Abstract divisorial spaces and arithmetic intersection numbers

TL;DR

This paper introduces abstract divisorial spaces to extend arithmetic intersection numbers to more general settings, including singular metrics and proper adelic base curves, building on prior work by Yuan, Zhang, Burgos, and Kramer.

Contribution

It generalizes arithmetic intersection numbers to the setting of proper adelic base curves using abstract divisorial spaces and allows more singular metrics at non-archimedean places.

Findings

Extended arithmetic intersection numbers to proper adelic base curves.

Allowed more singular metrics at non-archimedean places.

Provided a new framework for intersection theory in arithmetic geometry.

Abstract

Yuan and Zhang introduced arithmetic intersection numbers for adelic line bundles on quasi-projective varieties over a number field. Burgos and Kramer generalized this approach allowing more singular metrics at archimedean places. We introduce abstract divisorial spaces as a tool to generalize these arithmetic intersection numbers to the setting of a proper adelic base curve in the sense of Chen and Moriwaki. We also allow more singular metrics at non-archimedean places using relative mixed energy there as well.