$\mathcal{M}$-points of bounded height on toric varieties

We establish an asymptotic formula for the number of $\mathcal{M}$-points of bounded height on split toric varieties, for the height induced by any big and nef divisor class. This formula establishes new cases of the extension of Manin’s conjecture to $\mathcal{M}$-points, as introduced by the author. As a special case of our result, we strengthen the results obtained by Pieropan and Schindler on Campana points of bounded height on toric varieties. As another special case, we obtain an asymptotic for the number of weak Campana points of bounded height, which is novel even for projective space. We illustrate this result by giving an asymptotic for the number of points on projective space of bounded height for which the product of coordinates is powerful. Finally, we derive an asymptotic for the number of rational points in the image of a toric rational map, in the spirit of the Loughran-Smeets conjecture.