Nonarchimedean geometry, tropicalization, and metrics on curves

TL;DR

This paper develops techniques to compare analytifications and tropicalizations of algebraic varieties, extending multiplicity formulas and exploring metrics on curves, with applications to tropical geometry and invariants.

Contribution

It generalizes multiplicity formulas and investigates metric relationships on curves, advancing the understanding of tropicalization in nonarchimedean geometry.

Findings

Projection formula for tropical multiplicities

Generalization of Sturmfels-Tevelev multiplicity formula

Maps stabilize to isometries on finite subgraphs

Abstract

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to an isometry on finite subgraphs. Other applications include generalizations of Speyer's well-spacedness condition and the Katz-Markwig-Markwig results on tropical j-invariants.