DEFINABLE SETS OF BERKOVICH CURVES
DEFINABLE SETS OF BERKOVICH CURVES
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可定义的伯科维奇曲线集
DOI：
10.1017/s1474748019000495
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发表时间：
2018
影响因子：
0.9
通讯作者：
Jérôme Poineau
中科院分区：
文献类型：
--
作者：
Pablo Cubides Kovacsics;Jérôme Poineau
摘要在本文中，对于一大类$k$解析曲线，我们在函数上将可定义集与$k$解析曲线联系起来，并将它们之间的可定义映射联系到解析态射。给定一条$k$-解析曲线$X$，我们的联系允许我们得到几个常用的Berkovich解析几何概念的可定义版本，例如从一点发出的分支和在类型2的点上的剩余曲线。我们还刻画了$X$的可定义对应的可定义子集，并证明了它们满足与$X$的径向子集的双射关系。作为应用，我们恢复(并略微推广)了Temkin关于给定重数的点集关于$k$解析曲线的态射的半径的结果。在分析代数曲线的情况下，我们的构造也可以看作是Hrushovski和Loeser关于曲线的等定义性定理的显式版本。然而，我们的方法也可以应用于严格的$k$仿射曲线和它们之间的任意态射，这些目前不在它们的设定范围内。
Abstract In this article, we functorially associate definable sets to $k$ -analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$ -analytic curves. Given a $k$ -analytic curve $X$ , our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$ . As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$ -analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser’s theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$ -affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.