Mustafin varieties are flat degenerations of projective spaces, induced by a choice of an $n-$tuple of lattices in a vector space over a non-archimedean field. They were introduced by Mustafin 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, H\"abich, Sturmfels and Werner. In this paper, we introduce a new approach to Mustafin varieties in terms of images of rational maps, which were studied by Li. 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. This enables connections to various topics. In particular, we see that any multiview variety appears as an irreducible component of the special fiber of some Mustafin variety. Furthermore, we use an interpretation of Mustafin varieties as a moduli functor introduced by Faltings to relate them to certain moduli functors, called linked Grassmannians. These objects are featured in limit linear series theory. The focal point of study regarding linked Grassmannians are so-called \textit{simple points}. As a direct consequence of the new combinatorial description of Mustafin varieties, we prove that the simple points of linked Grassmannians are dense in every fiber. Finally, we use the connection to linked Grassmannians, to relate the special fibers of Mustafin varieties to certain local models of unitary Shimura varieties.

中文翻译：

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Mustafin 品种、模空间和热带几何

Mustafin 变体是射影空间的平面退化，由在非阿基米德场上的向量空间中选择 $n-$ 元组晶格引起。它们是由 Mustafin 在 70 年代引入的，目的是将 Mumford 在曲线非形式化方面的开创性工作推广到更高维度。这些变体具有丰富的组合结构，这可以从 Cartwright、H\"abich、Sturmfels 和 Werner 的开创性工作中看出。在本文中，我们介绍了一种根据有理图图像来研究 Mustafin 变体的新方法，这些方法是由Li. 应用热带相交理论和热带凸包计算，我们使用这种方法对 Mustafin 品种的特殊纤维的不可约分量进行了新的组合描述。这使得连接到各种主题。特别是，我们看到，任何多视图品种都表现为某些 Mustafin 品种的特殊纤维的不可还原成分。此外，我们将 Mustafin 变体解释为 Faltings 引入的模函子，将它们与某些模函子联系起来，称为链接 Grassmannians。这些对象在极限线性级数理论中有特色。关于链接 Grassmannians 的研究重点是所谓的 \textit{simple points}。作为 Mustafin 变体的新组合描述的直接结果，我们证明了连接的 Grassmannian 的简单点在每个纤维中都是密集的。最后，我们使用与链接 Grassmannians 的联系，将 Mustafin 品种的特殊纤维与某些单一的 Shimura 品种的当地模型联系起来。