The combinatorial Chow ring of products of graphs

TL;DR

This paper characterizes the structure of a Chow ring associated with products of graphs, linking it to non-Archimedean height pairings and providing formulas for arithmetic intersection numbers on products of curves.

Contribution

It offers a complete description of the Chow ring structure and proves a vanishing conjecture, advancing understanding of arithmetic intersection theory over non-Archimedean fields.

Findings

Complete description of the degree map

Vanishing results confirming Kolb's conjecture

Analytic formula for arithmetic intersection numbers

Abstract

We prove results describing the structure of a Chow ring associated to a product of graphs, which arises from the Gross-Schoen desingularization of a product of regular proper semi-stable curves over discrete valuation rings. By the works of Johannes Kolb and Shou-Wu Zhang, this ring controls the behavior of the non-Archimedean height pairing on products of smooth proper curves over non-Archimedean fields. We provide a complete description of the degree map, and prove vanishing results affirming a conjecture of Kolb, which, combined with his work, leads to an analytic formula for the arithmetic intersection number of adelic metrized line bundles on products of curves over complete discretely valued fields.