Campana points of bounded height on vector group compactifications

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 behavior 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 setup, we prove a log version of Manin's conjecture for klt Campana points on equivariant compactifications of vector groups. Contents

Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

“…Here, a type I thin subset is a set of the form Z (Q) ⊂ X (Q), where Z is a proper closed subvariety, and a type I I thin subset is a set of the form f (Y (Q)), where f : Y → X is a generically finite dominant morphism with dim Y = dim X , deg f ≥ 2 and Y geometrically integral. Theorem 1.2 illustrates that it is important to allow the possibility of removing thin sets of rational points from the statement of [11,Conj. 1.1].…”

“…The orbifold (P n−1 , ) is said to be smooth if the divisor r i=0 D i is strict normal crossings and it is said to be log-Fano if −K P n−1 , is ample, where K P n−1 , = K P n−1 + . Very recent work by Pieropan, Smeets, Tanimoto and Várilly-Alvarado [11] builds on the programme of Campana [4], by studying the distribution of Campana-points on vector group compactifications. Inspired by the Manin conjecture for rational points on Fano varieties [7], they formulate in [11,Conj.…”

“…Very recent work by Pieropan, Smeets, Tanimoto and Várilly-Alvarado [11] builds on the programme of Campana [4], by studying the distribution of Campana-points on vector group compactifications. Inspired by the Manin conjecture for rational points on Fano varieties [7], they formulate in [11,Conj. 1.1] a new prediction for the density of Campana-points of bounded height on arbitrary smooth log-Fano orbifolds.…”

“…gcd as B → ∞. (As a matter of fact, [11,Conj. 1.1] allows for the removal of a thin set of rational points from P n−1 (Q), a topic that we shall return to in our discussion of Theorem 1.2 below.)…”

Campana points and powerful values of norm forms

“…They capture the idea of rational points which are integral with respect to a weighted boundary divisor. These two notions have been termed Campana points and weak Campana points in the recent paper [27] of Pieropan, Smeets, Tanimoto and Várilly-Alvarado, in which the authors initiate a systematic quantitative study of points of the former type on smooth Campana orbifolds and prove a logarithmic version of Manin's conjecture for Campana points on vector group compactifications. The only other quantitative results in the literature are found in [5,6,26,32,33], and the former four of these indicate the close relationship between Campana points and m-full solutions of equations.…”