Metric uniformization of morphisms of Berkovich curves
Metric uniformization of morphisms of Berkovich curves
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伯科维奇曲线态射的度量均匀化
DOI：
10.1016/j.aim.2017.07.010
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
M. Temkin
中科院分区：
文献类型：
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作者：
M. Temkin
证明了代数闭域上拟光滑紧Berkovich曲线间态射f: Y→X的度量结构允许有限组合描述。特别是,一个足够大的骨架Γ=(ΓY,ΓX)的f, f集N, N≥Y多样性至少N的点周围的纤维径向ΓY半径变化的分段单项沿着ΓY在这种情况下,对于任何间隔l = [z, Y]⊂Y连接一个点z 1型骨架,限制f | l产生一个概要文件分段单项函数φY:[0, 1]→[0,1],只取决于类型2点Y∈ΓY特别是,f的度量结构由Γ和轮廓函数{φ y}族（y∈Γ y(2)）决定。我们证明了这个族在y上是分段单项式的，并自然地扩展到整个y。此外，我们将经典的高分枝群理论推广到任意实值域，并证明了φ y与H (y)/H （f (y)）的Herbrand函数重合。这给了Herbrand函数一个奇怪的几何解释，它也适用于非正态的甚至不可分的扩展。
We show that the metric structure of morphisms f: Y→ X between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Γ=(Γ Y, Γ X) of f, the sets N f,≥ n of points of Y of multiplicity at least n in the fiber are radial around Γ Y with the radius changing piecewise monomially along Γ Y. In this case, for any interval l=[z, y]⊂ Y connecting a point z of type 1 to the skeleton, the restriction f| l gives rise to a profile piecewise monomial function φ y:[0, 1]→[0, 1] that depends only on the type 2 point y∈ Γ Y. In particular, the metric structure of f is determined by Γ and the family of the profile functions {φ y} with y∈ Γ Y (2). We prove that this family is piecewise monomial in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that φ y coincides with the Herbrand function of H (y)/H (f (y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.