Semi-integral Brauer-Manin obstruction and quadric orbifold pairs (Q6544133)

The study of Campana points on varieties has gained a lot of attention in recent years. There are several definitions of Campana points, all having in common that they are integral points on a variety satisfying prescribed `intersection conditions' with respect to a boundary divisor. Sets of Campana points on a variety over a number field interpolate between the set of rational points and the set of integral points, and as such can offer potential insight into the latter. Classical conjectures and local-global principles for rational points have been formulated and studied for Campana points, e.g. the Manin Conjecture [\textit{M. Pieropan} et al., Proc. Lond. Math. Soc. (3) 123, No. 1, 57--101 (2021; Zbl 1479.11116); \textit{S. Streeter}, Math. Z. 301, No. 1, 627--664 (2022; Zbl 1516.11070); \textit{A. Shute}, Acta Arith. 204, No. 4, 317--346 (2022; Zbl 1517.11072)] and the Hilbert property and weak approximation [\textit{M. Nakahara} and \textit{S. Streeter}, Mich. Math. J. 74, No. 2, 227--252 (2024; Zbl 1548.14077)].\N\NIn the paper under review, the authors continue this line of research by defining the Hasse principle and strong and weak approximation for both Campana points and the closely related Darmon points (where Campana points correspond to points with \(m\)-full coordinates, Darmon points correspond to \(m^{\text{th}}\) powers), together referred to as semi-integral points.\N\NTo this end, one of the main results is the construction of a semi-integral version of the Brauer-Manin obstruction, done in Section 3. By choosing different boundary divisors, this semi-integral version recovers both the classical Brauer-Manin obstruction defined by \textit{Yu. I. Manin} [Actes Congr. Int. Math. 1970, No. 1, 401--411 (1971; Zbl 0239.14010)], as well as the integral Brauer-Manin obstruction introduced by \textit{J.-L. Colliot-Thélène} and \textit{F. Xu} [Compos. Math. 145, No. 2, 309--363 (2009; Zbl 1190.11036)]. In Theorem 1.1, the authors give conditions for a Campana orbifold to admit no Brauer-Manin obstruction to the Campana Hasse principle and the Darmon Hasse principle, as well as conditions such that there is a Brauer-Manin obstruction to Campana strong approximation and Darmon strong approximation.\N\NThe rest of the paper consists of the study of the Campana/Darmon Hasse principle and weak and strong approximation for two families of orbifolds: (1) where the ambient space is projective space with boundary divisor obtained from \(Q_m=(1-1/m)Q\) where \(Q\) is a smooth quadric hypersurface, and (2) where the ambient space is obtained from \(Q\) and the boundary divisor from the intersection of \(Q\) with a hyperplane.\N\NTheorem 1.3 shows that orbifolds in the first family satisfy Campana weak approximation but fail Darmon weak approximation if \(m\) is even. Orbifolds in the second family are shown to satisfy Campana weak approximation in Theorem 1.4, and conditions are given for these orbifolds to satisfy Campana and Darmon strong approximation and the Darmon Hasse Principle in Theorem 1.5. The paper finishes with a count of the failures of Darmon Hasse principle for the orbifolds in the second family where \(Q\) is a diagonal quadric in Theorem 1.8.