忠实热带化与$\overline{M}_{0,n}$的骨架

平面格拉斯曼的忠实热带化

Faithful tropicalization of the Grassmannian of planes

我们证明了热带投影平面的格拉斯曼同胚的一个封闭的子集的解析格拉斯曼在Berkovich的意义下，通过构建一个连续的部分的tropicalization地图。我们的主要工具是一个明确的描述代数坐标环的复曲面地层的格拉斯曼。我们确定了热带化图的纤维，并计算了所有复曲面地层的初始退化。因此，我们证明了热带投射格拉斯曼的所有点的热带重数等于一。最后，我们确定了一个分段线性结构的图像，我们的部分，对应于多面体结构的热带投影格拉斯曼。

Effective faithful tropicalizations associated to linear systems on curves

For a connected smooth projective curve $X$ of genus $g$, global sections of any line bundle $L$ with $°(L) \geq 2g+ 1$ give an embedding of the curve into projective space. We consider an analogous statement for a Berkovich skeleton in nonarchimedean geometry, in which projective space is replaced by tropical projective space, and an embedding is replaced by a homeomorphism onto its image preserving integral structures (called a faithful tropicalization). Let $K$ be an algebraically closed field which is complete with respect to a non-trivial nonarchimedean value. Suppose that $X$ is defined …

Faithful tropicalizations of elliptic curves using minimal models and inflection points

We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ admits a tropicalization that contains a cycle of length $-v(j(E))$. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is an inflection point that reduces to the singular point on the reducti…

Faithful tropicalization of hypertoric varieties

The hypertoric variety $\mathfrak{M}_{\mathcal{A}}$ defined by an affine arrangement $\mathcal{A}$ admits a natural tropicalization, induced by its embedding in a Lawrence toric variety. We explicitly describe the polyhedral structure of this tropicalization. Using a recent result of Gubler, Rabinoff, and Werner, we prove that there is a continuous section of the tropicalization map.

Effective faithful tropicalizations associated to adjoint linear systems

Let $R$ be a complete discrete valuation ring of equi-characteristic zero with fractional field $K$. Let $X$ be a connected, smooth projective variety of dimension $d$ over $K$, and let $L$ be an ample line bundle over $X$. We assume that there exist a regular strictly semistable model $\mathscr{X}$ of $X$ over $R$ and a relatively ample line bundle $\mathscr{L}$ over $\mathscr{X}$ with $\mathscr{L}|_{X} \cong L$. Let $S(\mathscr{X})$ be the skeleton associated to $\mathscr{X}$ in the Berkovich analytification $X^{\mathrm{an}}$ of $X$. In this article, we study when $S(\mathscr{X})$ is faithfu…

Symmetric powers of algebraic and tropical curves: a non-Archimedean perspective

We show that the non-Archimedean skeleton of the $d$-th symmetric power of a smooth projective algebraic curve $X$ is naturally isomorphic to the $d$-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of $X$. The retraction to the skeleton is precisely the specialization map for divisors. Moreover, we show that the process of tropicalization naturally commutes with the diagonal morphisms and the Abel-Jacobi map and we exhibit a faithful tropicalization for symmetric powers of curves. Finally, we prove a version of the Bieri-Groves Theorem that allows us, under…

有效忠实的热带化与嵌入用于阿贝尔簇

Effective faithful tropicalizations and embeddings for abelian varieties

本文通过非阿基米德theta函数实现阿贝尔簇的热带化，提供经典勒夫定理的热带版本，结合热带theta函数嵌入与非阿基米德theta函数提升，拓展了热带几何的应用范围。

曲线模空间的热带化

The tropicalization of the moduli space of curves

本文证明稳定曲线模叠的骨架自然等同于扩展热带曲线模空间，并构造了与代数重言映射相容的遗忘、粘合与胶合映射。

Skeletons of stable maps I: Rational curves in toric varieties

We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new…

Faithful realizability of tropical curves

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…

基于调和热带化的 Berkovich 空间上的形式

Forms on Berkovich spaces based on harmonic tropicalizations

本文引入了 Berkovich 空间的热带骨架，提出通过更一般的调和热带化拉回方法构建更大的微分形式层，在保持原有性质的同时改善了上同调行为。

A non-Archimedean analogue of Teichmüller space and its tropicalization

In this article we use techniques from tropical and logarithmic geometry to construct a non-Archimedean analogue of Teichmüller space $\overline{\mathcal{T}}_g$ whose points are pairs consisting of a stable projective curve over a non-Archimedean field and a Teichmüller marking of the topological fundamental group of its Berkovich analytification. This construction is closely related to and inspired by the classical construction of a non-Archimedean Schottky space for Mumford curves by Gerritzen and Herrlich. We argue that the skeleton of non-Archimedean Teichmüller space is precisely the trop…

Galois actions on analytifications and tropicalizations

This paper initiates a research program that seeks to recover algebro-geometric Galois representations from combinatorial data. We study tropicalizations equipped with symmetries coming from the Galois-action present on the lattice of $1$-parameter subgroups inside ambient Galois-twisted toric varieties. Over a Henselian field, the resulting tropicalization maps become Galois-equivariant. We call their images Galois-equivariant tropicalizations, and use them to construct a large supply of Galois representations in the tropical cellular cohomology groups of Itenberg, Katzarkov, Mikhalkin, and Z…

Buildings、赋值拟阵与热带线性空间

Buildings, valuated matroids, and tropical linear spaces

本文研究了 Affine Bruhat--Tits buildings 与赋值拟阵的联系，证明了半范数空间是所有热带化线性嵌入的极限，并验证了其作为通用可实现赋值拟阵的热带线性空间本质。