TIMS Taipei Number Theory Seminar (2017 Fall)

Analytification of an affine curve as the limit of tropicalizations of bounded degree

川口周教授
( 同志社大學 )

2017 - 09 - 15
13:00 - 15:00 (Fri.)
數學研究中心(原新數學館) 103

Let K be an algebraically closed field that is complete with respect to a nontrivial nonarchimedean valuation. In the paper ``Analytification is the limit of all tropicalizations,'' Payne showed among other things that if X is an affine variety over K, then tropicalizations of various embeddings of X into affine space give rise to analytification of X in the sense of Berkovich. In this talk, we explain that if X is an affine curve in A^N with smooth compactification in P^N, then embeddings of X of bounded degree give rise to analytification of X. For example, analytification of an affine plane curve of genus 1 is the limit of linear tropicalizations (as shown in Baker, Rabinoff and Payne), and analytification of an affine plane curve of genus 3 is the limit of quadratic tropicalizations. This is joint work with Kazuhiko Yamaki.