Nonarchimedean geometry, tropicalization, and metrics on curves

Understand

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.