Nonarchimedean geometry, tropicalization, and metrics on curves

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.

37 pages, 7 figures. To appear in Algebraic Geometry. Significantly different from v2: the numbering has changed, the former Section 5 was extracted and published separately, and much expository material and many examples in the remaining sections have been omitted