Semi-integral Brauer-Manin obstruction and quadric orbifold pairs
Vladimir Mitankin, Masahiro Nakahara, Sam Streeter
TL;DR
This work develops a semi-integral Brauer–Manin obstruction framework for Campana and Darmon points, bridging rational and integral local-global principles. It defines and analyzes semi-integral adelic spaces, proves a Harari-type formal lemma in this setting, and applies the obstruction to two orbifold families attached to quadric hypersurfaces, deriving Campana Hasse principles and various weak/strong approximation results, along with conditions under which Darmon obstructions occur. The paper also studies the behavior of semi-integral points under birational maps and provides a quantitative counting result that measures the gap between Darmon obstructions and integral obstructions on affine quadrics, yielding upper and lower bounds for the number of counterexamples to the integral Hasse principle in a diagonal quadric family. Overall, the results illuminate how stacky and orbifold structures influence semi-integral arithmetic and offer tools to compare semi-integral, integral, and rational obstructions in concrete geometric settings.
Abstract
We study local-global principles for two notions of semi-integral points, termed Campana points and Darmon points. In particular, we develop a semi-integral version of the Brauer-Manin obstruction interpolating between Manin's classical version for rational points and the integral version developed by Colliot-Thélène and Xu. We determine the status of local-global principles, and obstructions to them, in two families of orbifolds naturally associated to quadric hypersurfaces. Further, we establish a quantitative result measuring the failure of the semi-integral Brauer-Manin obstruction to account for its integral counterpart for affine quadrics.