$\mathcal{M}$-points of bounded height on toric varieties
Boaz Moerman
TL;DR
This work extends Manin's conjecture to $\mathcal{M}$-points on split toric varieties, providing a precise asymptotic $N_{(X,M),L,S}(B)=B^{a((X,M),L)}(Q(\log B)+O(B^{-\theta}))$ with an explicit leading constant. It develops the toric framework of toric pairs, Cox coordinates, and universal torsors to translate $\mathcal{M}$-point counts into Dirichlet-series problems, enabling Tauberian arguments that yield upper bounds and full leading-term formulas. The leading constant is interpreted adelically and shown to agree with Moe25conjecture, while special cases recover and strengthen results on Campana and weak Campana points, including new counts for powerful products in projective spaces. The methods unify geometric, analytic, and adelic techniques in a robust toric setting, offering sharp asymptotics across a broad class of height functions beyond the anticanonical height.
Abstract
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 our result by giving an asymptotic for the number of points on projective space of bounded height for which the product of coordinates is powerful.