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arXiv · 1002.1418

Mustafin Varieties

Dustin Cartwright·Mathias Häbich·Bernd Sturmfels·Annette Werner
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Abstract

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 two-dimensional building of PGL(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types.

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BibTeXRIS

Dustin Cartwright, Mathias Häbich, Bernd Sturmfels, Annette Werner. 2011-04-11. Mustafin Varieties. https://arxiv.org/abs/1002.1418

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