Nonarchimedean Uniformization

Mustafin, G A

doi:10.1070/sm1978v034n02abeh001156

Cited by 57 publications

(26 citation statements)

References 10 publications

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“…The induced complex is always a subcomplex of the reduction complex of M( ), even if is not convex. Indeed, if ⊃ is the metric convex closure of , then for any simplex in , the corresponding components in M( ) k intersect by [15], and hence, so do their images in M( ) k .…”

Section: Lemma 24 Letmentioning

confidence: 99%

“…This polytope has a vertex that is adjacent to four edges. Example 5.5 underscores the fact that the convex configurations considered by Mustafin [15] and Faltings [7] are very special. In the convex case, all primary components are smooth.…”

Section: Theorem 53 a Projective Variety X Arises As A Primary Compomentioning

confidence: 99%

“…Varieties of the form M( ) were investigated by Mustafin [15] in order to generalize Mumford's seminal work [16] on uniformization of curves to higher dimension. Mustafin primarily considered the case of convex subsets , as defined in the text prior to Theorem 2.10.…”

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confidence: 99%

“…In the previous discussions of these objects in [4,7,13], Mustafin varieties had been called "Deligne schemes," since for a convex set of vertices the corresponding Mustafin variety represents the so-called Deligne functor. We decided to name them after G. A. Mustafin, to recognize the contributions of [15], and we opted for "variety" over "scheme" because M( ) and its special fiber M( ) k are reduced for all lattice configurations ⊂ B d . Figure 1 shows two pictures representing Mustafin varieties for d = 3 and n = 4.…”

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confidence: 99%

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Mustafin varieties

Cartwright

Häbich

Sturmfels

et al. 2011

Sel. Math. New Ser.

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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 CohenMacaulay, 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 PG L(2) and in the 2-dimensional building of PG L(3). The latter yields the classification of Mustafin triangles into 38 combinatorial types. 758 D. Cartwright et al.Fig. 1 Special fibers of Mustafin varieties for d = 3 are degenerations of the projective plane. The two schemes depicted above arise from configurations of n = 4 points in B 3

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Section: Lemma 24 Letmentioning

confidence: 99%

Section: Theorem 53 a Projective Variety X Arises As A Primary Compomentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Mustafin varieties

Cartwright

Häbich

Sturmfels

et al. 2011

Sel. Math. New Ser.

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“…There is a 1-1 correspondance between these components and certain bounded convex subsets of the building B (See Theorem 7.5). In the last paragraph we describe the components of the reduction of Y 5 . Giving a reduction of Y s is equivalent to giving a formal scheme over L°, the ring of integers of L, that has as its generic fibre the analytic space Y s and as its closed fibre the reduction of Y s .…”

Section: Introductionmentioning

confidence: 99%

On the action of the unitary group on the projective plane over a local field

Voskuil¹

1997

J Aust Math Soc A

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Let G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ^ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.1991 Mathematics subject classification (Amer. Math. Soc): primary 32P05; secondary 14M17, 20G25.

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Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves

Poineau

Turchetti

2020

Lecture Notes in Mathematics

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Nonarchimedean Uniformization

Cited by 57 publications

References 10 publications

Mustafin varieties

Mustafin varieties

On the action of the unitary group on the projective plane over a local field

Berkovich Curves and Schottky Uniformization II: Analytic Uniformization of Mumford Curves

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