A guide to tropical modifications

Abstract

This paper surveys {\it tropical modifications}, which have already become a folklore in tropical geometry. Tropical modifications are used in tropical intersection theory, tropical Hodge theory, and in the study of singularities. They admit interpretations in various contexts, such as hyperbolic geometry, Berkovich spaces, and non-standard analysis. Our main goal is to mention different points of view, to give references, and to demonstrate the abilities of tropical modifications. We assume that the reader has already met ``tropical modifications'' somewhere and wants to understand them better. There are novelties here: a new obstruction to the realizability of non-transversal intersections and a tropical version of Weil's reciprocity law.

Figures (10)

• Figure 1: Example of a modification of a line along itself. Let $L_1,L_2$ be defined by $y=t^{-1},y=t^{-1}+t^{3}$ respectively. In each group of pictures, the bottom picture is the initial ${\mathbb T}^2$, the middle picture is $m_{L_1}{\mathbb T}^2$, the picture at the back is the projection to $X,Z$-plane. On the left we see the modification of $L_1$ along $L_1$, on the right we see the modification of $L_2$ along $L_1$. Red line is the result $m_{L_1}L_1$ (resp. $m_{L_1}L_2$) of the modification.
• Figure 2: Example of a modification along a line.
• Figure 3: Example of modification in the case of inflection point. The point $(0,0)$ on the bottom picture is the tropicalization of the inflection point. We modified the black curve along the blue curve, red parts are the parts becoming visible after the modification.
• Figure 4: The extended Newton polyhedron $\widetilde{{\mathcal{A}}}$ of the curve $C'$ is drawn in $\mathrm{(A)}$. The projection of its faces gives us the subdivision of the Newton polygon of $C'$; see $\mathrm{(B)}$. The tropical curve $\mathrm{Trop}(C')$ is drawn in $\mathrm{(C)}$. The vertices $A_1,A_2,A_3$ have coordinates $(-2,0),(1,0),(4,0)$. The edge $A_1A_2$ has weight $3$, while the edge $A_2A_3$ has weight $2$. The point $P$ is $(0,0)=\mathrm{Val}((1,1))$.
• Figure 5: Refer to Example \ref{['ex_singularexample']}. Left bottom picture represents a curve $C$. On top of it, a modification of it is depicted, with the projection of the latter on the $XZ$-plane. On the right side we see two other possible modification of $C$.

Theorems & Definitions (78)

• Definition 1.1
• Definition 1.2
• Remark 1.1
• Definition 1.3
• Proposition 1.1: cf. mikh2, 1.5 B,C
• Definition 1.4
• Proposition 1.2
• Proposition 1.3
• Definition 1.5
• Proposition 1.4