Nonarchimedean geometry, tropicalization, and metrics on curves

By Matthew Baker, Sam Payne, Joseph Rabinoff. 2011

191 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/04bcbf08993d1f220dad27828dfbc2148cd36ef5

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Citations in the corpus

Moduli of tropical plane curves (2014, methodology): “This is the Berkovich skeleton of the analytification of C as in [5].”
Tropical independence I: Shapes of divisors and a proof of the Gieseker–Petri theorem (2014, methodology): “Then Thuillier’s nonarchimedean analytic Poincaré–Lelong formula [Thuillier 2005; Baker et al. 2011] says that si (trop( f )) is the order of vanishing at xi of the reduction of f in κ(X i ).”
Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta (2013, unknown): “an admissible formal R scheme by [11], Proposition 3.”
Functorial tropicalization of logarithmic schemes: the case of constant coefficients (2013, background): “on 7 we give a detailed comparison of these two cases. These considerations togetherwith the above two theoremslead to the following incarnation of the principle of faithful tropicalization (see e.g. [BPR16], [GRW16], and [CHW14]). Corollary 1.3. Suppose that Yis a proper and scho¨n subvariety of a ”