An introduction to local tropicalization

TL;DR

The paper presents local tropicalization as a local analogue of the global tropicalization for embeddings $Y\hookrightarrow(\mathcal{X}_{\sigma},0)$, unifying four viewpoints—arc weight vectors, initial forms, semivaluations, and toric modifications—through a toric-geometric interpretation. It proves the four definitions are equivalent (Theorem) and develops a structure theorem: the local tropicalization is a conic, polyhedral fan whose maximal cones have dimension $\dim Y$ and on which initial ideals are constant. The work analyzes specific cases, including the local trop for interior principal divisors via Newton polyhedra, and the splice type surface singularities, showing Trop_loc corresponds to Newton fans or trees embedded in simplices, respectively. It also surveys variants and extensions, including extended local tropicalization and Ulirsch's logarithmic tropicalization, highlighting broader connections to log-geometry, non-Archimedean analytics, and $\,\mathbb{F}_1$-schemes. Overall, the framework provides a cohesive, geometry-driven approach to local tropical features of singularities and toric embeddings with practical implications for resolution and deformation theory.

Abstract

In this paper we explain four viewpoints on the local tropicalization of formal subgerms of toric germs, which is a local analog of the global tropicalization of subvarieties of algebraic tori. We start by illustrating some of those viewpoints for plane curve singularities, then we pass to arbitrary dimensions. We conclude by describing several variants and extensions of the notion of local tropicalization presented in this paper.

Figures (7)

• Figure 1: The minimal embedded resolution $\pi_0 \circ \pi_1 \circ \pi_2$ and a toric partial resolution $\tilde{\pi}$ of the cuspidal cubic $Y$ contained in the plane $S_0 = \mathbb{C}^2$. The exceptional curve created by blowing up the point $p_i$ is denoted by $E_i$ and its strict transform on the surface $S_j$ is denoted by $E_{i,j}$. The strict transforms of $Z(x)$ and $Z(y)$ are still denoted by $Z(x)$ and $Z(y)$. The strict transform of $Y$ on the surface $S_j$ is denoted by $Y_j$ and that on $\tilde{S}$ is denoted by $\tilde{Y}$.
• Figure 2: The fans corresponding to the toric surfaces of Figure \ref{['fig:embrescusp']} and their subdivisions corresponding to the toric morphisms of the same figure. We write the name of the corresponding irreducible component of the total transform of $Z(xy) \hookrightarrow S_0$. Inside each regular two-dimensional cone which gets subdivided, we indicate the name of the corresponding $0$-dimensional orbit. The right diagonal arrow is the toric modification which blows down the orbit closures $E_{0,3}$ and $E_{1,3}$. The left diagonal arrow blows then down the orbit closure $\tilde{E}_2$.
• Figure 3: The Minkowski sum $\mathrm{supp}\ f +\mathbb{R}_{\geq 0}^{2}$ of Definition \ref{['def:NP']} (on the left) and the Newton polygon $\mathcal{N}(f)$ of the series $f(x,y)=y^2 - 2 x^3 + x^2 y$ (on the right).
• Figure 4: The local tropicalization of the series $f(x,y)=y^2 - 2 x^3 + x^2 y$ is the ray spanned by the primitive vector $(2,3)$, which is indicated with a filled dot.
• Figure 5: The Newton polyhedron $\mathcal{N}(Y)$ of the Pham-Brieskorn singularity $Y \hookrightarrow (\mathbb{C}^3,0)$ defined by equation $x_{1}^{\alpha} + x_{2}^{\beta} + x_{3}^{\gamma}=0$ (on the left) and its local tropicalization (on the right).

Theorems & Definitions (53)

• definition 1
• definition 2
• definition 3
• proposition 4
• definition 5
• example 1
• remark 6
• proposition 7
• theorem 8
• remark 9