Campana points of bounded height on vector group compactifications

TL;DR

This paper develops a systematic study of Campana points, a special set of rational points on Fano orbifolds, and proves a log version of Manin's conjecture for these points on vector group compactifications.

Contribution

It introduces a new framework for analyzing Campana points with varying boundary weights and proves a log Manin's conjecture for klt Campana points on certain compactifications.

Findings

Established a Manin-type conjecture for Campana points.

Proved a log version of Manin's conjecture for klt Campana points.

Connected Campana points with existing conjectures and proved results in this setting.

Abstract

We initiate a systematic quantitative study of subsets of rational points that are integral with respect to a weighted boundary divisor on Fano orbifolds. We call the points in these sets Campana points. Earlier work of Campana and subsequently Abramovich shows that there are several reasonable competing definitions for Campana points. We use a version that delineates well different types of behaviour of points as the weights on the boundary divisor vary. This prompts a Manin-type conjecture on Fano orbifolds for sets of Campana points that satisfy a klt (Kawamata log terminal) condition. By importing work of Chambert-Loir and Tschinkel to our set-up, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups.