Skeletons of stable maps II: superabundant geometries
- Dhruv Ranganathan1Email author
https://doi.org/10.1186/s40687-017-0101-5
© The Author(s) 2017
Received: 23 September 2016
Accepted: 27 February 2017
Published: 1 June 2017
Abstract
We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov–Witten theory—the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer’s well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition.
1 Background
Central to the application of tropical techniques to questions in algebraic geometry are so-called lifting theorems. Given a “synthetic” tropical object, such as a weighted balanced polyhedral complex, one must understand whether this object is the tropicalization of an algebraic variety. We deal in this paper with the case of curves. The tropical lifting question in this setting asks, when does an embedded tropical curve in \({\mathbb R}^n\) arise as the tropicalization of an algebraic curve in a torus over a nonarchimedean field? This question becomes highly nontrivial in the so-called superabundant case and has been the primary obstacle to the application of tropical curve counting techniques in high genus settings. A tropical curve in \({\mathbb R}^n\) encodes the combinatorial data in a degenerate logarithmic stable map to a toric variety. If the tropical curve is superabundant, i.e., if the tropical deformation space is larger than expected, the obstruction group of this degenerate logarithmic map is nonzero. As a result, such a map may not deform and the tropical curve may fail to be realizable. See Sect. 2.2 for a precise definition of superabundance. The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7, 15, 19, 23–25, 29, 34].
In this note, we use recent technical breakthroughs in nonarchimedean geometry and the theory of logarithmic maps to provide a conceptual framework in which the realizability question may be approached. To demonstrate the efficacy of this framework, we use it to give simple proofs of three new results for lifting tropical curves. The same framework provides new insight into the structure of realizability conditions more globally—the locus of realizable tropical curves is given by a union of bend loci of a collection of tropical polynomials in the edge lengths of the tropical curve.
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
.
Theorem A
Let
for \({t\in [0,1)}\) be a continuously varying family of parameterized tropical curves. Let
denote the limit of this family in the moduli space of parameterized tropical curves. If
is realizable for all \(t\in [0,1)\), then the limiting map
is realizable.
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.
Theorem B
Let
be a parameterized tropical curve of genus 1 with a unique genus 1 vertex v. Assume that the star of v in
is realizable. If
is well-spaced, then
is realizable.
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].
Theorem C
Let \(n\ge 3\). There exist superabundant parameterized genus 1 tropical curves
that lift to algebraic curves but violate the well-spacedness condition.
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.
Theorem D
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].
Remark 1.1
In the months between when the first version of this paper appeared on ar\(\chi \)iv and the final publication, there has been additional progress on tropical realizability. The framework established here has been used by Jensen and the author to prove realization theorems for superabundant tropical curves in the “chain of cycles” combinatorial type, in arbitrary genus. This has in turn found applications in Brill–Noether theory, see [18, Theorems A & B].
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 Tropicalization for maps and their moduli
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
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].
Theorem 2.2
The moduli space \(\mathscr {L}_\Gamma (X)\) of minimal logarithmic stable maps with discrete data \(\Gamma \) is a proper logarithmic algebraic stack with projective coarse moduli space.
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


- (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)
- (3)



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.
Remark 2.4
The terminology of “smooth” here is motivated by the following fact. Given a tropical curve
, one can consider any family \(\mathscr {C}\) of marked prestable curves over a valuation ring \({{\mathrm{Spec}}}(R)\), such that the skeleton of \(\mathscr {C}\) is
. The general fiber \(\mathscr {C}_\eta \) of this family is smooth if and only if the nonleaf edge lengths of
are all finite, i.e., if
is smooth in the above sense. To see this, consider an edge e of
corresponding to a node q of \(\mathscr {C}_0\). Formally locally near q, the total family may be described as \(xy = f\), where \(f\in R\), and the valuation of f is identified with the length of e. The valuation of f is infinity if and only if f is zero. In turn f is zero if and only if the node q persists in the generic fiber of \(\mathscr {C}\), see [1, 7].
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

- (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.
- (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.
- (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 \(\Delta \).

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
Definition 2.7
The recession type of a combinatorial type \(\Theta \) is obtained from
by collapsing all bounded edges of
to a single vertex. That is,
is a single vertex with genus g and marked edges, and the marked edges are each decorated by a contact order.
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.
Proposition 2.8
Let \(\Theta \) be the combinatorial type of a tropical stable map. The set of all tropical curves
together with an identification with the type \(\Theta \) is parameterized by a cone \(\sigma _\Theta \). Further, there are finitely many combinatorial types with fixed recession type.
Proof





Definition 2.9
A combinatorial type \(\Theta \) is said to be superabundant if the dimension of \(\sigma _\Theta \) is strictly larger than the expected dimension.


Definition 2.10


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.
- (1)
The source type
and \(\Theta '\) is obtained from the source graph
of \(\Theta \) by a (possibly trivial) sequence of edge contractions \(\alpha : G\rightarrow G'\).
- (2)
Given any vertex
and a vertex v such that \(\alpha (v) = v'\), then the cone \(\sigma _{v'}\) is a face of \(\sigma _v\).
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



Theorem 2.11
(Gross and Siebert [17, Section 1.5]) Let \([f:C\rightarrow X]\) be a logarithmic (pre)-stable map over \({{\mathrm{Spec}}}(P\rightarrow {\mathbb C})\) with combinatorial type \(\Theta \). The map [f] is minimal if and only if the dual monoid \({\text {Hom}}(P,{\mathbb N})\) is isomorphic to the monoid \(T_\Theta ({\mathbb N})\).
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

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
We obtain the following as an immediate corollary.
Corollary 2.13
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.
Theorem 2.14
The stack \(\mathscr {L}^{\mathrm {pre}}_\Gamma (\mathscr {A}_X)\) is a logarithmically smooth algebraic stack, locally of finite type, and dimension \(3g-3+n\).
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.
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
Proof
We derive the first part of Theorem D as a corollary.
Corollary 2.16
Proof
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 Proofs of lifting theorems
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.
Remark 3.1
The essential content of the above theorem is that the realizability conditions are given by the bend loci of the equations that describe the moduli space of maps locally. By Vakil’s Murphy’s law, one should expect these equations to become arbitrarily complicated, and thus one should also expect arbitrarily high complexity on the piecewise-linear side.
3.2 Realizability of limits: Theorem A


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.
Definition 3.2
Let
be a superabundant genus 1 tropical stable map. Let H be a hyperplane containing the loop L and consider the subgraph
, the connected component of
containing L. Denote the 1-valent vertices of
by \(v_1,\ldots ,v_k\) and by \(\ell _i\) the distance from \(v_i\) to L. The map f is well-spaced with respect to
H if the minimum of the multiset of distances \(\{\ell _1,\ldots , \ell _k\}\) occurs at least twice.
The map f is said to be well-spaced if it is well-spaced with respect to every hyperplane containing L.
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




Remark 3.4
Let
be a genus 1 tropical stable map as in the statement of Theorem B, and let p be the genus 1 vertex. We have chosen to formulate the result with the hypothesis that the star of p in the modified curve
is realizable rather than placing a hypothesis on
itself. This choice has been made because the local realizability of
is often easier to check, since one has an explicit coordinate on the nodal \(\mathbb {P}^1\) in the special fiber corresponding to p, c.f. [15, Proposition 2.8]. One could instead impose that the star of p in
is realizable, and the result continues to hold, as seen in the remark above.
3.5 Proof of Theorem C





A connected component of the intersection of an embedded tropical curve in \({\mathbb R}^3\) with a plane H. The black vertices indicate the points at which the cycle component leaves H. The edges labeled \(e_t\) and \(e'_t\) have length equal to \((1-t)\)
Acknowledgements
During the preparation of this paper, I have benefited greatly from conversations with friends and colleagues including Dori Bejleri, Renzo Cavalieri, Dave Jensen, Sam Payne, Martin Ulirsch, and Ravi Vakil. I would like to thank David Speyer, in particular, for sharing his insights on realizability in the good reduction case, see Remark 3.3. I would also like to thank Dan Abramovich for pointing out a gap in an earlier version of this paper. During important phases of the project I was a student at Brown University and Yale University, and it is a pleasure to acknowledge these institutions here. The final version of this document was greatly improved as a result of the comments of two anonymous referees. Final revisions were completed, while the author was a member at the Institute for Advanced Study in Spring 2017. This research was partially supported by NSF Grant CAREER DMS-1149054 (PI: Sam Payne).
We use \(\mathscr {C}\), C, and
to denote families of curves, single curves and tropical curves, respectively, choosing notation that best approximates the shape of these objects as found in the wild. We thank Dan Abramovich for this most creative of suggestions.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Abramovich, D., Caporaso, L., Payne, S.: The tropicalization of the moduli space of curves. Ann. Sci. Éc. Norm. Supér. 48, 765–809 (2015)MathSciNetMATHGoogle Scholar
- Abramovich, D., Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs II. Asian J. Math. 18, 465–488 (2014)MathSciNetView ArticleMATHGoogle Scholar
- Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M., Sun, S.: Logarithmic geometry and moduli. In: Farkas, G., Morrison, I. (eds.) Handbook of moduli: Volume I. Advanced Lectures in Mathematics (ALM), vol. 24. International Press, Somerville, MA. ISBN 978-1-57146-257-2/pbk (2013)Google Scholar
- Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M., Wise, J.: Skeletons and fans of logarithmic structures. In: Nonarchimedean and Tropical Geometry. Based on Two Simons Symposia, Island of St. John, March 31–April 6, 2013 and Puerto Rico, February 1–7, 2015, pp. 287–336. Springer, Cham (2016)Google Scholar
- Abramovich, D., Wise, J.: Invariance in logarithmic Gromov–Witten theory. arXiv:1306.1222 (2013)
- Baker, M., Payne, S., Rabinoff, J.: On the structure of non-Archimedean analytic curves. In: Tropical and Non-archimedean geometry, vol. 605 of Contemp. Math., pp. 93–121. Amer. Math. Soc., Providence, RI (2013)Google Scholar
- Baker, M., Payne, S., Rabinoff, J.: Nonarchimedean geometry, tropicalization, and metrics on curves. Algebr. Geom. 3, 63–105 (2016)MathSciNetView ArticleMATHGoogle Scholar
- Bieri, R., Groves, J .R.: The geometry of the set of characters iduced by valuations. J. Reine Angew. Math. (Crelle’s Journal) 347, 168–195 (1984)MATHGoogle Scholar
- Cavalieri, R., Hampe, S., Markwig, H., Ranganathan, D.: Moduli spaces of rational weighted stable curves and tropical geometry. In: Forum of Mathematics, Sigma, vol. 4 (2016)Google Scholar
- Cavalieri, R., Markwig, H., Ranganathan, D.: Tropicalizing the space of admissible covers. Math. Ann. 364, 1275–1313 (2016)MathSciNetView ArticleMATHGoogle Scholar
- Cavalieri, R., Markwig, H., Ranganathan, D.: Tropical compactification and the Gromov–Witten theory of \(\mathbb{P}^1\). Sel. Math. arXiv:1410.2837 (to appear)
- Chan, M.: Topology of the tropical moduli spaces \(\overline{M}_{2,n}\). arXiv:1507.03878 (2015)
- Chan, M., Galatius, S., Payne, S.: The tropicalization of the moduli space of curves II: topology and applications. arXiv:1604.03176 (2016)
- Chen, Q.: Stable logarithmic maps to Deligne–Faltings pairs I. Ann. Math. 180, 341–392 (2014)MathSciNetMATHGoogle Scholar
- Cheung, M.-W., Fantini, L., Park, J., Ulirsch, M.: Faithful realizability of tropical curves. Int. Math. Res. Not., rnv269 (2015)Google Scholar
- Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology 36, 335–353 (1997)MathSciNetView ArticleMATHGoogle Scholar
- Gross, M., Siebert, B.: Logarithmic Gromov–Witten invariants. J. Am. Math. Soc. 26, 451–510 (2013)MathSciNetView ArticleMATHGoogle Scholar
- Jensen, D., Ranganathan, D.: Brill–Noether theory for curves of a fixed gonality. arXiv:1701.06579 (2017)
- Katz, E.: Lifting tropical curves in space and linear systems on graphs. Adv. Math. 230, 853–875 (2012)MathSciNetView ArticleMATHGoogle Scholar
- Len, Y., Ranganathan, D.: Enumerative geometry of elliptic curves on toric surfaces. arXiv:1510.08556 (2015)
- Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Volume 161 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2015)Google Scholar
- Mikhalkin, G.: Enumerative tropical geometry in \(\mathbb{R}^2\). J. Am. Math. Soc 18, 313–377 (2005)MathSciNetView ArticleMATHGoogle Scholar
- Nishinou, T.: Correspondence theorems for tropical curves. arXiv:0912.5090 (2009)
- Nishinou, T.: Describing tropical curves via algebraic geometry. arXiv:1503.06435 (2015)
- Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135, 1–51 (2006)MathSciNetView ArticleMATHGoogle Scholar
- Payne, S.: Analytification is the limit of all tropicalizations. Math. Res. Lett. 16, 543–556 (2009)MathSciNetView ArticleMATHGoogle Scholar
- Popescu-Pampu, P., Stepanov, D.: Local tropicalization. In: Brugallé, E., Cueto, M.A., Dickenstein, A., Feichtner, E-M., Itenberg, I. (eds.) Algebraic and combinatorial aspects of tropical geometry. Proceedings based on the CIEM workshop on tropical geometry, International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain, December 12–16, 2011. Contemporary Mathematics, vol. 589. American Mathematical Society (AMS), RI. ISBN 978-0-8218-9146-9/pbk (2013)Google Scholar
- Ranganathan, D.: Skeletons of stable maps I: Rational curves in toric varieties. J. Lond. Math. Soc. arXiv:1506.03754 (2015) (to appear)
- Ranganathan, D.: Superabundant curves and the Artin fan. Int. Math. Res. Not. (2016)Google Scholar
- Silverman, J.: The Arithmetic of Elliptic Curves, Applications of Mathematics. Springer, Berlin (1986)View ArticleMATHGoogle Scholar
- Speyer, D.E.: Parameterizing tropical curves. I: curves of genus zero and one. Algebra Number Theory 8, 963–998 (2014)MathSciNetView ArticleMATHGoogle Scholar
- The Stacks Project Authors: Stacks project. http://stacks.math.columbia.edu (2017)
- Thuillier, A.: Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels. Manuscr. Math. 123, 381–451 (2007)View ArticleGoogle Scholar
- Tyomkin, I.: Tropical geometry and correspondence theorems via toric stacks. Math. Ann. 353, 945–995 (2012)MathSciNetView ArticleMATHGoogle Scholar
- Ulirsch, M.: Functorial tropicalization of logarithmic schemes: the case of constant coefficients. arXiv:1310.6269 (2013)
- Ulirsch, M.: Tropical compactification in log-regular varieties. Math. Z. 280, 195–210 (2015)MathSciNetView ArticleMATHGoogle Scholar
- Ulirsch, M.: Non-archimedean geometry of Artin fans. arXiv:1603.07589 (2016)
- Ulirsch, M.: Tropicalization is a non-archimedean analytic stack quotient. Math. Res. Lett. arXiv:1410.2216 (to appear)
- Vakil, R.: Murphy’s law in algebraic geometry: badly-behaved deformation spaces. Invent. Math. 164, 569–590 (2006)MathSciNetView ArticleMATHGoogle Scholar
- Yu, T.Y.: Tropicalization of the moduli space of stable maps. Math. Z. 281, 1035–1059 (2015)MathSciNetView ArticleMATHGoogle Scholar
- Yu, T.Y.: Gromov compactness in non-archimedean analytic geometry, J. Reine Angew. Math. (Crelle’s Journal). arXiv:1401.6452 (to appear)