\), so we have: \[\begin{tikzcd} {\sf D}_g(A)(K)\arrow[hookrightarrow]{r}\arrow{d}& \overline{\sf D}_g(A)(K)\arrow{d}\\ {\sf D}_g^{\sf trop}(A)\arrow[hookrightarrow]{r}& \overline{\sf D}_g^{\sf trop}(A). \end{tikzcd}\]

We will use this diagram to complete the proof of our genus \(2\) results as follows. First, we will argue that certain points at infinity lie in the image of the right vertical map. We will then deduce the existence of points in the interior formally.

4.2.2 Realizability in the infinite faces↩︎

We now prove Theorem 24.

5 Generalizations via dimensional reduction↩︎

We have thus far examined maps \(f\colon \Gamma\to{\mathbb{R}}\). However, one is often interested in maps to higher-dimensional vector spaces. There is a simple bootstrapping technique that allows one to deduce statements about maps to \(\mathbb{R}^r\) from statements about balanced functions.

5.1 Setup↩︎

We recall some basic terminology. Let \(\Gamma\) be a tropical curve. A tropical map \(\Gamma\to{\mathbb{R}}^r\) is a map such that for any linear projection \({\mathbb{R}}^r\to{\mathbb{R}}\), the induced map \[\Gamma\to{\mathbb{R}}^r\to{\mathbb{R}}\] is balanced. It is sufficient to check this for the coordinate projections with respect to any coordinate system.

Recall that a balanced map \(f\colon \Gamma\to {\mathbb{R}}^r\) has a type. Loosely, this is the data obtained by discarding the metric data but retaining all other discrete data associated to \(f\). Formally, it consists of the following:

(i) The marked graph \(G\) underlying \(\Gamma\),

(ii) For each directed edge \(e\) of \(G\), the edge direction in \({\mathbb{R}}^r\), or equivalently, for every linear projection \({\mathbb{R}}^r\to {\mathbb{R}}\), the slope of the composite along \(e\),

(iii) For each marked ray \(\ell\) of \(G\), its edge direction, or equivalently, for every linear projection \({\mathbb{R}}^r\to {\mathbb{R}}\), the slope of the composite along \(\ell\), directed away from the root of the ray.

If we fix the type \(\Theta\), there is a space parameterizing tropical curves with a balanced map to \({\mathbb{R}}^r\) having this type. It is naturally isomorphic to \(\sigma^\circ\times {\mathbb{R}}^r\), where \(\sigma\) is a rational polyhedral cone and \(\sigma^\circ\) is its interior. The cone \(\sigma\) should be viewed as a conical subset \[\sigma\subset{\mathbb{R}}_{\geq 0}^E,\] where \(E\) is the edge set of \(G\), consisting of those assignments of edge lengths that support a continuous piecewise linear map of type \(\Theta\). In the interior, there is the additional condition that the edge lengths are positive.

It is slightly more convenient to work with the space of maps up to translation. Given a tropical curve \(\Gamma\) with underlying graph \(G\) and a compatible type \(\Theta\), the graph \(\Gamma\) may or may not admit a map to \({\mathbb{R}}^r\) of type \(\Theta\), and if it does, there is an \(\mathbb{R}^r\)-torsor worth of such maps.

It follows from the main result of [9] that there is a closed conical subset \(\mathsf M^{\sf real}_\Theta\) parameterizing those maps (again, up to translation) that are realizable. That is, they arise from tropicalizing a map \[\mathcal{C}\dashrightarrow \mathbb{G}_m^r\] that is regular outside a marked set of points \(p_1,\ldots,p_n\) and defined over a non-archimedean field \(K\). The tropicalization procedure is identical to the one presented in Section 1.4.2, applied coordinatewise.

5.2 The dimensional reduction↩︎

We now assume that the marked graph \(G\) underlying \(\Theta\) is trivalent with genus \(0\) vertices – the maximally degenerate case. Fix a linear projection \(\chi\colon \mathbb{R}^r\to \mathbb{R}\). There is an associated type \((\Theta,\chi)\), which is the type of a map to \({\mathbb{R}}\), obtained from a map of type \(\Theta\) by projecting. We therefore obtain a natural realizable subset \[\mathsf M^{\sf real}_{\Theta,\chi}\subset {\mathbb{R}}_{\geq 0}^E\] comprising those prescriptions of edge lengths such that the resulting map is realizable.

Fix a splitting of the torus \(\mathbb{G}_m^r\) corresponding to basis \(\chi_1,\ldots,\chi_r\) of the character lattice. Since we can compose a realization of \([F\colon \Gamma\to{\mathbb{R}}^r]\) with the character \(\chi_i\) to produce a realization of \([\chi_i\circ F]\) there is a containment: \[\mathsf M^{\sf real}_{\Theta} \subset \bigcap_{i=1}^r \mathsf M^{\sf real}_{\Theta,\chi_i}.\] The following statement allows us to conclude the reverse containment under some circumstances. The result is, in some sense, implicit in [8].

Recall that \(g\) denotes the genus of the marked graph \(G\), which by assumption is equal to the first Betti number of \(G\).

For example, by applying the theorem above, the \(r = 1\) case of the realizability of Speyer’s theorem, proved in Section 3, implies the general case of well-spacedness for trivalent domains. Similarly, the genus \(2\) results for \(r=1\) imply results in higher dimensional targets. However, one should note that there exist more exotic forms of non-realizability for tropical curves of higher genus. See recent work of Koyama [7].

It should be possible to modify the theorem to work without the maximal degeneracy hypothesis. However, our goal is to show the utility of the methods, we do not go into this here.

References↩︎