Submission date: 19 September 2018

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.

Mathematics Subject Classification: 14G22 (primary), 12J25, 03C98 (secondary)

Keywords and phrases:

Full text arXiv 1809.07156: pdf, ps.