Mark Luxton, Zhenhua Qu
We give an affirmative answer to a conjecture proposed by Tevelev in characteristic 0 case: any variety contains a schön very affine open subvariety. Also we show that any fan supported on the tropicalization of a schön very affine variety produces a schön compactification. Using toric schemes over a discrete valuation ring, we extend tropical compatifications to the non-constant coefficient case.
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Quasilinear tropical compactifications
本文提出拟线性热带紧化这一更广泛的热带紧化类，证明其具有 schön 性质且相交理论完全由热带扇决定，并应用于证明6条直线模空间与带标记三次曲面模空间的拟线性。
In this article we define a natural tropicalization procedure for closed subsets of log-regular varieties in the case of constant coefficients and study its basic properties. This framework allows us to generalize some of Tevelev's results on tropical compactification as well as Hacking's result on the cohomology of the link of a tropical variety to log-regular varieties.
We construct a family of compactifications of the affine cone of the Grassmannian variety of 2-planes. We show that both the tropical variety of the Plücker ideal and familiar valuations associated to the construction of Newton-Okounkov bodies for the Grassmannian variety can be recovered from these compactifications. In this way, we unite various perspectives for constructing toric degenerations of flag varieties.
Degenerations in tropical compactifications and tropical intersection theory of $\overline{M}_{0,n}$
本文提出热带紧化中子簇极限环的公式，扩展了Dickenstein-Feichtner-Sturmfels和Katz的结果，并应用于$\overline{M}_{0,n}$的热带化研究，引入了新的热带$ψ$-超曲面。
Given a smooth Mori dream space $X$ we construct a model dominating all the small $\mathbb{Q}$-factorial modifications via tropicalization. This construction allows us to recover a Minkowski basis for the Newton-Okounkov bodies of divisors on $X$ and hence the movable cone of $X$. The existence of such basis allows us to prove the polyhedrality of the global Newton-Okounkov body and the existence of toric degenerations.
We extend tropicalization and tropical compactification of subvarieties of algebraic tori to subvarieties of spherical homogeneous spaces. Given a tropical compactification of a subvariety, we show that the support of the colored fan of the ambient spherical variety agrees with the tropicalization of the subvariety. The proof is based on our equivariant version of the attening by blow-up theorem. We provide many examples.
We describe the stabilizers of points in the Bruhat-Tits building of the group SL with tropical geometry. There are several compactifications of this building associated to algebraic representations of SL. We show that the fans used to compactify apartments in this theory are given by tropical Schur polynomials.
We describe a canonical compactification of a polyhedral complex in Euclidean space. When the recession cones of the polyhedral complex form a fan, the compactified polyhedral complex is a subspace of a tropical toric variety. In this case, the procedure is analogous to the tropical compactifications of subvarieties of tori. We give an analysis of the combinatorial structure of the compactification and show that its Hasse diagram can be computed via Ganter's algorithm. Our algorithm is implemented in and shipped with polymake.
The logarithmic generalised Kummer locus and its tropicalisation
本文为广义Kummer型IHS簇的退化构造了分离对数紧化，通过热带化映射和热带广义Kummer构造，利用对数Hilbert概形推广了传统构造，并描述了边界纤维及多面体结构。
We relate some features of Bruhat-Tits buildings and their compactifications to tropical geometry. If G is a semisimple group over a suitable non-Archimedean field, the stabilizers of points in the Bruhat-Tits building of G and in some of its compactifications are described by tropical linear algebra. The compactifications we consider arise from algebraic representations of G. We show that the fan which is used to compactify an apartment in this theory is given by the weight polytope of the representation and that it is related to the tropicalization of the hypersurface given by the character …
Cohomologically tropical varieties
本文研究了cohomologically tropical流形的定义与性质，证明了schön流形为cohomologically tropical当且仅当满足wunderschön条件和tropical化为tropical上同调manifold，并扩展了相关Hodge数的结果。
This is mostly* a non-technical exposition of the joint work arXiv:1212.0373 with Caporaso and Payne. Topics include: Moduli of Riemann surfaces / algebraic curves; Deligne-Mumford compactification; Dual graphs and the combinatorics of the compactification; Tropical curves and their moduli; Non-archimedean geometry and comparison. * Maybe the last section is technical.
We give a functorial normal crossing compactification of the moduli of smooth marked cubic surfaces entirely analogous to the Grothendieck-Knudsen compactification $M_{0,n} \subset \bar{M}_{0,n}$.
These lecture notes are based on lectures given by the author at the summer school "Arrangements in Pyrénées" in June 2012. We survey and compare various compactifications of complex hyperplane arrangement complements. In particular, we review the Gel'fand-MacPherson construction, Kapranov's visible contours compactification, and De Concini and Procesi's wonderful compactification. We explain how these constructions are unified by some ideas from the modern origins of tropical geometry. The paper contains a few new arguments intended to make the presentation as self-contained as possible.
We use tropical and nonarchimedean geometry to study the moduli space of genus $0$ stable maps to $\mathbb{P}^1$ relative to two points. This space is exhibited as a tropical compactification in a toric variety. Moreover, the fan of this toric variety may be interpreted as a moduli space for tropical relative stable maps with the same discrete data. As a consequence, we confirm an expectation of Bertram and the first two authors, that the tropical Hurwitz cycles are tropicalizations of classical Hurwitz cycles. As a second application, we obtain a full descendant correspondence for genus $0$ r…
In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We want to show how some properties of the boundary becomes straightforward if looked at from this point of view. For example…
Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogues of t…
We provide a moduli-theoretic framework for the collapsing of Ricci-flat Kahler metrics via compactification of moduli varieties of Morgan-Shalen and Satake type. In patricular, we use it to study the Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters, especially for K3 surfaces.
We study a compactification of the moduli space of theta characteristics, giving a modular interpretation of the geometric points and describing the boundary stratification. This space is different from the moduli space of spin curves. The modular description and the boundary stratification of the new compactification are encoded by a tropical moduli space. We show that this tropical moduli space is a refinement of the moduli space of spin tropical curves. We describe explicitly the induced decomposition of its cones.
We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are often studied as "tropical curves", are obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic metrics of the Riemann surfaces. This phenomenon can be also seen as an archemidean analogue of the tropicalization of Berkovich analytification of $M_{g}$ (cf., [ACP]…
Each Gröbner stratum of a tropical variety is a connected set of points, all of which induce the same initial subscheme. The Gröbner stratification is a coarsening of the decomposition into Gröbner polyhedra, and has the advantage that it does not depend on a choice of compactification. We give an example of a curve over a field with non-trivial valuation whose Gröbner stratification is strictly finer than the coarsest polyhedral decomposition of the tropical variety. We also show that the Gröbner stratification of a locally matroidal tropical variety is completely determined by the underlying…
Tropical Feichtner-Yuzvinsky and positivity criterion for fans
本文建立单纯扇形 Chow 环与其典范紧化热带上同调中间次数部分的同构，证明热带版 Kleiman 丰富性判据，并给出拟阵 Chow 环的新表示。
We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. This work is analogous to the first of our series (available at arXiv:1406.7772v2), which compactified the moduli of curves by attaching the moduli of metrized graphs. Then, we also explicitly specify the Gromov-Hausdorff limits along holomorphic family of abelian varieties and sho…
Let $Ω$ be the complement of a connected, essential hyperplane arrangement. We prove that every dominant endomorphism of $Ω$ extends to an endomorphism of the tropical compactification $X$ of $Ω$ associated to the Bergman fan structure on the tropicalization of $Ω$. This generalizes a previous result by Rémy, Thuillier and the second author which states that every automorphism of Drinfeld's half-space over a finite field $\mathbb{F}_q$ extends to an automorphism of the successive blow-up of projective space at all $\mathbb{F}_q$-rational linear subspaces. This successive blow-up is in fact the…
Gauss--Bonnet for CSM classes of very affine manifolds, and tropical isotopies
该研究提出无需紧化的技术研究极仿射簇，从仿射示性类性质推导出极仿射流形CSM类高斯-博内恒等式的一般形式，推广至非环面不变1-形式，还得到热带同痕定理等结果。
Given a finitely generated algebra $A$, it is a fundamental question whether $A$ has a full rank discrete (Krull) valuation $\mathfrak{v}$ with finitely generated value semigroup. We give a necessary and sufficient condition for this, in terms of tropical geometry of $A$. In the course of this we introduce the notion of a Khovanskii basis for $(A, \mathfrak{v})$ which provides a framework for far extending Gröbner theory on polynomial algebras to general finitely generated algebras. In particular, this makes a direct connection between the theory of Newton-Okounkov bodies and tropical geometry…
We give a complete description of the cohomology ring $A^*(\overline Z)$ of a compactification of a linear subvariety $Z$ of a torus in a smooth toric variety whose fan $Σ$ is supported on the tropicalization of $Z$. It turns out that cocycles on $\overline Z$ canonically correspond to Minkowski weights on $Σ$ and that the cup product is described by the intersection product on the tropical matroid variety $\operatorname{Trop}(Z)$.
We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the "tropical motivic nearby fiber." This invariant specializes in the schon case to the Hodge-Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge-Deligne polynomial in the cases of schon hypersurfaces and smooth tropical varieties. We also deduce a formula for the Euler characteristic of a gen…
We find restrictions on the topology of tropical varieties that arise from a certain natural class of varieties. We develop a theory of tropical degenerations that is a nonconstant coefficient analogue of Tevelev's theory of tropical compactifications, and use it to construct normal crossings degenerations of a subvariety X of a torus, under mild hypotheses on X. These degenerations allow us to construct a natural, "multiplicity-free" parameterization of Trop(X) by a topological space Γ_X. We give a geometric interpretation of the cohomology of Γ_X in terms of the action of a monodromy operato…
In this text, we merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety. Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of…
The trigonal construction and the second moment of the tropical Prym variety
本文利用热带三角构造计算亏格不超过4的热带曲线双重覆盖的Prym簇第二矩，结果由多项式与分段多项式项组成，并将后者与Prym--Torelli映射的扩展问题相关联。
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 $K_X$…
On Bands and Limit Theorems in Tropical Geometry
本文基于Baker-Jin-Lorscheid提出的带理论，研究了带方案在热带化、解析化和F₁几何中的应用，并证明了通过带方案的极限得到Payne极限定理的方案论增强。
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of Ardila-Klivans who showed that the Bergman c…
In this paper, we give some categorical description of the general spectrum functor, defining it as an adjoint of a global section functor. The general spectrum functor includes that of $F_1$ and of semirings.
We define a tropical version $\F_d(\trop X)$ of the Fano Scheme $\F_d(X)$ of a projective variety $X\subseteq \mathbb P^n$ and prove that $\F_d(\trop X)$ is the support of a polyhedral complex contained in $\trop \Grp(d,n)$. In general $\trop \F_d(X)\subseteq \F_d(\trop X)$ but we construct linear spaces $L$ such that $\trop \F_1(X)\subsetneq \F_1(\trop X)$ and show that for a toric variety $\trop \F_d(X)=\F_d(\trop X)$.
We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme struct…
In her Ph.D. thesis, Mainò introduced the notion of enriched structure on stable curves and constructed their moduli space. In this paper we give a tropical notion of enriched structure on tropical curves and construct a moduli space parametrizing these objects. Moreover, we use this construction to give a toric description of the scheme parametrizing enriched structures on a fixed stable curve.
In this article we study the tropicalization of the Hilbert scheme and its suitability as a parameter space for tropical varieties. We prove that the points of the tropicalization of the Hilbert scheme have a tropical variety naturally associated to them. To prove this, we find a bound on the degree of the elements of a tropical basis of an ideal in terms of its Hilbert polynomial. As corollary, we prove that the set of tropical varieties defined over an algebraically closed valued field only depends on the characteristic pair of the field and the image group of the valuation. In conclusion, w…
For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…
We determine the tropicalizations of very affine surfaces over a valued field that are obtained from del Pezzo surfaces of degree 5, 4 and 3 by removing their (-1)-curves. On these tropical surfaces, the boundary divisors are represented by trees at infinity. These trees are glued together according to the Petersen, Clebsch and Schläfli graphs, respectively. There are 27 trees on each tropical cubic surface, attached to a bounded complex with up to 73 polygons. The maximal cones in the 4-dimensional moduli fan reveal two generic types of such surfaces.
Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal
本文提出基于横截拟阵的Macaulay热带理想构造，将主理想典范扩展为热带理想，具有普适性质，并生成不可实现的超曲面方案，与Maclagan-Rincón的直线构造在特定情形一致。
Non-archimedean SYZ fibrations via tropical contractions
本文通过热带收缩映射，证明了Batyrev-Borisov完全交的环退化中，对偶相交复形与Berkovich解析化的基本骨架同构，并构造了仿射环面纤维化。
From any monoid scheme $X$ (also known as an $\mathbb{F}_1$-scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) $X_S$ by scalar extension to an idempotent semifield $S$. We prove that for a given irreducible monoid scheme $X$ (satisfying some mild conditions) and an idempotent semifield $S$, the Picard group $Pic(X)$ of $X$ is stable under scalar extension to $S$ (and in fact to any field $K$). In other words, we show that the groups $Pic(X)$ and $Pic(X_S)$ (and $Pic(X_K)$) are isomorphic. In particular, if $X_\mathbb{C}$ is a toric variety, then $Pic(X)$ is the s…
An enumerative problem on a variety $V$ is usually solved by reduction to intersection theory in the cohomology of a compactification of $V$. However, if the problem is invariant under a "nice" group action on $V$ (so that $V$ is spherical), then many authors suggested a better home for intersection theory: the direct limit of the cohomology rings of all equivariant compactifications of $V$. We call this limit the affine cohomology of $V$ and construct affine characteristic classes of subvarieties of a complex torus, taking values in the affine cohomology of the torus. This allows us to make t…
We give a Thurston-like definition for laminations on higher Teichmuller spaces associated to a surface $S$ and a semi-simple group $G$ for $G-SL_m$ and $PGL_m$. The case $G=SL_2$ or $PGL_2$ corresponds to the classical theory of laminations. Our construction involves positive configurations of points in the affine building. We show that these laminations are parameterized by the tropical points of the spaces $\X_{G,S}$ and $\A_{G,S}$ of Fock and Goncharov. Finally, we explain how these laminations give a compactification of higher Teichmuller spaces.
We study the relationship between min-plus, max-plus and Euclidean convexity for subsets of $\mathbb{R}^n$. We introduce a construction which associates to any max-plus convex set with compact projectivisation a canonical matrix called its dominator. The dominator is a Kleene star whose max-plus column space is the min-plus convex hull of the original set. We apply this to show that a set which is any two of (i) a max-plus polytope, (ii) a min-plus polytope and (iii) a Euclidean polytope must also be the third. In particular, these results answer a question of Sergeev, Schneider and Butkovic a…
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