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DEFINABLE SETS OF BERKOVICH CURVES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 4 / July 2021
- Published online by Cambridge University Press:
- 11 October 2019, pp. 1275-1339
- Print publication:
- July 2021
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- Article
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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.
