Mustafin varieties, moduli spaces and tropical geometry
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Marvin Anas Hahn
© The Author(s) 2020
· Binglin Li
Mustafin varieties, moduli spaces and tropical geometry Received: 18 April 2019 / Accepted: 3 August 2020 Abstract. Mustafin varieties are flat degenerations of projective spaces, induced by a set of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin (Math USSR-Sbornik 34(2):187, 1978) in the 70s in order to generalise Mumford’s groundbreaking work on the unformisation of curves to higher dimension. These varieties have a rich combinatorial structure as can be seen in pioneering work of Cartwright et al. (Selecta Math 17(4):757–793, 2011). In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied in Li (IMRN, 2017). Applying tropical intersection theory and tropical convex hull computations, we use this method to give a new combinatorial description of the irreducible components of the special fibers of Mustafin varieties. Finally, we outline a first application of our results in limit linear series theory.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Tropical geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Bruhat-Tits buildings and tropical convexity . . . . . . . . . . . . . . . . . 2.3. Images of rational maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Mustafin varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Special fibers of Mustafin varieties . . . . . . . . . . . . . . . . . . . . . . . . r () . . . . . . . . . . . . . . . . 3.1. Constructing the varieties M() and M 3.2. Proof of Theorem 1.2 (1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Proof of Theorem 1.2 (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Classification of the irreducible components of special fibers of Mustafin varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. A first application: prelinked Grassmannians . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Hahn (B): Department of Mathematics, University of Tübingen, 72076 Tübingen, Germany. e-mail: [email protected] B. Lin: Department of Statistics, University of Georgia, Athens, GA 30602, USA. e-mail: [email protected] Mathematics Subject Classification: Primary 14T05 · 14D06 · 14D20; Secondary 20E42 · 20G25 · 52B99 · 14G35
https://doi.org/10.1007/s00229-020-01237-8
M. A. Hahn, B. Li
1. Introduction Mustafin varieties are flat degenerations of projective spaces induced by choosing a set of lattices in a K -vector space V . These objects were introduced by Mustafin in [22] in order to generalise Mumford’s groundbreaking work on uniformisation of curves to higher dimensions [21]. Since then they have been repeatedly studied under the name Delig
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