Mustafin varieties
Mustafin varieties
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穆斯塔芬品种
DOI：
10.1007/s00029-011-0075-x
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
A. Werner
中科院分区：
文献类型：
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作者：
Dustin A. Cartwright;Mathias Häbich;B. Sturmfels;A. Werner
Mustafin簇是由Bruhat-Tits建筑中的点配置引起的射影空间的退化。这种特殊的纤维被简化成Cohen-Macaulay和它的不可约成分，形成了有趣的组合模式。对于位于一个单元中的构型，这些模式是缩放单形的规则混合细分，而穆斯塔法芬变体是从这样的细分构建的扭曲的维罗纳变体。这将我们的研究与热带和复曲面几何学联系起来。对于一般的构形，特殊纤维的不可约分量是有理簇，任何射影空间沿着线性子空间排列的爆破都可能发生。Mustafin品种的详细研究是在PGL（2）的Bruhat-Tits树和PGL（3）的二维建筑的配置。后者将Mustafin三角形分为38种组合类型。
A Mustafin variety is a degeneration of projective space induced by a point configuration in a Bruhat-Tits building. The special fiber is reduced and Cohen-Macaulay, and its irreducible components form interesting combinatorial patterns. For configurations that lie in one apartment, these patterns are regular mixed subdivisions of scaled simplices, and the Mustafin variety is a twisted Veronese variety built from such a subdivision. This connects our study to tropical and toric geometry. For general configurations, the irreducible components of the special fiber are rational varieties, and any blow-up of projective space along a linear subspace arrangement can arise. A detailed study of Mustafin varieties is undertaken for configurations in the Bruhat-Tits tree of PGL(2) and in the 2-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types.