2019/12/05 by Trevor Gunn, Gunn, Trevor, Philipp Jell +1
Computer Science · Mathematics · #14T05 (Primary) 14G22 #32P05 (Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1912.02648
openalex publication_date 2019/12/05 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the tropicalization has nice properties. We construct, for any Mumford curve X, a map to a three-dimensional torus, such that the tropicalization is isometric to a subgraph of the Berkovich space X\rm an, called the extended skeleton. In this case, we say the tropicalization is "fully faithful." Additionally, given a map X to a toric variety Y, which induces a fully faithful tropicalization, we show that we can extend the map to X → Y × (P1)n such that the new tropicalization is smooth and fully faithful.