2023/04/21 by Kiumars Kaveh, Kaveh, Kiumars, Christopher Manon +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #14J60 (Secondary) #14M25 (Primary) #14T15 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Nonlinear Waves and Solitons #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2304.11211
openalex publication_date 2023/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A toric vector bundle E is a torus equivariant vector bundle on a toric variety. We give a valuation theoretic and tropical point of view on toric vector bundles. We present three (equivalent) classifications of toric vector bundles, which should be regarded as repackagings of the Klyachko data of compatible ℤ-filtrations of a toric vector bundle: (1) as piecewise linear maps to space of ℤ-valued valuations, (2) as valuations with values in the semifield of piecewise linear functions, and (3) as points in tropical linear ideals over the semifield of piecewise linear functions. Moreover, we interpret the known criteria for ampleness and global generation of E as convexity conditions on its piecewise linear map in (1). Finally, using (2) we associate to E a collection of polytopes indexed by elements of a certain (representable) matroid encoding the dimensions of weight spaces of global sections of E. This recovers and extends the Di Rocco-Jabbusch-Smith matriod and parliament of polytopes of E. This is a follow up paper to arXiv:1806.05613.