Time：15:00-16:00, Wednesday, August 2025
Venue：E4-233
Speaker：Paul Helminck, Tohoku University
Title：An algorithm for computing the dual intersection graph of a semistable model
Abstract：Semistable models and their associated dual intersection graphs play an important role in understanding the arithmetic of a curve defined over a number field. It for instance arises in the Chabauty-Kim method for finding rational points, as well formulas for heights and effective versions of the Bogomolov conjecture.
In this talk, I will give a full algorithm to calculate these dual intersection graphs for a curve defined over a number field. The main idea is to represent the curve as a finite covering of the projective line, and then to reconstruct the graph of our target curve from a graph associated to the projective line. More generally, I will explain how to group-theoretically reconstruct the relative poset structure of a finite covering of normal connected Noetherian schemes. In this algorithm, one finds various types of monodromy that lead to different posets, even though most of the initial data is the same. I will give several examples of this phenomenon. To explicitly calculate the necessary data in the reconstruction algorithm, we introduce multivariate symbolic generalizations of the Newton-Puiseux algorithm. These allow us to compute with finite coverings of normal schemes without calculating any integral closures. I will give several examples to show how this algorithm works in practice.
