Manin's conjecture for $\mathcal{M}$-points

Abstract

We initiate a general quantitative study of sets of $\mathcal{M}$-points, which are special subsets of rational points, generalizing Campana points, Darmon points, and squarefree solutions of Diophantine equations. We propose an asymptotic formula for the number of $\mathcal{M}$-points of bounded height on rationally connected varieties, extending Manin's conjecture as well as its generalization to Campana points by Pieropan, Smeets, Tanimoto and V\'arilly-Alvarado. Finally, we show that the conjecture explains several previously established results in arithmetic statistics.