Paolo Tripoli
Given a tropical fan curve $Σ$ and a family of algebraic curves $X \rightarrow \mathbb{A}^k$ we define the realization locus $\mathop{Real}_Σ\subseteq \mathbb{A}^k$ as the set of fibers $X_a$ whose tropicalization is $Σ$. We produce an algorithm that describes the Zariski closure of $\mathop{Real}_Σ$ by imposing algebraic conditions for each ray of $Σ$.
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Let E be a plane in an algebraic torus over an algebraically closed field. Given a balanced 1-dimensional fan C in the tropicalization of E, i.e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic…
Let X be a plane in a torus over an algebraically closed field K, with tropicalization the matroidal fan Sigma. In this paper we present an algorithm which completely solves the question whether a given one-dimensional balanced polyhedral complex in Sigma is relatively realizable, i.e. whether it is the tropicalization of an algebraic curve, over the field of Puiseux series over K, in X. The algorithm implies that the space of all such relatively realizable curves of fixed degree is an abstract polyhedral set. In the case when X is a general plane in 3-space, we use the idea of this algorithm …
We study whether a given tropical curve $Γ$ in $\mathbb{R}^n$ can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by $Γ$. We give an affirmative answer to this question for a large class of tropical curves that includes all trivalent tropical curves, but also many tropical curves of higher valence. We then deduce that for every metric graph $G$ with rational edge lengths there exists a smooth algebraic curve in a toric variety whose analytification has skeleton $G$, and the corresponding tropicalization is faithful. Our approach…
We present results on the relative realizability of infinite families of lines on general smooth tropical cubic surfaces. Inspired by the problem of relative realizability of lines on surfaces, we investigate the information we can derive tropically from the Brundu-Logar normal form of smooth cubic surfaces. In particular, we prove that for a residue field of characteristic $\neq 2$ the tropicalization of the Brundu-Logar normal form is not smooth. We also take first steps in investigating the behavior of the tropicalized lines.
Combinatorics of Hurwitz degenerations and tropical realizability
研究热带曲线上平衡函数的可实现性，通过组合比较半稳定极限定理，为亏格二曲线上的超丰函数建立新的充分准则，并引入降维技术将映射到 R^r 的结论归结为映射到 R 的情形。
To each prime ideal in a polynomial ring over a field we associate an algebraic matroid and show that it is preserved under tropicalization. This gives a necessary condition for a tropical variety to be set-theoretically realizable from a prime ideal. We also show that there are infinitely many Bergman fans that are not set-theoretically realizable as the tropicalization of any ideal.
Tropical linear systems and the realizability problem
本文研究抽象 tropical 曲线几何，探索其完全线性系统与 tropical 子模，基于 Baker-Norine 秩建立子模维数下界，并刻画可实现典范除子，解决了 tropical 几何与代数对应物连接的根本问题。
Realizability of tropical pluri-canonical divisors
本文针对抽象热带曲线与有效多重典范除子构成的配对，提出了纯组合判据以判定其可实现性，通过引入热带正规覆盖将问题归约，推广了热带典范除子相关研究。
We use recent results by Bainbridge-Chen-Gendron-Grushevsky-Moeller on compactifications of strata of abelian differentials to give a comprehensive solution to the realizability problem for effective tropical canonical divisors in equicharacteristic zero. Given a pair $(Γ, D)$ consisting of a stable tropical curve $Γ$ and a divisor $D$ in the canonical linear system on $Γ$, we give a purely combinatorial condition to decide whether there is a smooth curve $X$ over a non-Archimedean field whose stable reduction has $Γ$ as its dual tropical curve together with a effective canonical divisor $K_X$…
Motivated by the realizability problem for principal tropical divisors with a fixed ramification profile, we explore the tropical geometry of the double ramification locus in $\mathcal{M}_{g,n}$.There are two ways to define a tropical analogue of the double ramification locus: one as a locus of principal divisors, the other as a locus of finite effective ramified covers of a tree. We show that both loci admit a structure of a generalized cone complex in $M_{g,n}^{trop}$, with the latter contained in the former. We prove that the locus of principal divisors has cones of codimension zero in $M_{…
Rigidity and Realizability for Tropical Curves in Dimension 3
本研究利用伪全纯圆盘边界的$H_1(L)$类提出拉格朗日三维流形的无障碍判据，证明具裤对分解的刚性热带曲线的拉格朗日提升无障碍，并在同调镜像对称成立的前提下推出镜像阿贝尔三维流形中存在对应实现。
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 3 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…
Realizability in tropical geometry and unobstructedness of Lagrangian submanifolds
本文提出热带子簇的 $B$-可实现性概念，证明若拉格朗日子流形提升具有定义良好的 Floer 上同调且可通过界定上链变为无障碍，则该热带子簇是 $B$-可实现的，并应用于证明亏格零热带曲线及热带阿贝尔曲面内热带曲线的可实现性。
We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of n-marked abstract rational tropical curves.
Abelian tropical covers
该研究利用热带曲线上层的上同调群分类有限阿贝尔群的调和热带覆叠，给出基于扩展同调群局部单值数据拼接的可实现性判据，还以模p整数群为例关联了图论无处零流问题。
Tropical geometry gives a bound on the ranks of divisors on curves in terms of the combinatorics of the dual graph of a degeneration. We show that for a family of examples, curves realizing this bound might only exist over certain characteristics or over certain fields of definition. Our examples also apply to the theory of metrized complexes and weighted graphs. These examples arise by relating the lifting problem to matroid realizability. We also give a proof of Mnëv universality with explicit bounds on the size of the matroid, which may be of independent interest.
Constructions of superabundant tropical curves in higher genus
本文构造了亏格3和4中不可实现的超丰热带曲线新例子，其形变空间盈余维数为1，结合对数曲线空间的维数计算证明了其不可实现性。
Brodsky, Joswig, Morrison and Sturmfels showed that not all abstract tropical curves of genus $3$ can be realized as a tropicalization of a quartic in the euclidean plane. In this article, we focus on the interior of the maximal cones in the moduli space and classify all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, we show that these are exactly those which are not realizably hyperelliptic. Our approach is constructive: For any not realizably hyperelliptic curve, we explicitly construct a realizable model of the tropica…
\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin's approximation…
In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the obstruction cohomology class of degenerate algebraic curves. This enables us to judge whether a given degenerate curve can be deformed to a smooth curve or not in variety of situations. In this paper, we apply it to curves of genus one on degeneration of toric varieties. In particular, we obtain the necessary and sufficient condition for the realizability of tr…
Families of KP solutions associated with tropical curves having nontrivial weights
本研究推广Agostini等人的KP解构建方案，从带非平凡权重的热带曲线对应的退化黎曼曲面构造准周期KP解族，经正则化热带极限得到孤子与黎曼θ函数混合表达的通用KP解公式。
This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of singular curves of genus $1$. This volume focuses on logarithmic Gromov--Witten theory and tropical geometry. We construct a logarithmically nonsingular moduli space of genus $1$ curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich--Chen--Gross--Siebert space of logarithmic stable maps and produces an enumerative genus $1$ curve counting theory. We describe the non-archimedean analytic skeleton of this moduli space and, a…
We study the asymptotic behavior of volume forms on a degenerating family of compact complex manifolds. Under rather general conditions, we prove that the volume forms converge in a natural sense to a Lebesgue-type measure on a certain simplicial complex. In particular, this provides a measure-theoretic version of a conjecture by Kontsevich--Soibelman and Gross--Wilson, bearing on maximal degenerations of Calabi--Yau manifolds.
Differential forms and cohomology in tropical and complex geometry
本文改进 Ducros、Hrushovski 和 Loeser 的微分形式映射，为复射影簇建立热带上同调到 Dolbeault 上同调的自然映射，并证明其与半代数集及加权热带化上的积分相容。
We introduce a tropical geometric framework that allows us to define $ψ$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $ψ$ classes for some one-dimensional families of genus-one tropical curves.
We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric vari…
We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies continuously in families of tropical plane curves.
We give a simple expression for the integral of the canonical holomorphic volume form in degenerating families of varieties constructed from wall structures and with central fiber a union of toric varieties. The cycles to integrate over are constructed from tropical 1-cycles in the intersection complex of the central fiber. One application is a proof that the mirror map for the canonical formal families of Calabi-Yau varieties constructed by Gross and the second author is trivial. We also show that these families are the completion of an analytic family, without reparametrization, and that the…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by fixing some algebraic or geometric invariants (such as degree, genus and types of singularities), the problem usually takes the form of "how many curves of that class pass through a configuration of n points in general position?" Tropical Geometry deals with certain piecewise-linear complexes, which arise as degeneration of families of complex algebraic varieti…
We study the tropicalization of intersections of plane curves, under the assumption that they have the same tropicalization. We show that the set of tropical divisors that arise in this manner is a pure dimensional balanced polyhedral complex and compute its dimension. When the genus is at most 1, we show that all the tropical divisors that move in the expected dimension are realizable. As part of the proof, we introduce a combinatorial tool for explicitly constructing large families of realizable tropical divisors.
Duality of curves is one of the important aspects of the ``classical'' algebraic geometry. In this paper, using this foundation, the duality of tropical polynomials is constructed to introduce the duality of Non-Archimedean curves. Using the development of ``mechanism'' which is based on ``distortion'' values and their matrices, we discuss some aspects refereing to quadrics with respect to their dual objects. This topic includes also the induced dual subdivision of Newton Polytope and its compatible properties. Finally, a regularity of tropical curves in the duality sense is generally defined …
We construct polyhedral families of valuations on the branching algebra of a morphism of reductive groups. This establishes a connection between the combinatorial rules for studying a branching problem and the tropical geometry of the branching algebra. In the special case when the branching problem comes from the inclusion of a Levi subgroup or a diagonal subgroup, we use the dual canonical basis of Lusztig and Kashiwara to build toric deformations of the branching algebra.
We construct a space classifying divisor classes of a fixed degree on all tropical curves of a fixed combinatorial type and show that the function taking a divisor class to its rank is upper semicontinuous. We extend the definition of the Brill-Noether rank of a metric graph to tropical curves and use the upper semicontinuity of the rank function on divisors to show that the Brill-Noether rank varies upper semicontinuously in families of tropical curves. Furthermore, we present a specialization lemma relating the Brill-Noether rank of a tropical curve with the dimension of the Brill-Noether lo…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.
In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear tropical varieties from Ardila and Klivans and from Feichtner and Sturmfels. We show that a singularity tropicalises either to a vertex of higher valence or of higher multiplicity, or to an edge of higher weight. We then classify maximal dimensional types of singular tropical curves. For those, the singularity is either a crossing of two edges, or a 3-valent ver…
Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the Gromov-Witten theory of toric surfaces. After a short survey of the subject (based on lectures of Pandharipande at the 2009 Geometry summer school in Lisbon), we prove new results about the rays and symmetries of scattering diagrams of commutators (including previous conjectures by Gross-Siebert and Kontsevich). Where possible, we present both the quiver and Grom…
Let $C$ be a nodal curve, and let $E$ be a union of semistable subcurves of $C$. We consider the problem of contracting the connected components of $E$ to singularities in a way that preserves the genus of $C$ and makes sense in families, so that this contraction may induce maps between moduli spaces of curves. In order to do this, we introduce the notion of mesa curve, a nodal curve $C$ with a logarithmic structure and a piecewise linear function $\overlineλ$ on the tropicalization of $C$. This piecewise linear function determines a subcurve $E$. We then construct a contraction of $E$ inside …
Tropical limit of hyperbolic amoebas of complex analytic surfaces
本文证明了复解析曲面在SL_2(C)中经SU(2)商并缩放后像的收敛性，即双曲阿米巴的热带极限。
We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical order-by-order description of the degeneration via families of tropical trees. This gives complete control of the B-model side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. W…
Euler characteristic and signature of real semi-stable degenerations
本文通过动机性近纤维方法，证明了非奇异实热带完全交集中实部欧拉示性数等于复部符号差，并推广至具有Q-非奇异热带极限的实解析族。
We study a class of meromorphic connections $\nabla(Z)$ on $\mathbb{P}^1$, parametrised by the central charge $Z$ of a stability condition, with values in a Lie algebra of formal vector fields on a torus. Their definition is motivated by the work of Gaiotto, Moore and Neitzke on wall-crossing and three-dimensional field theories. Our main results concern two limits of the families $\nabla(Z)$ as we rescale the central charge $Z \mapsto RZ$. In the $R \to 0$ "conformal limit" we recover a version of the connections introduced by Bridgeland and Toledano Laredo (and so the Joyce holomorphic gener…
Given $d$ row vectors of $n$ tropical numbers, $d<n$, the tropical Stiefel map constructs a version of their row space, whose Plücker coordinates are tropical determinants. We explicitly describe the fibers of this map. From the viewpoint of matroid theory, the tropical Stiefel map defines a generalization of transversal matroids in the valuated context, and our results are the valuated generalizations of theorems of Brualdi and Dinolt, Mason and others on the set of all set families that present a given transversal matroid. We show that a connected valuated matroid is transversal if and only …
In this article, we explore the connections between nonnegativity, the theory of $A$-discriminants, and tropical geometry. For an integral support set $A \subset \mathbb{Z}^n$, we cover the boundary of the sonc-cone by semi-algebraic sets that are parametrized by families of tropical hypersurfaces. As an application, we characterization generic support sets for which the sonc-cone is equal to the sparse nonnegativity cone, and we describe a semi-algebraic stratification of the boundary of the sonc-cone in the univariate case.
Let $k$ be a non-Archimedean rational valued field. We construct the moduli space of linearly rigidified polarized analytic tori over $k$ that admit rigid-analytic uniformization by an algebraic torus and observe that it is in definable rigid subanalytic bijection with a $PGL_N$-bundle over a polyhedral domain in an algebraic torus. We use this observation to prove that the Hrushovski-Kazhdan motivic volume of a non-Archimedean semi-algebraic family of Abelian varieties admitting such a uniformization fibrewise vanishes. This question is motivated by the conjectural geometric interpretation of…
The moduli space $Δ_{g,w}$ of tropical $w$-weighted stable curves of volume $1$ is naturally identified with the dual complex of the divisor of singular curves in Hassett's spaces of $w$-weighted stable curves. If at least two of the weights are $1$, we prove that $Δ_{0,w}$ is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space $Δ_{0,w}$ is disconnected and for which the fundamental group of $Δ…
Maximal Mumford Curves from Planar Graphs
本文研究极大Mumford曲线的构造，通过在典范曲线的Hilbert概型中利用平面图曲线的一阶形变，构建了具有全维数的MM曲线族。
Given a multivariate complex polynomial ${p\in\mathbb{C}[z_1,\ldots,z_n]}$, the imaginary projection $\mathcal{I}(p)$ of $p$ is defined as the projection of the variety $\mathcal{V}(p)$ onto its imaginary part. We focus on studying the imaginary projection of complex polynomials and we state explicit results for certain families of them with arbitrarily large degree or dimension. Then, we restrict to complex conic sections and give a full characterization of their imaginary projections, which generalizes a classification for the case of real conics. That is, given a bivariate complex polynomia…
Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gröbner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algeb…
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