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

TL;DR

This paper introduces a broad framework for counting special rational points called $\

Contribution

It extends Manin's conjecture to a new class of points called $\

Findings

The proposed asymptotic formula aligns with known results.

The conjecture explains several established arithmetic statistics results.

Generalizes existing conjectures to $\

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.